Daily Papers

We introduce WAM-Flow, a vision-language-action (VLA) model that casts ego-trajectory planning as discrete flow matching over a structured token space. In contrast to autoregressive decoders, WAM-Flow performs fully parallel, bidirectional denoising, enabling coarse-to-fine refinement with a tunable compute-accuracy trade-off. Specifically, the approach combines a metric-aligned numerical tokenizer that preserves scalar geometry via triplet-margin learning, a geometry-aware flow objective and a simulator-guided GRPO alignment that integrates safety, ego progress, and comfort rewards while retaining parallel generation. A multi-stage adaptation converts a pre-trained auto-regressive backbone (Janus-1.5B) from causal decoding to non-causal flow model and strengthens road-scene competence through continued multimodal pretraining. Thanks to the inherent nature of consistency model training and parallel decoding inference, WAM-Flow achieves superior closed-loop performance against autoregressive and diffusion-based VLA baselines, with 1-step inference attaining 89.1 PDMS and 5-step inference reaching 90.3 PDMS on NAVSIM v1 benchmark. These results establish discrete flow matching as a new promising paradigm for end-to-end autonomous driving. The code will be publicly available soon.

In our previous work, we introduced CosyVoice, a multilingual speech synthesis model based on supervised discrete speech tokens. By employing progressive semantic decoding with two popular generative models, language models (LMs) and Flow Matching, CosyVoice demonstrated high prosody naturalness, content consistency, and speaker similarity in speech in-context learning. Recently, significant progress has been made in multi-modal large language models (LLMs), where the response latency and real-time factor of speech synthesis play a crucial role in the interactive experience. Therefore, in this report, we present an improved streaming speech synthesis model, CosyVoice 2, which incorporates comprehensive and systematic optimizations. Specifically, we introduce finite-scalar quantization to improve the codebook utilization of speech tokens. For the text-speech LM, we streamline the model architecture to allow direct use of a pre-trained LLM as the backbone. In addition, we develop a chunk-aware causal flow matching model to support various synthesis scenarios, enabling both streaming and non-streaming synthesis within a single model. By training on a large-scale multilingual dataset, CosyVoice 2 achieves human-parity naturalness, minimal response latency, and virtually lossless synthesis quality in the streaming mode. We invite readers to listen to the demos at https://funaudiollm.github.io/cosyvoice2.

In this work, we propose an accurate and real-time optical flow and disparity estimation model by fusing pairwise input images in the proposed non-causal selective state space for dense perception tasks. We propose a non-causal Mamba block-based model that is fast and efficient and aptly manages the constraints present in a real-time applications. Our proposed model reduces inference times while maintaining high accuracy and low GPU usage for optical flow and disparity map generation. The results and analysis, and validation in real-life scenario justify that our proposed model can be used for unified real-time and accurate 3D dense perception estimation tasks. The code, along with the models, can be found at https://github.com/vimstereo/DensePerceptNCSSD

Generating realistic, dyadic talking head video requires ultra-low latency. Existing chunk-based methods require full non-causal context windows, introducing significant delays. This high latency critically prevents the immediate, non-verbal feedback required for a realistic listener. To address this, we present DyStream, a flow matching-based autoregressive model that could generate video in real-time from both speaker and listener audio. Our method contains two key designs: (1) we adopt a stream-friendly autoregressive framework with flow-matching heads for probabilistic modeling, and (2) We propose a causal encoder enhanced by a lookahead module to incorporate short future context (e.g., 60 ms) to improve quality while maintaining low latency. Our analysis shows this simple-and-effective method significantly surpass alternative causal strategies, including distillation and generative encoder. Extensive experiments show that DyStream could generate video within 34 ms per frame, guaranteeing the entire system latency remains under 100 ms. Besides, it achieves state-of-the-art lip-sync quality, with offline and online LipSync Confidence scores of 8.13 and 7.61 on HDTF, respectively. The model, weights and codes are available.

Causal inference is essential for data-driven decision making across domains such as business engagement, medical treatment and policy making. However, research on causal discovery has evolved separately from inference methods, preventing straight-forward combination of methods from both fields. In this work, we develop Deep End-to-end Causal Inference (DECI), a single flow-based non-linear additive noise model that takes in observational data and can perform both causal discovery and inference, including conditional average treatment effect (CATE) estimation. We provide a theoretical guarantee that DECI can recover the ground truth causal graph under standard causal discovery assumptions. Motivated by application impact, we extend this model to heterogeneous, mixed-type data with missing values, allowing for both continuous and discrete treatment decisions. Our results show the competitive performance of DECI when compared to relevant baselines for both causal discovery and (C)ATE estimation in over a thousand experiments on both synthetic datasets and causal machine learning benchmarks across data-types and levels of missingness.

Multi-agent systems (MAS) are critical for automating complex tasks, yet their practical deployment is severely hampered by the challenge of failure attribution. Current diagnostic tools, which rely on statistical correlations, are fundamentally inadequate; on challenging benchmarks like Who\&When, state-of-the-art methods achieve less than 15\% accuracy in locating the root-cause step of a failure. To address this critical gap, we introduce the first failure attribution framework for MAS grounded in multi-granularity causal inference. Our approach makes two key technical contributions: (1) a performance causal inversion principle, which correctly models performance dependencies by reversing the data flow in execution logs, combined with Shapley values to accurately assign agent-level blame; (2) a novel causal discovery algorithm, CDC-MAS, that robustly identifies critical failure steps by tackling the non-stationary nature of MAS interaction data. The framework's attribution results directly fuel an automated optimization loop, generating targeted suggestions whose efficacy is validated via counterfactual simulations. Evaluations on the Who\&When and TRAIL benchmarks demonstrate a significant leap in performance. Our method achieves up to 36.2\% step-level accuracy. Crucially, the generated optimizations boost overall task success rates by an average of 22.4\%. This work provides a principled and effective solution for debugging complex agent interactions, paving the way for more reliable and interpretable multi-agent systems.

Autoregressively trained transformers have brought a profound revolution to the world, especially with their in-context learning (ICL) ability to address downstream tasks. Recently, several studies suggest that transformers learn a mesa-optimizer during autoregressive (AR) pretraining to implement ICL. Namely, the forward pass of the trained transformer is equivalent to optimizing an inner objective function in-context. However, whether the practical non-convex training dynamics will converge to the ideal mesa-optimizer is still unclear. Towards filling this gap, we investigate the non-convex dynamics of a one-layer linear causal self-attention model autoregressively trained by gradient flow, where the sequences are generated by an AR process x_{t+1} = W x_t. First, under a certain condition of data distribution, we prove that an autoregressively trained transformer learns W by implementing one step of gradient descent to minimize an ordinary least squares (OLS) problem in-context. It then applies the learned W for next-token prediction, thereby verifying the mesa-optimization hypothesis. Next, under the same data conditions, we explore the capability limitations of the obtained mesa-optimizer. We show that a stronger assumption related to the moments of data is the sufficient and necessary condition that the learned mesa-optimizer recovers the distribution. Besides, we conduct exploratory analyses beyond the first data condition and prove that generally, the trained transformer will not perform vanilla gradient descent for the OLS problem. Finally, our simulation results verify the theoretical results.

Flow models are effective at progressively generating realistic images, but they generally struggle to capture long-range dependencies during the generation process as they compress all the information from previous time steps into a single corrupted image. To address this limitation, we propose integrating autoregressive modeling -- known for its excellence in modeling complex, high-dimensional joint probability distributions -- into flow models. During training, at each step, we construct causally-ordered sequences by sampling multiple images from the same semantic category and applying different levels of noise, where images with higher noise levels serve as causal predecessors to those with lower noise levels. This design enables the model to learn broader category-level variations while maintaining proper causal relationships in the flow process. During generation, the model autoregressively conditions the previously generated images from earlier denoising steps, forming a contextual and coherent generation trajectory. Additionally, we design a customized hybrid linear attention mechanism tailored to our modeling approach to enhance computational efficiency. Our approach, termed ARFlow, under 400k training steps, achieves 14.08 FID scores on ImageNet at 128 * 128 without classifier-free guidance, reaching 4.34 FID with classifier-free guidance 1.5, significantly outperforming the previous flow-based model SiT's 9.17 FID. Extensive ablation studies demonstrate the effectiveness of our modeling strategy and chunk-wise attention design.

Despite Flow Matching and diffusion models having emerged as powerful generative paradigms for continuous variables such as images and videos, their application to high-dimensional discrete data, such as language, is still limited. In this work, we present Discrete Flow Matching, a novel discrete flow paradigm designed specifically for generating discrete data. Discrete Flow Matching offers several key contributions: (i) it works with a general family of probability paths interpolating between source and target distributions; (ii) it allows for a generic formula for sampling from these probability paths using learned posteriors such as the probability denoiser (x-prediction) and noise-prediction (epsilon-prediction); (iii) practically, focusing on specific probability paths defined with different schedulers considerably improves generative perplexity compared to previous discrete diffusion and flow models; and (iv) by scaling Discrete Flow Matching models up to 1.7B parameters, we reach 6.7% Pass@1 and 13.4% Pass@10 on HumanEval and 6.7% Pass@1 and 20.6% Pass@10 on 1-shot MBPP coding benchmarks. Our approach is capable of generating high-quality discrete data in a non-autoregressive fashion, significantly closing the gap between autoregressive models and discrete flow models.

Normalizing Flows (NFs) have been established as a principled framework for generative modeling. Standard NFs consist of a forward process and a reverse process: the forward process maps data to noise, while the reverse process generates samples by inverting it. Typical NF forward transformations are constrained by explicit invertibility, ensuring that the reverse process can serve as their exact analytic inverse. Recent developments in TARFlow and its variants have revitalized NF methods by combining Transformers and autoregressive flows, but have also exposed causal decoding as a major bottleneck. In this work, we introduce Bidirectional Normalizing Flow (BiFlow), a framework that removes the need for an exact analytic inverse. BiFlow learns a reverse model that approximates the underlying noise-to-data inverse mapping, enabling more flexible loss functions and architectures. Experiments on ImageNet demonstrate that BiFlow, compared to its causal decoding counterpart, improves generation quality while accelerating sampling by up to two orders of magnitude. BiFlow yields state-of-the-art results among NF-based methods and competitive performance among single-evaluation ("1-NFE") methods. Following recent encouraging progress on NFs, we hope our work will draw further attention to this classical paradigm.

Normalizing flows (NFs) are end-to-end likelihood-based generative models for continuous data, and have recently regained attention with encouraging progress on image generation. Yet in the video generation domain, where spatiotemporal complexity and computational cost are substantially higher, state-of-the-art systems almost exclusively rely on diffusion-based models. In this work, we revisit this design space by presenting STARFlow-V, a normalizing flow-based video generator with substantial benefits such as end-to-end learning, robust causal prediction, and native likelihood estimation. Building upon the recently proposed STARFlow, STARFlow-V operates in the spatiotemporal latent space with a global-local architecture which restricts causal dependencies to a global latent space while preserving rich local within-frame interactions. This eases error accumulation over time, a common pitfall of standard autoregressive diffusion model generation. Additionally, we propose flow-score matching, which equips the model with a light-weight causal denoiser to improve the video generation consistency in an autoregressive fashion. To improve the sampling efficiency, STARFlow-V employs a video-aware Jacobi iteration scheme that recasts inner updates as parallelizable iterations without breaking causality. Thanks to the invertible structure, the same model can natively support text-to-video, image-to-video as well as video-to-video generation tasks. Empirically, STARFlow-V achieves strong visual fidelity and temporal consistency with practical sampling throughput relative to diffusion-based baselines. These results present the first evidence, to our knowledge, that NFs are capable of high-quality autoregressive video generation, establishing them as a promising research direction for building world models. Code and generated samples are available at https://github.com/apple/ml-starflow.

Conventional diffusion models typically relies on a fixed forward process, which implicitly defines complex marginal distributions over latent variables. This can often complicate the reverse process' task in learning generative trajectories, and results in costly inference for diffusion models. To address these limitations, we introduce Neural Flow Diffusion Models (NFDM), a novel framework that enhances diffusion models by supporting a broader range of forward processes beyond the fixed linear Gaussian. We also propose a novel parameterization technique for learning the forward process. Our framework provides an end-to-end, simulation-free optimization objective, effectively minimizing a variational upper bound on the negative log-likelihood. Experimental results demonstrate NFDM's strong performance, evidenced by state-of-the-art likelihood estimation. Furthermore, we investigate NFDM's capacity for learning generative dynamics with specific characteristics, such as deterministic straight lines trajectories. This exploration underscores NFDM's versatility and its potential for a wide range of applications.

Quantifying the causal influence of input features within neural networks has become a topic of increasing interest. Existing approaches typically assess direct, indirect, and total causal effects. This work treats NNs as structural causal models (SCMs) and extends our focus to include intrinsic causal contributions (ICC). We propose an identifiable generative post-hoc framework for quantifying ICC. We also draw a relationship between ICC and Sobol' indices. Our experiments on synthetic and real-world datasets demonstrate that ICC generates more intuitive and reliable explanations compared to existing global explanation techniques.

Neural Ordinary Differential Equations (NODEs) often struggle to adapt to new dynamic behaviors caused by parameter changes in the underlying physical system, even when these dynamics are similar to previously observed behaviors. This problem becomes more challenging when the changing parameters are unobserved, meaning their value or influence cannot be directly measured when collecting data. To address this issue, we introduce Neural Context Flow (NCF), a robust and interpretable Meta-Learning framework that includes uncertainty estimation. NCF uses Taylor expansion to enable contextual self-modulation, allowing context vectors to influence dynamics from other domains while also modulating themselves. After establishing theoretical guarantees, we empirically test NCF and compare it to related adaptation methods. Our results show that NCF achieves state-of-the-art Out-of-Distribution performance on 5 out of 6 linear and non-linear benchmark problems. Through extensive experiments, we explore the flexible model architecture of NCF and the encoded representations within the learned context vectors. Our findings highlight the potential implications of NCF for foundational models in the physical sciences, offering a promising approach to improving the adaptability and generalization of NODEs in various scientific applications. Our code is openly available at https://github.com/ddrous/ncflow.

Large language model (LLM) agents frequently fail on multi-step tasks involving reasoning, tool use, and environment interaction. While such failures are typically logged or retried heuristically, they contain structured signals about where execution broke down. We introduce CausalFlow, an interventional framework that converts failed agent traces into minimal counterfactual repairs and reusable supervision. CausalFlow models execution traces as sequential chains of dependent steps and computes Causal Responsibility Scores(CRS) via step-level counterfactual intervention to identify failure-inducing steps. For these steps, we generate minimally edited repairs that flip the final outcome to success, producing validated contrastive pairs of the form (wrong step, corrected step). CausalFlow supports two complementary uses: targeted test-time repair that recovers from failures with minimal behavioral drift, and training-time supervision suitable for offline preference optimization or reward modeling. Across four benchmarks spanning mathematical reasoning, code generation, question answering, and medical browsing, CausalFlow converts failed executions into validated minimal repairs with high minimality and causal-consensus scores, and demonstrates that causal attribution is necessary for reliable improvement across diverse agent tasks, outperforming heuristic refinement in complex retrieval settings while producing more localized repairs throughout. These results demonstrate that interventional analysis over structured execution traces provides a principled and scalable mechanism for transforming agent failures into reliability gains and learning-ready supervision.

Diffusion models are commonly interpreted as learning the score function, i.e., the gradient of the log-density of noisy data. However, this assumption implies that the target of learning is a conservative vector field, which is not enforced by the neural network architectures used in practice. We present numerical evidence that trained diffusion networks violate both integral and differential constraints required of true score functions, demonstrating that the learned vector fields are not conservative. Despite this, the models perform remarkably well as generative mechanisms. To explain this apparent paradox, we advocate a new theoretical perspective: diffusion training is better understood as flow matching to the velocity field of a Wasserstein Gradient Flow (WGF), rather than as score learning for a reverse-time stochastic differential equation. Under this view, the "probability flow" arises naturally from the WGF framework, eliminating the need to invoke reverse-time SDE theory and clarifying why generative sampling remains successful even when the neural vector field is not a true score. We further show that non-conservative errors from neural approximation do not necessarily harm density transport. Our results advocate for adopting the WGF perspective as a principled, elegant, and theoretically grounded framework for understanding diffusion generative models.

Time-series causal discovery (TSCD) is a fundamental problem of machine learning. However, existing synthetic datasets cannot properly evaluate or predict the algorithms' performance on real data. This study introduces the CausalTime pipeline to generate time-series that highly resemble the real data and with ground truth causal graphs for quantitative performance evaluation. The pipeline starts from real observations in a specific scenario and produces a matching benchmark dataset. Firstly, we harness deep neural networks along with normalizing flow to accurately capture realistic dynamics. Secondly, we extract hypothesized causal graphs by performing importance analysis on the neural network or leveraging prior knowledge. Thirdly, we derive the ground truth causal graphs by splitting the causal model into causal term, residual term, and noise term. Lastly, using the fitted network and the derived causal graph, we generate corresponding versatile time-series proper for algorithm assessment. In the experiments, we validate the fidelity of the generated data through qualitative and quantitative experiments, followed by a benchmarking of existing TSCD algorithms using these generated datasets. CausalTime offers a feasible solution to evaluating TSCD algorithms in real applications and can be generalized to a wide range of fields. For easy use of the proposed approach, we also provide a user-friendly website, hosted on www.causaltime.cc.

We propose PROSE-FD, a zero-shot multimodal PDE foundational model for simultaneous prediction of heterogeneous two-dimensional physical systems related to distinct fluid dynamics settings. These systems include shallow water equations and the Navier-Stokes equations with incompressible and compressible flow, regular and complex geometries, and different buoyancy settings. This work presents a new transformer-based multi-operator learning approach that fuses symbolic information to perform operator-based data prediction, i.e. non-autoregressive. By incorporating multiple modalities in the inputs, the PDE foundation model builds in a pathway for including mathematical descriptions of the physical behavior. We pre-train our foundation model on 6 parametric families of equations collected from 13 datasets, including over 60K trajectories. Our model outperforms popular operator learning, computer vision, and multi-physics models, in benchmark forward prediction tasks. We test our architecture choices with ablation studies.

Neural processes (NPs) are a class of models that learn stochastic processes directly from data and can be used for inference, sampling and conditional sampling. We introduce a new NP model based on flow matching, a generative modeling paradigm that has demonstrated strong performance on various data modalities. Following the NP training framework, the model provides amortized predictions of conditional distributions over any arbitrary points in the data. Compared to previous NP models, our model is simple to implement and can be used to sample from conditional distributions using an ODE solver, without requiring auxiliary conditioning methods. In addition, the model provides a controllable tradeoff between accuracy and running time via the number of steps in the ODE solver. We show that our model outperforms previous state-of-the-art neural process methods on various benchmarks including synthetic 1D Gaussian processes data, 2D images, and real-world weather data.

We introduce a new paradigm for generative modeling built on Continuous Normalizing Flows (CNFs), allowing us to train CNFs at unprecedented scale. Specifically, we present the notion of Flow Matching (FM), a simulation-free approach for training CNFs based on regressing vector fields of fixed conditional probability paths. Flow Matching is compatible with a general family of Gaussian probability paths for transforming between noise and data samples -- which subsumes existing diffusion paths as specific instances. Interestingly, we find that employing FM with diffusion paths results in a more robust and stable alternative for training diffusion models. Furthermore, Flow Matching opens the door to training CNFs with other, non-diffusion probability paths. An instance of particular interest is using Optimal Transport (OT) displacement interpolation to define the conditional probability paths. These paths are more efficient than diffusion paths, provide faster training and sampling, and result in better generalization. Training CNFs using Flow Matching on ImageNet leads to consistently better performance than alternative diffusion-based methods in terms of both likelihood and sample quality, and allows fast and reliable sample generation using off-the-shelf numerical ODE solvers.

This monograph presents the core principles that have guided the development of diffusion models, tracing their origins and showing how diverse formulations arise from shared mathematical ideas. Diffusion modeling starts by defining a forward process that gradually corrupts data into noise, linking the data distribution to a simple prior through a continuum of intermediate distributions. The goal is to learn a reverse process that transforms noise back into data while recovering the same intermediates. We describe three complementary views. The variational view, inspired by variational autoencoders, sees diffusion as learning to remove noise step by step. The score-based view, rooted in energy-based modeling, learns the gradient of the evolving data distribution, indicating how to nudge samples toward more likely regions. The flow-based view, related to normalizing flows, treats generation as following a smooth path that moves samples from noise to data under a learned velocity field. These perspectives share a common backbone: a time-dependent velocity field whose flow transports a simple prior to the data. Sampling then amounts to solving a differential equation that evolves noise into data along a continuous trajectory. On this foundation, the monograph discusses guidance for controllable generation, efficient numerical solvers, and diffusion-motivated flow-map models that learn direct mappings between arbitrary times. It provides a conceptual and mathematically grounded understanding of diffusion models for readers with basic deep-learning knowledge.

Flow-based generative models achieve state-of-the-art sample quality, but require the expensive solution of a differential equation at inference time. Flow map models, commonly known as consistency models, encompass many recent efforts to improve inference-time efficiency by learning the solution operator of this differential equation. Yet despite their promise, these models lack a unified description that clearly explains how to learn them efficiently in practice. Here, building on the methodology proposed in Boffi et. al. (2024), we present a systematic algorithmic framework for directly learning the flow map associated with a flow or diffusion model. By exploiting a relationship between the velocity field underlying a continuous-time flow and the instantaneous rate of change of the flow map, we show how to convert any distillation scheme into a direct training algorithm via self-distillation, eliminating the need for pre-trained teachers. We introduce three algorithmic families based on different mathematical characterizations of the flow map: Eulerian, Lagrangian, and Progressive methods, which we show encompass and extend all known distillation and direct training schemes for consistency models. We find that the novel class of Lagrangian methods, which avoid both spatial derivatives and bootstrapping from small steps by design, achieve significantly more stable training and higher performance than more standard Eulerian and Progressive schemes. Our methodology unifies existing training schemes under a single common framework and reveals new design principles for accelerated generative modeling. Associated code is available at https://github.com/nmboffi/flow-maps.

We introduce CaLMFlow (Causal Language Models for Flow Matching), a novel framework that casts flow matching as a Volterra integral equation (VIE), leveraging the power of large language models (LLMs) for continuous data generation. CaLMFlow enables the direct application of LLMs to learn complex flows by formulating flow matching as a sequence modeling task, bridging discrete language modeling and continuous generative modeling. Our method implements tokenization across space and time, thereby solving a VIE over these domains. This approach enables efficient handling of high-dimensional data and outperforms ODE solver-dependent methods like conditional flow matching (CFM). We demonstrate CaLMFlow's effectiveness on synthetic and real-world data, including single-cell perturbation response prediction, showcasing its ability to incorporate textual context and generalize to unseen conditions. Our results highlight LLM-driven flow matching as a promising paradigm in generative modeling, offering improved scalability, flexibility, and context-awareness.

Few-step diffusion or flow-based generative models typically distill a velocity-predicting teacher into a student that predicts a shortcut towards denoised data. This format mismatch has led to complex distillation procedures that often suffer from a quality-diversity trade-off. To address this, we propose policy-based flow models (pi-Flow). pi-Flow modifies the output layer of a student flow model to predict a network-free policy at one timestep. The policy then produces dynamic flow velocities at future substeps with negligible overhead, enabling fast and accurate ODE integration on these substeps without extra network evaluations. To match the policy's ODE trajectory to the teacher's, we introduce a novel imitation distillation approach, which matches the policy's velocity to the teacher's along the policy's trajectory using a standard ell_2 flow matching loss. By simply mimicking the teacher's behavior, pi-Flow enables stable and scalable training and avoids the quality-diversity trade-off. On ImageNet 256^2, it attains a 1-NFE FID of 2.85, outperforming MeanFlow of the same DiT architecture. On FLUX.1-12B and Qwen-Image-20B at 4 NFEs, pi-Flow achieves substantially better diversity than state-of-the-art few-step methods, while maintaining teacher-level quality.

We present Piecewise Rectified Flow (PeRFlow), a flow-based method for accelerating diffusion models. PeRFlow divides the sampling process of generative flows into several time windows and straightens the trajectories in each interval via the reflow operation, thereby approaching piecewise linear flows. PeRFlow achieves superior performance in a few-step generation. Moreover, through dedicated parameterizations, the obtained PeRFlow models show advantageous transfer ability, serving as universal plug-and-play accelerators that are compatible with various workflows based on the pre-trained diffusion models. The implementations of training and inference are fully open-sourced. https://github.com/magic-research/piecewise-rectified-flow

Recent work on the sequence universality of State Space Models (SSMs) has introduced efficient, maximally expressive continuous-time approaches for time-series modelling. While these works focus on discriminative settings, we extend this perspective to generative time-series modelling by proving that maximally expressive Structured Linear Controlled Differential Equations (SLiCEs) are universal time-series generators, in the sense that they can approximate the induced path laws of continuous causal pushforwards on compact latent sets in W_infty. Building on these theoretical results, we propose Generative SLiCEs (G-SLiCEs), a maximally expressive continuous-time model for flow matching on path-space. Empirically, we show that expressivity improves performance in probabilistic forecasting and downstream tasks, while retaining the advantages of continuous-time models such as generalising to arbitrary observation grids. This is particularly beneficial for irregular grids, where fixed-grid models often struggle.

It is commonplace to encounter heterogeneous or nonstationary data, of which the underlying generating process changes across domains or over time. Such a distribution shift feature presents both challenges and opportunities for causal discovery. In this paper, we develop a framework for causal discovery from such data, called Constraint-based causal Discovery from heterogeneous/NOnstationary Data (CD-NOD), to find causal skeleton and directions and estimate the properties of mechanism changes. First, we propose an enhanced constraint-based procedure to detect variables whose local mechanisms change and recover the skeleton of the causal structure over observed variables. Second, we present a method to determine causal orientations by making use of independent changes in the data distribution implied by the underlying causal model, benefiting from information carried by changing distributions. After learning the causal structure, next, we investigate how to efficiently estimate the "driving force" of the nonstationarity of a causal mechanism. That is, we aim to extract from data a low-dimensional representation of changes. The proposed methods are nonparametric, with no hard restrictions on data distributions and causal mechanisms, and do not rely on window segmentation. Furthermore, we find that data heterogeneity benefits causal structure identification even with particular types of confounders. Finally, we show the connection between heterogeneity/nonstationarity and soft intervention in causal discovery. Experimental results on various synthetic and real-world data sets (task-fMRI and stock market data) are presented to demonstrate the efficacy of the proposed methods.

We introduce a framework for generating highly multimodal datasets with explicitly calculable mutual information between modalities. This enables the construction of benchmark datasets that provide a novel testbed for systematic studies of mutual information estimators and multimodal self-supervised learning techniques. Our framework constructs realistic datasets with known mutual information using a flow-based generative model and a structured causal framework for generating correlated latent variables.

This paper reports the first brain-inspired large language model (BriLLM). This is a non-Transformer, non-GPT, non-traditional machine learning input-output controlled generative language model. The model is based on the Signal Fully-connected flowing (SiFu) definition on the directed graph in terms of the neural network, and has the interpretability of all nodes on the graph of the whole model, instead of the traditional machine learning model that only has limited interpretability at the input and output ends. In the language model scenario, the token is defined as a node in the graph. A randomly shaped or user-defined signal flow flows between nodes on the principle of "least resistance" along paths. The next token or node to be predicted or generated is the target of the signal flow. As a language model, BriLLM theoretically supports infinitely long n-gram models when the model size is independent of the input and predicted length of the model. The model's working signal flow provides the possibility of recall activation and innate multi-modal support similar to the cognitive patterns of the human brain. At present, we released the first BriLLM version in Chinese, with 4000 tokens, 32-dimensional node width, 16-token long sequence prediction ability, and language model prediction performance comparable to GPT-1. More computing power will help us explore the infinite possibilities depicted above.

Classifier-free guidance is a key component for enhancing the performance of conditional generative models across diverse tasks. While it has previously demonstrated remarkable improvements for the sample quality, it has only been exclusively employed for diffusion models. In this paper, we integrate classifier-free guidance into Flow Matching (FM) models, an alternative simulation-free approach that trains Continuous Normalizing Flows (CNFs) based on regressing vector fields. We explore the usage of Guided Flows for a variety of downstream applications. We show that Guided Flows significantly improves the sample quality in conditional image generation and zero-shot text-to-speech synthesis, boasting state-of-the-art performance. Notably, we are the first to apply flow models for plan generation in the offline reinforcement learning setting, showcasing a 10x speedup in computation compared to diffusion models while maintaining comparable performance.

We propose a unified, few-step generative modeling framework based on cumulative flow maps for long-range transport in probability space, inspired by flow-map techniques for physical transport and dynamics. At its core is a cumulative-flow abstraction that connects local, instantaneous updates with finite-time transport, enabling generative models to reason about global state transitions. This perspective yields a unified few-step framework built on cumulative transport and cumulative parameterization that applies broadly to existing diffusion- and flow-based models without being tied to a specific prediction instantiation. Our formulation supports few-step and even one-step generation while preserving synthesis quality, requiring only minimal changes to time embeddings and training objectives, and no increase in model capacity. We demonstrate its effectiveness across diverse tasks, including image generation, geometric distribution modeling, joint prediction, and SDF generation, with reduced inference cost.

Neural Flows efficiently model irregular multivariate time series by directly learning ODE solution trajectories with neural networks, bypassing step-by-step numerical solvers. Despite their efficiency, many existing approaches treat variables independently, leaving inter-variable interactions underexplored. Moreover, their one-step mapping makes interaction modeling inherently challenging, as it removes the iterative refinement of interactions during learning. To address this challenge, we propose one-step Graph-Structured Neural Flows (GSNF), which introduce two auxiliary-trajectory self-supervision strategies to strengthen interaction learning: (i) interaction-aware trajectory generation via re-initialization, which induces trajectory divergence to expose graph-induced interactions, with a theoretically derived lower bound on divergence; and (ii) reverse-time trajectory generation, which enforces forward-backward consistency to regularize graph learning, enabled by flow invertibility. Experiments on five real-world datasets show that GSNF achieves state-of-the-art classification performance with highly competitive training time and memory usage.

A faithful and interpretable explanation of an AI model's behavior and internal structure is a high-level explanation that is human-intelligible but also consistent with the known, but often opaque low-level causal details of the model. We argue that the theory of causal abstraction provides the mathematical foundations for the desired kinds of model explanations. In causal abstraction analysis, we use interventions on model-internal states to rigorously assess whether an interpretable high-level causal model is a faithful description of an AI model. Our contributions in this area are: (1) We generalize causal abstraction to cyclic causal structures and typed high-level variables. (2) We show how multi-source interchange interventions can be used to conduct causal abstraction analyses. (3) We define a notion of approximate causal abstraction that allows us to assess the degree to which a high-level causal model is a causal abstraction of a lower-level one. (4) We prove constructive causal abstraction can be decomposed into three operations we refer to as marginalization, variable-merge, and value-merge. (5) We formalize the XAI methods of LIME, causal effect estimation, causal mediation analysis, iterated nullspace projection, and circuit-based explanations as special cases of causal abstraction analysis.

Streaming speech recognition architectures are employed for low-latency, real-time applications. Such architectures are often characterized by their causality. Causal architectures emit tokens at each frame, relying only on current and past signal, while non-causal models are exposed to a window of future frames at each step to increase predictive accuracy. This dichotomy amounts to a trade-off for real-time Automatic Speech Recognition (ASR) system design: profit from the low-latency benefit of strictly-causal architectures while accepting predictive performance limitations, or realize the modeling benefits of future-context models accompanied by their higher latency penalty. In this work, we relax the constraints of this choice and present the Adaptive Non-Causal Attention Transducer (ANCAT). Our architecture is non-causal in the traditional sense, but executes in a low-latency, streaming manner by dynamically choosing when to rely on future context and to what degree within the audio stream. The resulting mechanism, when coupled with our novel regularization algorithms, delivers comparable accuracy to non-causal configurations while improving significantly upon latency, closing the gap with their causal counterparts. We showcase our design experimentally by reporting comparative ASR task results with measures of accuracy and latency on both publicly accessible and production-scale, voice-assistant datasets.

Numerous biological and physical processes can be modeled as systems of interacting entities evolving continuously over time, e.g. the dynamics of communicating cells or physical particles. Learning the dynamics of such systems is essential for predicting the temporal evolution of populations across novel samples and unseen environments. Flow-based models allow for learning these dynamics at the population level - they model the evolution of the entire distribution of samples. However, current flow-based models are limited to a single initial population and a set of predefined conditions which describe different dynamics. We argue that multiple processes in natural sciences have to be represented as vector fields on the Wasserstein manifold of probability densities. That is, the change of the population at any moment in time depends on the population itself due to the interactions between samples. In particular, this is crucial for personalized medicine where the development of diseases and their respective treatment response depends on the microenvironment of cells specific to each patient. We propose Meta Flow Matching (MFM), a practical approach to integrating along these vector fields on the Wasserstein manifold by amortizing the flow model over the initial populations. Namely, we embed the population of samples using a Graph Neural Network (GNN) and use these embeddings to train a Flow Matching model. This gives MFM the ability to generalize over the initial distributions unlike previously proposed methods. We demonstrate the ability of MFM to improve prediction of individual treatment responses on a large scale multi-patient single-cell drug screen dataset.

We introduce a new approach to functional causal modeling from observational data, called Causal Generative Neural Networks (CGNN). CGNN leverages the power of neural networks to learn a generative model of the joint distribution of the observed variables, by minimizing the Maximum Mean Discrepancy between generated and observed data. An approximate learning criterion is proposed to scale the computational cost of the approach to linear complexity in the number of observations. The performance of CGNN is studied throughout three experiments. Firstly, CGNN is applied to cause-effect inference, where the task is to identify the best causal hypothesis out of Xrightarrow Y and Yrightarrow X. Secondly, CGNN is applied to the problem of identifying v-structures and conditional independences. Thirdly, CGNN is applied to multivariate functional causal modeling: given a skeleton describing the direct dependences in a set of random variables X = [X_1, ldots, X_d], CGNN orients the edges in the skeleton to uncover the directed acyclic causal graph describing the causal structure of the random variables. On all three tasks, CGNN is extensively assessed on both artificial and real-world data, comparing favorably to the state-of-the-art. Finally, CGNN is extended to handle the case of confounders, where latent variables are involved in the overall causal model.

Continuous normalizing flows (CNFs) are an attractive generative modeling technique, but they have been held back by limitations in their simulation-based maximum likelihood training. We introduce the generalized conditional flow matching (CFM) technique, a family of simulation-free training objectives for CNFs. CFM features a stable regression objective like that used to train the stochastic flow in diffusion models but enjoys the efficient inference of deterministic flow models. In contrast to both diffusion models and prior CNF training algorithms, CFM does not require the source distribution to be Gaussian or require evaluation of its density. A variant of our objective is optimal transport CFM (OT-CFM), which creates simpler flows that are more stable to train and lead to faster inference, as evaluated in our experiments. Furthermore, we show that when the true OT plan is available, our OT-CFM method approximates dynamic OT. Training CNFs with CFM improves results on a variety of conditional and unconditional generation tasks, such as inferring single cell dynamics, unsupervised image translation, and Schr\"odinger bridge inference.

We formulate a hierarchical rectified flow to model data distributions. It hierarchically couples multiple ordinary differential equations (ODEs) and defines a time-differentiable stochastic process that generates a data distribution from a known source distribution. Each ODE resembles the ODE that is solved in a classic rectified flow, but differs in its domain, i.e., location, velocity, acceleration, etc. Unlike the classic rectified flow formulation, which formulates a single ODE in the location domain and only captures the expected velocity field (sufficient to capture a multi-modal data distribution), the hierarchical rectified flow formulation models the multi-modal random velocity field, acceleration field, etc., in their entirety. This more faithful modeling of the random velocity field enables integration paths to intersect when the underlying ODE is solved during data generation. Intersecting paths in turn lead to integration trajectories that are more straight than those obtained in the classic rectified flow formulation, where integration paths cannot intersect. This leads to modeling of data distributions with fewer neural function evaluations. We empirically verify this on synthetic 1D and 2D data as well as MNIST, CIFAR-10, and ImageNet-32 data. Our code is available at: https://riccizz.github.io/HRF/.

We present a formulation of flow matching as variational inference, which we refer to as variational flow matching (VFM). Based on this formulation we develop CatFlow, a flow matching method for categorical data. CatFlow is easy to implement, computationally efficient, and achieves strong results on graph generation tasks. In VFM, the objective is to approximate the posterior probability path, which is a distribution over possible end points of a trajectory. We show that VFM admits both the CatFlow objective and the original flow matching objective as special cases. We also relate VFM to score-based models, in which the dynamics are stochastic rather than deterministic, and derive a bound on the model likelihood based on a reweighted VFM objective. We evaluate CatFlow on one abstract graph generation task and two molecular generation tasks. In all cases, CatFlow exceeds or matches performance of the current state-of-the-art models.

Flow-based methods have achieved significant success in various generative modeling tasks, capturing nuanced details within complex data distributions. However, few existing works have exploited this unique capability to resolve fine-grained structural details beyond generation tasks. This paper presents a flow-inspired framework for representation learning. First, we demonstrate that a rectified flow trained using independent coupling is zero everywhere at t=0.5 if and only if the source and target distributions are identical. We term this property the zero-flow criterion. Second, we show that this criterion can certify conditional independence, thereby extracting sufficient information from the data. Third, we translate this criterion into a tractable, simulation-free loss function that enables learning amortized Markov blankets in graphical models and latent representations in self-supervised learning tasks. Experiments on both simulated and real-world datasets demonstrate the effectiveness of our approach. The code reproducing our experiments can be found at: https://github.com/probabilityFLOW/zfe.

In real-world scenarios, the application of reinforcement learning is significantly challenged by complex non-stationarity. Most existing methods attempt to model changes in the environment explicitly, often requiring impractical prior knowledge of environments. In this paper, we propose a new perspective, positing that non-stationarity can propagate and accumulate through complex causal relationships during state transitions, thereby compounding its sophistication and affecting policy learning. We believe that this challenge can be more effectively addressed by implicitly tracing the causal origin of non-stationarity. To this end, we introduce the Causal-Origin REPresentation (COREP) algorithm. COREP primarily employs a guided updating mechanism to learn a stable graph representation for the state, termed as causal-origin representation. By leveraging this representation, the learned policy exhibits impressive resilience to non-stationarity. We supplement our approach with a theoretical analysis grounded in the causal interpretation for non-stationary reinforcement learning, advocating for the validity of the causal-origin representation. Experimental results further demonstrate the superior performance of COREP over existing methods in tackling non-stationarity problems.

In this paper, we address the problem of human trajectory forecasting, which aims to predict the inherently multi-modal future movements of humans based on their past trajectories and other contextual cues. We propose a novel motion prediction conditional flow matching model, termed MoFlow, to predict K-shot future trajectories for all agents in a given scene. We design a novel flow matching loss function that not only ensures at least one of the K sets of future trajectories is accurate but also encourages all K sets of future trajectories to be diverse and plausible. Furthermore, by leveraging the implicit maximum likelihood estimation (IMLE), we propose a novel distillation method for flow models that only requires samples from the teacher model. Extensive experiments on the real-world datasets, including SportVU NBA games, ETH-UCY, and SDD, demonstrate that both our teacher flow model and the IMLE-distilled student model achieve state-of-the-art performance. These models can generate diverse trajectories that are physically and socially plausible. Moreover, our one-step student model is 100 times faster than the teacher flow model during sampling. The code, model, and data are available at our project page: https://moflow-imle.github.io

Deep learning models are mostly used in an offline inference fashion. However, this strongly limits the use of these models inside audio generation setups, as most creative workflows are based on real-time digital signal processing. Although approaches based on recurrent networks can be naturally adapted to this buffer-based computation, the use of convolutions still poses some serious challenges. To tackle this issue, the use of causal streaming convolutions have been proposed. However, this requires specific complexified training and can impact the resulting audio quality. In this paper, we introduce a new method allowing to produce non-causal streaming models. This allows to make any convolutional model compatible with real-time buffer-based processing. As our method is based on a post-training reconfiguration of the model, we show that it is able to transform models trained without causal constraints into a streaming model. We show how our method can be adapted to fit complex architectures with parallel branches. To evaluate our method, we apply it on the recent RAVE model, which provides high-quality real-time audio synthesis. We test our approach on multiple music and speech datasets and show that it is faster than overlap-add methods, while having no impact on the generation quality. Finally, we introduce two open-source implementation of our work as Max/MSP and PureData externals, and as a VST audio plugin. This allows to endow traditional digital audio workstation with real-time neural audio synthesis on a laptop CPU.

We introduce Lagrangian Flow Networks (LFlows) for modeling fluid densities and velocities continuously in space and time. By construction, the proposed LFlows satisfy the continuity equation, a PDE describing mass conservation in its differentiable form. Our model is based on the insight that solutions to the continuity equation can be expressed as time-dependent density transformations via differentiable and invertible maps. This follows from classical theory of the existence and uniqueness of Lagrangian flows for smooth vector fields. Hence, we model fluid densities by transforming a base density with parameterized diffeomorphisms conditioned on time. The key benefit compared to methods relying on numerical ODE solvers or PINNs is that the analytic expression of the velocity is always consistent with changes in density. Furthermore, we require neither expensive numerical solvers, nor additional penalties to enforce the PDE. LFlows show higher predictive accuracy in density modeling tasks compared to competing models in 2D and 3D, while being computationally efficient. As a real-world application, we model bird migration based on sparse weather radar measurements.

Diffusion models have emerged as a promising alternative to autoregressive models in modeling discrete categorical data. Yet diffusion models that directly work on discrete data space do not fully exploit the power of iterative refinement, as the signals are lost during the transition between discrete states. Existing continuous diffusion models for discrete data have limited performance compared to discrete approaches, and the unclear link between them restricts the development of diffusion models for discrete data. In this work, we propose a continuous diffusion model for language modeling that incorporates the geometry of the underlying categorical distribution. We establish a connection between the discrete diffusion and continuous flow on the statistical manifold, and building on the analogy, we introduce a simple design for the diffusion process that generalizes previous discrete diffusion models. We further propose a simulation-free training framework based on radial symmetry and a simple technique to address the high dimensionality of the manifold. Comprehensive experiments on language modeling benchmarks and other modalities show that our method outperforms existing discrete diffusion models and approaches the performance of autoregressive models. Codes available at https://github.com/harryjo97/RDLM{https://github.com/harryjo97/RDLM}.

One-step generative modeling has emerged as a leading approach for amortizing the inference cost of diffusion and flow-matching models. Among distillation-free methods, MeanFlow training is notoriously unstable, with non-decreasing loss and unbounded gradient variance. In this work, we establish a theory that attributes this pathology to a misuse of the conditional velocity field. We show that the conditional velocity plays two distinct statistical roles in the loss: both as an unbiased regression target and as a Monte Carlo control variate in a Jacobi-vector product, with the original MeanFlow loss assigning the wrong coefficient to the latter. We derive the optimal coefficient in closed form and show that a family of fixes in concurrent works corresponds to different practical realizations of the same optimum. A controlled sweep of this coefficient on two-dimensional benchmarks and on a latent Diffusion Transformer recovers the predicted bias-variance ordering. Our DiT experiment also reveals a quantitative FID-MSE landscape mismatch. Specifically, although the gradient-MSE is minimized at an interior coefficient value near β!=!0.94, the coefficient that minimizes FID prefers to use conditional velocity directly at the unbiased corner. Our analysis therefore explains why MeanFlow is unstable and unifies its concurrent remedies, and shows that the variance-optimal coefficient need not coincide with the quality-optimal one.

Causal discovery, i.e., inferring underlying cause-effect relationships from observations of a scene or system, is an inherent mechanism in human cognition, but has been shown to be highly challenging to automate. The majority of approaches in the literature aiming for this task consider constrained scenarios with fully observed variables or data from stationary time-series. In this work we aim for causal discovery in a more general class of scenarios, scenes with non-stationary behavior over time. For our purposes we here regard a scene as a composition objects interacting with each other over time. Non-stationarity is modeled as stationarity conditioned on an underlying variable, a state, which can be of varying dimension, more or less hidden given observations of the scene, and also depend more or less directly on these observations. We propose a probabilistic deep learning approach called State-Dependent Causal Inference (SDCI) for causal discovery in such conditionally stationary time-series data. Results in two different synthetic scenarios show that this method is able to recover the underlying causal dependencies with high accuracy even in cases with hidden states.

Causal discovery for dynamical systems poses a major challenge in fields where active interventions are infeasible. Most methods used to investigate these systems and their associated benchmarks are tailored to deterministic, low-dimensional and weakly nonlinear time-series data. To address these limitations, we present CausalDynamics, a large-scale benchmark and extensible data generation framework to advance the structural discovery of dynamical causal models. Our benchmark consists of true causal graphs derived from thousands of coupled ordinary and stochastic differential equations as well as two idealized climate models. We perform a comprehensive evaluation of state-of-the-art causal discovery algorithms for graph reconstruction on systems with noisy, confounded, and lagged dynamics. CausalDynamics consists of a plug-and-play, build-your-own coupling workflow that enables the construction of a hierarchy of physical systems. We anticipate that our framework will facilitate the development of robust causal discovery algorithms that are broadly applicable across domains while addressing their unique challenges. We provide a user-friendly implementation and documentation on https://kausable.github.io/CausalDynamics.

Current end-to-end autonomous driving systems are fundamentally limited by a mismatch between temporal causal reasoning and global trajectory consistency. Autoregressive (AR) models capture interaction-aware temporal dependencies via causal factorization, but their step-wise decoding leads to error accumulation and suboptimal global structure. In contrast, diffusion models optimize trajectories globally but lack explicit causal constraints, making them unreliable in interactive and safety-critical scenarios. This dichotomy reveals a deeper issue: existing methods treat causal modeling and global optimization as separate paradigms, without a principled way to unify them within a single trajectory distribution. To address this, we propose ChainFlow-VLA, which unifies causal generation and global refinement within a unified probabilistic framework. We formulate planning as a mixture over AR-induced modes and learn Vision-Language Model (VLM)-conditioned residual distributions over these modes. An autoregressive generator (Chain) produces a discrete set of causal trajectory modes, followed by a diffusion-based refiner (Flow) that leverages VLM hidden states as semantic priors to perform mode-conditioned correction in residual space while preserving causal structure. This straightforward conditioning seamlessly injects high-level scene understanding into fine-grained trajectory adjustments. Experiments demonstrate that ChainFlow-VLA achieves robust planning in ambiguous and long-tail scenarios, achieving a state-of-the-art score of 94.85 on the NAVSIM v1 leaderboard, matching human-level performance (94.8). Code will be available at https://github.com/AFARI-Research/ChainFlow-VLA.

Flow-based generative models are powerful exact likelihood models with efficient sampling and inference. Despite their computational efficiency, flow-based models generally have much worse density modeling performance compared to state-of-the-art autoregressive models. In this paper, we investigate and improve upon three limiting design choices employed by flow-based models in prior work: the use of uniform noise for dequantization, the use of inexpressive affine flows, and the use of purely convolutional conditioning networks in coupling layers. Based on our findings, we propose Flow++, a new flow-based model that is now the state-of-the-art non-autoregressive model for unconditional density estimation on standard image benchmarks. Our work has begun to close the significant performance gap that has so far existed between autoregressive models and flow-based models. Our implementation is available at https://github.com/aravindsrinivas/flowpp

Causal graphs provide a high-level language for making mechanisms transparent. Recent work uses Large Language Models (LLMs) to recover causal graphs of external-world processes. Instead, in this paper, we use causal graphs to model LLM inference itself, providing stakeholders with a transparent view of how the model perceives and organizes high-level concepts to produce a prediction. We propose a four-phase method for constructing such graphs. Given a target LLM and a set of textual examples, our method discovers class-discriminative, human-interpretable concepts and maps each input to LLM-perceived concept states. We then introduce an MCMC-inspired counterfactual augmentation procedure that expands the sparse observational data through chains of counterfactuals. This enables stable causal discovery with σ-CG, yielding informative, interpretable graphs. We apply our method to three LLMs across disease diagnosis, sentiment analysis, and LLM-as-a-judge classification tasks. We evaluate the learned graphs for predictive fidelity and structural stability, and the MCMC-inspired augmentation for convergence and downstream utility. Our results show that the discovered causal graphs capture meaningful dependencies consistent with LLMs' reasoning. Together, this paper provides a foundation for concept-level explainability of LLMs.

Standard flow matching scales well but typically relies on an unstructured source distribution, limiting its ability to learn interpretable latent structure. Latent-variable models, by contrast, capture structure but often sacrifice generative quality. We bridge this gap by proposing Structured Coupling for Flow Matching (SCFM), a cooperative framework that augments flow matching with structured latent representation learning. By introducing structured latent variables and exogenous noise into the source, SCFM jointly learns a structured prior (via latent variable modeling) and a continuous transport map (via flow matching). It uses a shared time-dependent recognition network for both latent variable model variational inference and intermediate-time flow velocity estimation. This yields a structurally informed yet unconditional, simulation-free flow model, where the latent variable model can also assist flow sampling. Empirically, SCFM facilitates unsupervised latent representation learning for clustering, disentanglement and downstream tasks, while remaining competitive with flow matching in sample quality, showing that meaningful structure can be learned without sacrificing generative fidelity.

We present OneFlow, the first non-autoregressive multimodal model that enables variable-length and concurrent mixed-modal generation. Unlike autoregressive models that enforce rigid causal ordering between text and image generation, OneFlow combines an insertion-based Edit Flow for discrete text tokens with Flow Matching for image latents. OneFlow enables concurrent text-image synthesis with hierarchical sampling that prioritizes content over grammar. Through controlled experiments across model sizes from 1B to 8B, we demonstrate that OneFlow outperforms autoregressive baselines on both generation and understanding tasks while using up to 50% fewer training FLOPs. OneFlow surpasses both autoregressive and diffusion-based approaches while unlocking new capabilities for concurrent generation, iterative refinement, and natural reasoning-like generation.

Existing machine learning methods for causal inference usually estimate quantities expressed via the mean of potential outcomes (e.g., average treatment effect). However, such quantities do not capture the full information about the distribution of potential outcomes. In this work, we estimate the density of potential outcomes after interventions from observational data. For this, we propose a novel, fully-parametric deep learning method called Interventional Normalizing Flows. Specifically, we combine two normalizing flows, namely (i) a nuisance flow for estimating nuisance parameters and (ii) a target flow for parametric estimation of the density of potential outcomes. We further develop a tractable optimization objective based on a one-step bias correction for efficient and doubly robust estimation of the target flow parameters. As a result, our Interventional Normalizing Flows offer a properly normalized density estimator. Across various experiments, we demonstrate that our Interventional Normalizing Flows are expressive and highly effective, and scale well with both sample size and high-dimensional confounding. To the best of our knowledge, our Interventional Normalizing Flows are the first proper fully-parametric, deep learning method for density estimation of potential outcomes.

Answering counterfactual queries has many important applications such as knowledge discovery and explainability, but is challenging when causal variables are unobserved and we only see a projection onto an observation space, for instance, image pixels. One approach is to recover the latent Structural Causal Model (SCM), but this typically needs unrealistic assumptions, such as linearity of the causal mechanisms. Another approach is to use na\"ive ML approximations, such as generative models, to generate counterfactual samples; however, these lack guarantees of accuracy. In this work, we strive to strike a balance between practicality and theoretical guarantees by focusing on a specific type of causal query called domain counterfactuals, which hypothesizes what a sample would have looked like if it had been generated in a different domain (or environment). Concretely, by only assuming invertibility, sparse domain interventions and access to observational data from different domains, we aim to improve domain counterfactual estimation both theoretically and practically with less restrictive assumptions. We define domain counterfactually equivalent models and prove necessary and sufficient properties for equivalent models that provide a tight characterization of the domain counterfactual equivalence classes. Building upon this result, we prove that every equivalence class contains a model where all intervened variables are at the end when topologically sorted by the causal DAG. This surprising result suggests that a model design that only allows intervention in the last k latent variables may improve model estimation for counterfactuals. We then test this model design on extensive simulated and image-based experiments which show the sparse canonical model indeed improves counterfactual estimation over baseline non-sparse models.

Causal Discovery plays a pivotal role in revealing relationships among observed variables, particularly in the temporal setup. While the majority of CD methods rely on synthetic data for evaluation, and recently for training, these fall short in accurately mirroring real-world scenarios; an effect even more evident in temporal data. Generation techniques depending on simplified assumptions on causal structure, effects and time, limit the quality and diversity of the simulated data. In this work, we introduce Temporal Causal-based Simulation (TCS), a robust framework for generating realistic time-series data and their associated temporal causal graphs. The approach is structured in three phases: estimating the true lagged causal structure of the data, approximating the functional dependencies between variables and learning the noise distribution of the corresponding causal model, each part of which can be explicitly tailored based on data assumptions and characteristics. Through an extensive evaluation process, we highlight that single detection methods for generated data discrimination prove inadequate, accentuating it as a multifaceted challenge. For this, we detail a Min-max optimization phase that draws on AutoML techniques. Our contributions include a flexible, model-agnostic pipeline for generating realistic temporal causal data, a thorough evaluation setup which enhances the validity of the generated datasets and insights into the challenges posed by realistic data generation. Through experiments involving not only real but also semi-synthetic and purely synthetic datasets, we demonstrate that while sampling realistic causal data remains a complex task, our method enriches the domain of generating sensible causal-based temporal data.

Few-step video generation has been significantly advanced by consistency distillation. However, the performance of consistency-distilled models often degrades as more sampling steps are allocated at test time, limiting their effectiveness for any-step video diffusion. This limitation arises because consistency distillation replaces the original probability-flow ODE trajectory with a consistency-sampling trajectory, weakening the desirable test-time scaling behavior of ODE sampling. To address this limitation, we introduce AnyFlow, the first any-step video diffusion distillation framework based on flow maps. Instead of distilling a model for only a few fixed sampling steps, AnyFlow optimizes the full ODE sampling trajectory. To this end, we shift the distillation target from endpoint consistency mapping (z_{t}rightarrow z_{0}) to flow-map transition learning (z_{t}rightarrow z_{r}) over arbitrary time intervals. We further propose Flow Map Backward Simulation, which decomposes a full Euler rollout into shortcut flow-map transitions, enabling efficient on-policy distillation that reduces test-time errors (i.e., discretization error in few-step sampling and exposure bias in causal generation). Extensive experiments across both bidirectional and causal architectures, at scales ranging from 1.3B to 14B parameters, demonstrate that AnyFlow achieves performance matches or surpasses consistency-based counterparts in the few-step regime, while scaling with sampling step budgets.

Causal effect estimation from observational data is fundamental across various applications. However, selecting an appropriate estimator from dozens of specialized methods demands substantial manual effort and domain expertise. We present CausalPFN, a single transformer that amortizes this workflow: trained once on a large library of simulated data-generating processes that satisfy ignorability, it infers causal effects for new observational datasets out-of-the-box. CausalPFN combines ideas from Bayesian causal inference with the large-scale training protocol of prior-fitted networks (PFNs), learning to map raw observations directly to causal effects without any task-specific adjustment. Our approach achieves superior average performance on heterogeneous and average treatment effect estimation benchmarks (IHDP, Lalonde, ACIC). Moreover, it shows competitive performance for real-world policy making on uplift modeling tasks. CausalPFN provides calibrated uncertainty estimates to support reliable decision-making based on Bayesian principles. This ready-to-use model does not require any further training or tuning and takes a step toward automated causal inference (https://github.com/vdblm/CausalPFN).

Longitudinal brain analysis is essential for understanding healthy aging and identifying pathological deviations. Longitudinal registration of sequential brain MRI underpins such analyses. However, existing methods are limited by reliance on densely sampled time series, a trade-off between accuracy and temporal smoothness, and an inability to prospectively forecast future brain states. To overcome these challenges, we introduce TimeFlow, a learning-based framework for longitudinal brain MRI registration. TimeFlow uses a U-Net backbone with temporal conditioning to model neuroanatomy as a continuous function of age. Given only two scans from an individual, TimeFlow estimates accurate and temporally coherent deformation fields, enabling non-linear extrapolation to predict future brain states. This is achieved by our proposed inter-/extra-polation consistency constraints applied to both the deformation fields and deformed images. Remarkably, these constraints preserve temporal consistency and continuity without requiring explicit smoothness regularizers or densely sampled sequential data. Extensive experiments demonstrate that TimeFlow outperforms state-of-the-art methods in terms of both future timepoint forecasting and registration accuracy. Moreover, TimeFlow supports novel biological brain aging analyses by differentiating neurodegenerative trajectories from normal aging without requiring segmentation, thereby eliminating the need for labor-intensive annotations and mitigating segmentation inconsistency. TimeFlow offers an accurate, data-efficient, and annotation-free framework for longitudinal analysis of brain aging and chronic diseases, capable of forecasting brain changes beyond the observed study period.

Temporal point processes are the dominant paradigm for modeling sequences of events happening at irregular intervals. The standard way of learning in such models is by estimating the conditional intensity function. However, parameterizing the intensity function usually incurs several trade-offs. We show how to overcome the limitations of intensity-based approaches by directly modeling the conditional distribution of inter-event times. We draw on the literature on normalizing flows to design models that are flexible and efficient. We additionally propose a simple mixture model that matches the flexibility of flow-based models, but also permits sampling and computing moments in closed form. The proposed models achieve state-of-the-art performance in standard prediction tasks and are suitable for novel applications, such as learning sequence embeddings and imputing missing data.

The traditional i.i.d.-based learning paradigm faces inherent challenges in addressing causal relationships, which has become increasingly evident with the rise of applications in causal representation learning. Our understanding of causality naturally requires a perspective as the creator rather than observer, as the ``what...if'' questions only hold within the possible world we conceive. The traditional perspective limits capturing dynamic causal outcomes and leads to compensatory efforts such as the reliance on hidden confounders. This paper lays the groundwork for the new perspective, which enables the relation-first modeling paradigm for causality. Also, it introduces the Relation-Indexed Representation Learning (RIRL) as a practical implementation, supported by experiments that validate its efficacy.

Modern continuous-time generative models typically induce V-shaped flows: each sample travels independently along a nearly straight trajectory from the prior to the data. Although effective, this independent movement overlooks the hierarchical structures that exist in real-world data. To address this, we introduce Y-shaped generative flows, a framework in which samples travel together along shared pathways before branching off to target-specific endpoints. Our formulation is theoretically justified, yet remains practical, requiring only minimal modifications to standard velocity-driven models. We implement this through a scalable, neural network-based training objective. Experiments on synthetic, image, and biological datasets demonstrate that our method recovers hierarchy-aware structures, improves distributional metrics over strong flow-based baselines, and reaches targets in fewer steps.

Flow matching is a recent state-of-the-art framework for generative modeling based on ordinary differential equations (ODEs). While closely related to diffusion models, it provides a more general perspective on generative modeling. Although inverse problem solving has been extensively explored using diffusion models, it has not been rigorously examined within the broader context of flow models. Therefore, here we extend the diffusion inverse solvers (DIS) - which perform posterior sampling by combining a denoising diffusion prior with an likelihood gradient - into the flow framework. Specifically, by driving the flow-version of Tweedie's formula, we decompose the flow ODE into two components: one for clean image estimation and the other for noise estimation. By integrating the likelihood gradient and stochastic noise into each component, respectively, we demonstrate that posterior sampling for inverse problem solving can be effectively achieved using flows. Our proposed solver, Flow-Driven Posterior Sampling (FlowDPS), can also be seamlessly integrated into a latent flow model with a transformer architecture. Across four linear inverse problems, we confirm that FlowDPS outperforms state-of-the-art alternatives, all without requiring additional training.

Estimating the causal effects of a spatially-varying intervention on a spatially-varying outcome may be subject to non-local confounding (NLC), a phenomenon that can bias estimates when the treatments and outcomes of a given unit are dictated in part by the covariates of other nearby units. In particular, NLC is a challenge for evaluating the effects of environmental policies and climate events on health-related outcomes such as air pollution exposure. This paper first formalizes NLC using the potential outcomes framework, providing a comparison with the related phenomenon of causal interference. Then, it proposes a broadly applicable framework, termed "weather2vec", that uses the theory of balancing scores to learn representations of non-local information into a scalar or vector defined for each observational unit, which is subsequently used to adjust for confounding in conjunction with causal inference methods. The framework is evaluated in a simulation study and two case studies on air pollution where the weather is an (inherently regional) known confounder.

Normalizing flows provide a general mechanism for defining expressive probability distributions, only requiring the specification of a (usually simple) base distribution and a series of bijective transformations. There has been much recent work on normalizing flows, ranging from improving their expressive power to expanding their application. We believe the field has now matured and is in need of a unified perspective. In this review, we attempt to provide such a perspective by describing flows through the lens of probabilistic modeling and inference. We place special emphasis on the fundamental principles of flow design, and discuss foundational topics such as expressive power and computational trade-offs. We also broaden the conceptual framing of flows by relating them to more general probability transformations. Lastly, we summarize the use of flows for tasks such as generative modeling, approximate inference, and supervised learning.

Conventional causal discovery methods rely on centralized data, which is inconsistent with the decentralized nature of data in many real-world situations. This discrepancy has motivated the development of federated causal discovery (FCD) approaches. However, existing FCD methods may be limited by their potentially restrictive assumptions of identifiable functional causal models or homogeneous data distributions, narrowing their applicability in diverse scenarios. In this paper, we propose a novel FCD method attempting to accommodate arbitrary causal models and heterogeneous data. We first utilize a surrogate variable corresponding to the client index to account for the data heterogeneity across different clients. We then develop a federated conditional independence test (FCIT) for causal skeleton discovery and establish a federated independent change principle (FICP) to determine causal directions. These approaches involve constructing summary statistics as a proxy of the raw data to protect data privacy. Owing to the nonparametric properties, FCIT and FICP make no assumption about particular functional forms, thereby facilitating the handling of arbitrary causal models. We conduct extensive experiments on synthetic and real datasets to show the efficacy of our method. The code is available at https://github.com/lokali/FedCDH.git.

Generating high-quality time series data has emerged as a critical research topic due to its broad utility in supporting downstream time series mining tasks. A major challenge lies in modeling the intrinsic stochasticity of temporal dynamics, as real-world sequences often exhibit random fluctuations and localized variations. While diffusion models have achieved remarkable success, their generation process is computationally inefficient, often requiring hundreds to thousands of expensive function evaluations per sample. Flow matching has emerged as a more efficient paradigm, yet its conventional ordinary differential equation (ODE)-based formulation fails to explicitly capture stochasticity, thereby limiting the fidelity of generated sequences. By contrast, stochastic differential equation (SDE) are naturally suited for modeling randomness and uncertainty. Motivated by these insights, we propose TimeFlow, a novel SDE-based flow matching framework that integrates a encoder-only architecture. Specifically, we design a component-wise decomposed velocity field to capture the multi-faceted structure of time series and augment the vanilla flow-matching optimization with an additional stochastic term to enhance representational expressiveness. TimeFlow is flexible and general, supporting both unconditional and conditional generation tasks within a unified framework. Extensive experiments across diverse datasets demonstrate that our model consistently outperforms strong baselines in generation quality, diversity, and efficiency.

Understanding how events in a scenario causally connect with each other is important for effectively modeling and reasoning about events. But event reasoning remains a difficult challenge, and despite recent advances, Large Language Models (LLMs) still struggle to accurately identify causal connections between events. This struggle leads to poor performance on deeper reasoning tasks like event forecasting and timeline understanding. To address this challenge, we investigate the generation of causal event graphs (e.g., A enables B) as a parallel mechanism to help LLMs explicitly represent causality during inference. This paper evaluates both how to generate correct graphs as well as how graphs can assist reasoning. We propose a collaborative approach to causal graph generation where we use LLMs to simulate experts that focus on specific semantic relations. The experts engage in multiple rounds of discussions which are then consolidated by a final expert. Then, to demonstrate the utility of causal graphs, we use them on multiple downstream applications, and also introduce a new explainable event prediction task that requires a causal chain of events in the explanation. These explanations are more informative and coherent than baseline generations. Finally, our overall approach not finetuned on any downstream task, achieves competitive results with state-of-the-art models on both forecasting and next event prediction tasks.

Fair machine learning seeks to identify and mitigate biases in predictions against unfavorable populations characterized by demographic attributes, such as race and gender. Recently, a few works have extended fairness to graph data, such as social networks, but most of them neglect the causal relationships among data instances. This paper addresses the prevalent challenge in fairness-aware ML algorithms, which typically assume Independent and Identically Distributed (IID) data. We tackle the overlooked domain of non-IID, graph-based settings where data instances are interconnected, influencing the outcomes of fairness interventions. We base our research on the Network Structural Causal Model (NSCM) framework and posit two main assumptions: Decomposability and Graph Independence, which enable the computation of interventional distributions in non-IID settings using the do-calculus. Based on that, we develop the Message Passing Variational Autoencoder for Causal Inference (MPVA) to compute interventional distributions and facilitate causally fair node classification through estimated interventional distributions. Empirical evaluations on semi-synthetic and real-world datasets demonstrate that MPVA outperforms conventional methods by effectively approximating interventional distributions and mitigating bias. The implications of our findings underscore the potential of causality-based fairness in complex ML applications, setting the stage for further research into relaxing the initial assumptions to enhance model fairness.

We propose a principled and effective framework for one-step generative modeling. We introduce the notion of average velocity to characterize flow fields, in contrast to instantaneous velocity modeled by Flow Matching methods. A well-defined identity between average and instantaneous velocities is derived and used to guide neural network training. Our method, termed the MeanFlow model, is self-contained and requires no pre-training, distillation, or curriculum learning. MeanFlow demonstrates strong empirical performance: it achieves an FID of 3.43 with a single function evaluation (1-NFE) on ImageNet 256x256 trained from scratch, significantly outperforming previous state-of-the-art one-step diffusion/flow models. Our study substantially narrows the gap between one-step diffusion/flow models and their multi-step predecessors, and we hope it will motivate future research to revisit the foundations of these powerful models.

We study the class of location-scale or heteroscedastic noise models (LSNMs), in which the effect Y can be written as a function of the cause X and a noise source N independent of X, which may be scaled by a positive function g over the cause, i.e., Y = f(X) + g(X)N. Despite the generality of the model class, we show the causal direction is identifiable up to some pathological cases. To empirically validate these theoretical findings, we propose two estimators for LSNMs: an estimator based on (non-linear) feature maps, and one based on neural networks. Both model the conditional distribution of Y given X as a Gaussian parameterized by its natural parameters. When the feature maps are correctly specified, we prove that our estimator is jointly concave, and a consistent estimator for the cause-effect identification task. Although the the neural network does not inherit those guarantees, it can fit functions of arbitrary complexity, and reaches state-of-the-art performance across benchmarks.

Market generators using deep generative models have shown promise for synthetic financial data generation, but existing approaches lack causal reasoning capabilities essential for counterfactual analysis and risk assessment. We propose a Time-series Neural Causal Model VAE (TNCM-VAE) that combines variational autoencoders with structural causal models to generate counterfactual financial time series while preserving both temporal dependencies and causal relationships. Our approach enforces causal constraints through directed acyclic graphs in the decoder architecture and employs the causal Wasserstein distance for training. We validate our method on synthetic autoregressive models inspired by the Ornstein-Uhlenbeck process, demonstrating superior performance in counterfactual probability estimation with L1 distances as low as 0.03-0.10 compared to ground truth. The model enables financial stress testing, scenario analysis, and enhanced backtesting by generating plausible counterfactual market trajectories that respect underlying causal mechanisms.

Recent advances in artificial intelligence reveal the limits of purely predictive systems and call for a shift toward causal and collaborative reasoning. Drawing inspiration from the revolution of Grothendieck in mathematics, we introduce the relativity of causal knowledge, which posits structural causal models (SCMs) are inherently imperfect, subjective representations embedded within networks of relationships. By leveraging category theory, we arrange SCMs into a functor category and show that their observational and interventional probability measures naturally form convex structures. This result allows us to encode non-intervened SCMs with convex spaces of probability measures. Next, using sheaf theory, we construct the network sheaf and cosheaf of causal knowledge. These structures enable the transfer of causal knowledge across the network while incorporating interventional consistency and the perspective of the subjects, ultimately leading to the formal, mathematical definition of relative causal knowledge.

While the popularity of physics-informed neural networks (PINNs) is steadily rising, to this date PINNs have not been successful in simulating dynamical systems whose solution exhibits multi-scale, chaotic or turbulent behavior. In this work we attribute this shortcoming to the inability of existing PINNs formulations to respect the spatio-temporal causal structure that is inherent to the evolution of physical systems. We argue that this is a fundamental limitation and a key source of error that can ultimately steer PINN models to converge towards erroneous solutions. We address this pathology by proposing a simple re-formulation of PINNs loss functions that can explicitly account for physical causality during model training. We demonstrate that this simple modification alone is enough to introduce significant accuracy improvements, as well as a practical quantitative mechanism for assessing the convergence of a PINNs model. We provide state-of-the-art numerical results across a series of benchmarks for which existing PINNs formulations fail, including the chaotic Lorenz system, the Kuramoto-Sivashinsky equation in the chaotic regime, and the Navier-Stokes equations in the turbulent regime. To the best of our knowledge, this is the first time that PINNs have been successful in simulating such systems, introducing new opportunities for their applicability to problems of industrial complexity.

We investigate causal computations taking sequences of inputs to sequences of outputs where the nth output depends on the first n inputs only. We model these in category theory via a construction taking a Cartesian category C to another category St(C) with a novel trace-like operation called "delayed trace", which misses yanking and dinaturality axioms of the usual trace. The delayed trace operation provides a feedback mechanism in St(C) with an implicit guardedness guarantee. When C is equipped with a Cartesian differential operator, we construct a differential operator for St(C) using an abstract version of backpropagation through time, a technique from machine learning based on unrolling of functions. This obtains a swath of properties for backpropagation through time, including a chain rule and Schwartz theorem. Our differential operator is also able to compute the derivative of a stateful network without requiring the network to be unrolled.

Normalizing flows are powerful non-parametric statistical models that function as a hybrid between density estimators and generative models. Current learning algorithms for normalizing flows assume that data points are sampled independently, an assumption that is frequently violated in practice, which may lead to erroneous density estimation and data generation. We propose a likelihood objective of normalizing flows incorporating dependencies between the data points, for which we derive a flexible and efficient learning algorithm suitable for different dependency structures. We show that respecting dependencies between observations can improve empirical results on both synthetic and real-world data, and leads to higher statistical power in a downstream application to genome-wide association studies.

The multi-step denoising process in diffusion and Flow Matching models causes major efficiency issues, which motivates research on few-step generation. We present Solution Flow Models (SoFlow), a framework for one-step generation from scratch. By analyzing the relationship between the velocity function and the solution function of the velocity ordinary differential equation (ODE), we propose a Flow Matching loss and a solution consistency loss to train our models. The Flow Matching loss allows our models to provide estimated velocity fields for Classifier-Free Guidance (CFG) during training, which improves generation performance. Notably, our consistency loss does not require the calculation of the Jacobian-vector product (JVP), a common requirement in recent works that is not well-optimized in deep learning frameworks like PyTorch. Experimental results indicate that, when trained from scratch using the same Diffusion Transformer (DiT) architecture and an equal number of training epochs, our models achieve better FID-50K scores than MeanFlow models on the ImageNet 256x256 dataset.

Flow-based generative models have become a strong framework for high-quality generative modeling, yet pretrained models are rarely used in their vanilla conditional form: conditional samples without guidance often appear diffuse and lack fine-grained detail due to the smoothing effects of neural networks. Existing guidance techniques such as classifier-free guidance (CFG) improve fidelity but double the inference cost and typically reduce sample diversity. We introduce Momentum Guidance (MG), a new dimension of guidance that leverages the ODE trajectory itself. MG extrapolates the current velocity using an exponential moving average of past velocities and preserves the standard one-evaluation-per-step cost. It matches the effect of standard guidance without extra computation and can further improve quality when combined with CFG. Experiments demonstrate MG's effectiveness across benchmarks. Specifically, on ImageNet-256, MG achieves average improvements in FID of 36.68% without CFG and 25.52% with CFG across various sampling settings, attaining an FID of 1.597 at 64 sampling steps. Evaluations on large flow-based models like Stable Diffusion 3 and FLUX.1-dev further confirm consistent quality enhancements across standard metrics.

Explainability methods for NLP systems encounter a version of the fundamental problem of causal inference: for a given ground-truth input text, we never truly observe the counterfactual texts necessary for isolating the causal effects of model representations on outputs. In response, many explainability methods make no use of counterfactual texts, assuming they will be unavailable. In this paper, we show that robust causal explainability methods can be created using approximate counterfactuals, which can be written by humans to approximate a specific counterfactual or simply sampled using metadata-guided heuristics. The core of our proposal is the Causal Proxy Model (CPM). A CPM explains a black-box model N because it is trained to have the same actual input/output behavior as N while creating neural representations that can be intervened upon to simulate the counterfactual input/output behavior of N. Furthermore, we show that the best CPM for N performs comparably to N in making factual predictions, which means that the CPM can simply replace N, leading to more explainable deployed models. Our code is available at https://github.com/frankaging/Causal-Proxy-Model.

Temporal causal representation learning methods assume that causal mechanisms switch instantaneously between discrete domains, yet real-world systems often exhibit continuous mechanism transitions. For example, a vehicle's dynamics evolve gradually through a turning maneuver, and human gait shifts smoothly from walking to running. We formalize this setting by modeling transitional mechanisms as convex combinations of finitely many atomic mechanisms, governed by time-varying mixing coefficients. Our theoretical contributions establish that both the latent causal variables and the continuous mixing trajectory are jointly identifiable. We further propose TRACE, a Mixture-of-Experts framework where each expert learns one atomic mechanism during training, enabling recovery of mechanism trajectories at test time. This formulation generalizes to intermediate mechanism states never observed during training. Experiments on synthetic and real-world data demonstrate that TRACE recovers mixing trajectories with up to 0.99 correlation, substantially outperforming discrete-switching baselines.

Autoregressive models are among the best performing neural density estimators. We describe an approach for increasing the flexibility of an autoregressive model, based on modelling the random numbers that the model uses internally when generating data. By constructing a stack of autoregressive models, each modelling the random numbers of the next model in the stack, we obtain a type of normalizing flow suitable for density estimation, which we call Masked Autoregressive Flow. This type of flow is closely related to Inverse Autoregressive Flow and is a generalization of Real NVP. Masked Autoregressive Flow achieves state-of-the-art performance in a range of general-purpose density estimation tasks.

Causal discovery, the process of recovering underlying causal structures from observational data, is a fundamental pursuit across scientific disciplines. Over the past decades, numerous algorithms have been developed to tackle this challenge through workflows tailored to the specific causal mechanisms underlying each type of dataset, demonstrating effectiveness across a wide range of applications. However, as the volume and heterogeneity of real-world data continue to grow, this dataset-specific approach inevitably leads to a fragmented, test-driven paradigm that struggles to scale to the demands of modern scientific discovery. To address this, we formulate the Causal Discovery Foundation Model (CDFM) as a unified, general-purpose framework for zero-shot structural inference. To ensure reliable generalization across unknown domains, we first investigate the theoretical boundaries of causal identifiability, revealing the indispensable role of causal prior mechanisms in this process. Building on these insights, we formulate a principled variational framework that treats unknown causal mechanisms as latent variables and mathematically decomposes the intractable marginal likelihood into distinct, tractable learning modules. The variational decomposition provides a conceptual design principle for the architecture design of CDFM, while comprehensive causal knowledge guides the large-scale synthesis of our pretraining data. By pretraining on a massive, highly diverse space of synthetic structural causal models, CDFM successfully internalizes complex statistical asymmetries. Extensive experiments demonstrate that CDFM consistently outperforms traditional algorithms, driving a paradigm shift toward a general-purpose causal discovery foundation model.

Autoregressive generative models naturally generate variable-length sequences, while non-autoregressive models struggle, often imposing rigid, token-wise structures. We propose Edit Flows, a non-autoregressive model that overcomes these limitations by defining a discrete flow over sequences through edit operationsx2013insertions, deletions, and substitutions. By modeling these operations within a Continuous-time Markov Chain over the sequence space, Edit Flows enable flexible, position-relative generation that aligns more closely with the structure of sequence data. Our training method leverages an expanded state space with auxiliary variables, making the learning process efficient and tractable. Empirical results show that Edit Flows outperforms both autoregressive and mask models on image captioning and significantly outperforms the mask construction in text and code generation.

Evaluating retrieval-augmented generation (RAG) pipelines requires corpora where ground truth is knowable, temporally structured, and cross-artifact properties that real-world datasets rarely provide cleanly. Existing resources such as the Enron corpus carry legal ambiguity, demographic skew, and no structured ground truth. Purely LLM-generated synthetic data solves the legal problem but introduces a subtler one: the generating model cannot be prevented from hallucinating facts that contradict themselves across documents.We present OrgForge, an open-source multi-agent simulation framework that enforces a strict physics-cognition boundary: a deterministic Python engine maintains a SimEvent ground truth bus; large language models generate only surface prose, constrained by validated proposals. An actor-local clock enforces causal timestamp correctness across all artifact types, eliminating the class of timeline inconsistencies that arise when timestamps are sampled independently per document. We formalize three graph-dynamic subsystems stress propagation via betweenness centrality, temporal edge-weight decay, and Dijkstra escalation routing that govern organizational behavior independently of any LLM. Running a configurable N-day simulation, OrgForge produces interleaved Slack threads, JIRA tickets, Confluence pages, Git pull requests, and emails, all traceable to a shared, immutable event log. We additionally describe a causal chain tracking subsystem that accumulates cross-artifact evidence graphs per incident, a hybrid reciprocal-rank-fusion recurrence detector for identifying repeated failure classes, and an inbound/outbound email engine that routes vendor alerts, customer complaints, and HR correspondence through gated causal chains with probabilistic drop simulation. OrgForge is available under the MIT license.

Normalising Flows are non-parametric statistical models characterised by their dual capabilities of density estimation and generation. This duality requires an inherently invertible architecture. However, the requirement of invertibility imposes constraints on their expressiveness, necessitating a large number of parameters and innovative architectural designs to achieve good results. Whilst flow-based models predominantly rely on neural-network-based transformations for expressive designs, alternative transformation methods have received limited attention. In this work, we present Ferumal flow, a novel kernelised normalising flow paradigm that integrates kernels into the framework. Our results demonstrate that a kernelised flow can yield competitive or superior results compared to neural network-based flows whilst maintaining parameter efficiency. Kernelised flows excel especially in the low-data regime, enabling flexible non-parametric density estimation in applications with sparse data availability.

We introduce a new family of deep neural network models. Instead of specifying a discrete sequence of hidden layers, we parameterize the derivative of the hidden state using a neural network. The output of the network is computed using a black-box differential equation solver. These continuous-depth models have constant memory cost, adapt their evaluation strategy to each input, and can explicitly trade numerical precision for speed. We demonstrate these properties in continuous-depth residual networks and continuous-time latent variable models. We also construct continuous normalizing flows, a generative model that can train by maximum likelihood, without partitioning or ordering the data dimensions. For training, we show how to scalably backpropagate through any ODE solver, without access to its internal operations. This allows end-to-end training of ODEs within larger models.

Physics-informed neural networks (PINNs) leverage neural-networks to find the solutions of partial differential equation (PDE)-constrained optimization problems with initial conditions and boundary conditions as soft constraints. These soft constraints are often considered to be the sources of the complexity in the training phase of PINNs. Here, we demonstrate that the challenge of training (i) persists even when the boundary conditions are strictly enforced, and (ii) is closely related to the Kolmogorov n-width associated with problems demonstrating transport, convection, traveling waves, or moving fronts. Given this realization, we describe the mechanism underlying the training schemes such as those used in eXtended PINNs (XPINN), curriculum regularization, and sequence-to-sequence learning. For an important category of PDEs, i.e., governed by non-linear convection-diffusion equation, we propose reformulating PINNs on a Lagrangian frame of reference, i.e., LPINNs, as a PDE-informed solution. A parallel architecture with two branches is proposed. One branch solves for the state variables on the characteristics, and the second branch solves for the low-dimensional characteristics curves. The proposed architecture conforms to the causality innate to the convection, and leverages the direction of travel of the information in the domain. Finally, we demonstrate that the loss landscapes of LPINNs are less sensitive to the so-called "complexity" of the problems, compared to those in the traditional PINNs in the Eulerian framework.

Despite the progress of foundation models, knowledge-based reasoning remains a persistent challenge due to their limited capacity for knowledge recall and inference. Existing methods primarily focus on encouraging these models to plan and solve problems or extensively sample reasoning chains independently. However, these methods often overlook conceptual errors and inferential fallacies, inevitably leading to a series of notorious issues such as misleading conclusions, cognitive biases, and reduced decision quality. While explicit modeling of causality is argued to hold promise in addressing these issues, contemporary research efforts have thus far fallen short in achieving causality-based foundation models. Drawing inspiration from the orchestration of diverse specialized agents collaborating to tackle intricate tasks, we propose a framework named Causal-Consistency Chain-of-Thought (CaCo-CoT) that harnesses multi-agent collaboration to bolster the faithfulness and causality of foundation models, involving a set of reasoners and evaluators. These agents collaboratively work within a reasoning-and-consensus paradigm to improve faithfulness. The reasoners are tasked with generating reasoning chains for knowledge-intensive problems by mimicking human causal reasoning. Meanwhile, the evaluator scrutinizes the causal consistency of a reasoner's reasoning chain from a non-causal and a counterfactual perspective. Our framework demonstrates significant superiority over state-of-the-art methods through extensive and comprehensive evaluations across text-based and multi-modal knowledge reasoning tasks (e.g., science question answering and commonsense reasoning).

Survival analysis, or time-to-event modelling, is a classical statistical problem that has garnered a lot of interest for its practical use in epidemiology, demographics or actuarial sciences. Recent advances on the subject from the point of view of machine learning have been concerned with precise per-individual predictions instead of population studies, driven by the rise of individualized medicine. We introduce here a conditional normalizing flow based estimate of the time-to-event density as a way to model highly flexible and individualized conditional survival distributions. We use a novel hierarchical formulation of normalizing flows to enable efficient fitting of flexible conditional distributions without overfitting and show how the normalizing flow formulation can be efficiently adapted to the censored setting. We experimentally validate the proposed approach on a synthetic dataset as well as four open medical datasets and an example of a common financial problem.

Given an unconditional generative model and a predictor for a target property (e.g., a classifier), the goal of training-free guidance is to generate samples with desirable target properties without additional training. As a highly efficient technique for steering generative models toward flexible outcomes, training-free guidance has gained increasing attention in diffusion models. However, existing methods only handle data in continuous spaces, while many scientific applications involve both continuous and discrete data (referred to as multimodality). Another emerging trend is the growing use of the simple and general flow matching framework in building generative foundation models, where guided generation remains under-explored. To address this, we introduce TFG-Flow, a novel training-free guidance method for multimodal generative flow. TFG-Flow addresses the curse-of-dimensionality while maintaining the property of unbiased sampling in guiding discrete variables. We validate TFG-Flow on four molecular design tasks and show that TFG-Flow has great potential in drug design by generating molecules with desired properties.

The concept of causal abstraction got recently popularised to demystify the opaque decision-making processes of machine learning models; in short, a neural network can be abstracted as a higher-level algorithm if there exists a function which allows us to map between them. Notably, most interpretability papers implement these maps as linear functions, motivated by the linear representation hypothesis: the idea that features are encoded linearly in a model's representations. However, this linearity constraint is not required by the definition of causal abstraction. In this work, we critically examine the concept of causal abstraction by considering arbitrarily powerful alignment maps. In particular, we prove that under reasonable assumptions, any neural network can be mapped to any algorithm, rendering this unrestricted notion of causal abstraction trivial and uninformative. We complement these theoretical findings with empirical evidence, demonstrating that it is possible to perfectly map models to algorithms even when these models are incapable of solving the actual task; e.g., on an experiment using randomly initialised language models, our alignment maps reach 100% interchange-intervention accuracy on the indirect object identification task. This raises the non-linear representation dilemma: if we lift the linearity constraint imposed to alignment maps in causal abstraction analyses, we are left with no principled way to balance the inherent trade-off between these maps' complexity and accuracy. Together, these results suggest an answer to our title's question: causal abstraction is not enough for mechanistic interpretability, as it becomes vacuous without assumptions about how models encode information. Studying the connection between this information-encoding assumption and causal abstraction should lead to exciting future work.

A central goal of science is to produce valid explanations of complex systems: high-level causal accounts that faithfully reflect the behavior of lower-level mechanisms. Yet no consensus exists on how to measure whether a proposed high-level explanation is actually valid. We introduce a benchmark of ten complex systems spanning both discrete and continuous state spaces, as well as static and dynamical regimes, each equipped with consensual ground-truth causal explanations and invalid contrastive conditions. Within a unified causal abstraction framework, we systematically evaluate over thirty candidate metrics drawn from observational, functional, information-theoretic, and causal families. Our results show that only the latter reliably discriminates valid from invalid abstractions, and only when incorporating faithfulness testing over unmapped variables. Building on these findings, we introduce the Causal Abstraction Error (CAE), a continuous validity metric with an explicit faithfulness test, which passes all discrimination tests across every system and can converge with as few as 30 sampled interventions. We offer it as a general-purpose metric for the discovery and validation of high-level explanations.

In the physical world we inhabit, space and time are fundamentally continuous. However, existing machine learning paradigms for world modeling are largely confined to discrete-time prediction, thereby exhibiting significant inefficiency in capturing the dynamics of physical world. We introduce Physical-Time Flow (PT-Flow), a novel approach that learns a continuous latent velocity field operating in physical time. Crucially, the underlying dynamics of sequential data are parameterized by an ordinary differential equation (ODE) embedded in a well-structured representation space. Under this paradigm, the prediction of future can be recast as temporal integration via an ODE solver in the compressed latent space. Building upon PT-Flow, we construct ODEWorld, a continuous-time latent world model that is both efficient and versatile. By extracting time-variant features and enforcing ODE properties on both the dynamical representation space and the latent velocity field, ODEWorld effectively addresses the long-standing representation collapse issue in latent world model literature. This also enables high-quality image reconstruction even after long-horizon prediction. Moreover, its continuous nature allows for arbitrary temporal resolution and even backward prediction, which is impossible for most discrete-time models. Lastly, ODEWorld can provide rich planning-oriented information to facilitate downstream policy learning. Comprehensive experiments demonstrate that ODEWorld successfully reconciles planning-conducive dynamics abstraction with visual realism, excelling in both video generation and robotic control. https://dstate.github.io/odeworld_website/{Project Website}.

Can evolving networks be inferred and modeled without directly observing their nodes and edges? In many applications, the edges of a dynamic network might not be observed, but one can observe the dynamics of stochastic cascading processes (e.g., information diffusion, virus propagation) occurring over the unobserved network. While there have been efforts to infer networks based on such data, providing a generative probabilistic model that is able to identify the underlying time-varying network remains an open question. Here we consider the problem of inferring generative dynamic network models based on network cascade diffusion data. We propose a novel framework for providing a non-parametric dynamic network model--based on a mixture of coupled hierarchical Dirichlet processes-- based on data capturing cascade node infection times. Our approach allows us to infer the evolving community structure in networks and to obtain an explicit predictive distribution over the edges of the underlying network--including those that were not involved in transmission of any cascade, or are likely to appear in the future. We show the effectiveness of our approach using extensive experiments on synthetic as well as real-world networks.

Flow matching (FM) is a general framework for defining probability paths via Ordinary Differential Equations (ODEs) to transform between noise and data samples. Recent approaches attempt to straighten these flow trajectories to generate high-quality samples with fewer function evaluations, typically through iterative rectification methods or optimal transport solutions. In this paper, we introduce Consistency Flow Matching (Consistency-FM), a novel FM method that explicitly enforces self-consistency in the velocity field. Consistency-FM directly defines straight flows starting from different times to the same endpoint, imposing constraints on their velocity values. Additionally, we propose a multi-segment training approach for Consistency-FM to enhance expressiveness, achieving a better trade-off between sampling quality and speed. Preliminary experiments demonstrate that our Consistency-FM significantly improves training efficiency by converging 4.4x faster than consistency models and 1.7x faster than rectified flow models while achieving better generation quality. Our code is available at: https://github.com/YangLing0818/consistency_flow_matching

We explore a generative machine learning-based approach for estimating multi-dimensional probability density functions (PDFs) in a target sample using a statistically independent but related control sample - a common challenge in particle physics data analysis. The generative model must accurately reproduce individual observable distributions while preserving the correlations between them, based on the input multidimensional distribution from the control sample. Here we present a conditional normalizing flow model (CNF) based on a chain of bijectors which learns to transform unpaired simulation events to data events. We assess the performance of the CNF model in the context of LHC Higgs to diphoton analysis, where we use the CNF model to convert a Monte Carlo diphoton sample to one that models data. We show that the CNF model can accurately model complex data distributions and correlations. We also leverage the recently popularized Modified Differential Multiplier Method (MDMM) to improve the convergence of our model and assign physical meaning to usually arbitrary loss-function parameters.

Learning causal structure from observational data often assumes that we observe independent and identically distributed (i.\,i.\,d) data. The traditional approach aims to find a graphical representation that encodes the same set of conditional independence relationships as those present in the observed distribution. It is known that under i.\,i.\,d assumption, even with infinite data, there is a limit to how fine-grained a causal structure we can identify. To overcome this limitation, recent work has explored using data originating from different, related environments to learn richer causal structure. These approaches implicitly rely on the independent causal mechanisms (ICM) principle, which postulates that the mechanism giving rise to an effect given its causes and the mechanism which generates the causes do not inform or influence each other. Thus, components of the causal model can independently change from environment to environment. Despite its wide application in machine learning and causal inference, there is a lack of statistical formalization of the ICM principle and how it enables identification of richer causal structures from grouped data. Here we present new causal de Finetti theorems which offer a first statistical formalization of ICM principle and show how causal structure identification is possible from exchangeable data. Our work provides theoretical justification for a broad range of techniques leveraging multi-environment data to learn causal structure.

Recent advances in artificial intelligence (AI) have produced highly capable and controllable systems. This creates unprecedented opportunities for structured reasoning as well as collaboration among multiple AI systems and humans. To fully realize this potential, it is essential to develop a principled way of designing and studying such structured interactions. For this purpose, we introduce the conceptual framework of Flows: a systematic approach to modeling complex interactions. Flows are self-contained building blocks of computation, with an isolated state, communicating through a standardized message-based interface. This modular design allows Flows to be recursively composed into arbitrarily nested interactions, with a substantial reduction of complexity. Crucially, any interaction can be implemented using this framework, including prior work on AI--AI and human--AI interactions, prompt engineering schemes, and tool augmentation. We demonstrate the potential of Flows on the task of competitive coding, a challenging task on which even GPT-4 struggles. Our results suggest that structured reasoning and collaboration substantially improve generalization, with AI-only Flows adding +21 and human--AI Flows adding +54 absolute points in terms of solve rate. To support rapid and rigorous research, we introduce the aiFlows library. The library comes with a repository of Flows that can be easily used, extended, and composed into novel, more complex Flows. The aiFlows library is available at https://github.com/epfl-dlab/aiflows. Data and Flows for reproducing our experiments are available at https://github.com/epfl-dlab/cc_flows.

Causal reasoning is a cornerstone of how humans interpret the world. To model and reason about causality, causal graphs offer a concise yet effective solution. Given the impressive advancements in language models, a crucial question arises: can they really understand causal graphs? To this end, we pioneer an investigation into language models' understanding of causal graphs. Specifically, we develop a framework to define causal graph understanding, by assessing language models' behaviors through four practical criteria derived from diverse disciplines (e.g., philosophy and psychology). We then develop CLEAR, a novel benchmark that defines three complexity levels and encompasses 20 causal graph-based tasks across these levels. Finally, based on our framework and benchmark, we conduct extensive experiments on six leading language models and summarize five empirical findings. Our results indicate that while language models demonstrate a preliminary understanding of causal graphs, significant potential for improvement remains. Our project website is at https://github.com/OpenCausaLab/CLEAR.

Institutional investors have been increasing the allocation of the illiquid alternative assets such as private equity funds in their portfolios, yet there exists a very limited literature on cash flow forecasting of illiquid alternative assets. The net cash flow of private equity funds typically follow a J-curve pattern, however the timing and the size of the contributions and distributions depend on the investment opportunities. In this paper, we develop a benchmark model and present two novel approaches (direct vs. indirect) to predict the cash flows of private equity funds. We introduce a sliding window approach to apply on our cash flow data because different vintage year funds contain different lengths of cash flow information. We then pass the data to an LSTM/ GRU model to predict the future cash flows either directly or indirectly (based on the benchmark model). We further integrate macroeconomic indicators into our data, which allows us to consider the impact of market environment on cash flows and to apply stress testing. Our results indicate that the direct model is easier to implement compared to the benchmark model and the indirect model, but still the predicted cash flows align better with the actual cash flows. We also show that macroeconomic variables improve the performance of the direct model whereas the impact is not obvious on the indirect model.

Latent or continuous chain-of-thought methods replace explicit textual rationales with a number of internal latent steps, but these intermediate computations are difficult to evaluate beyond correlation-based probes. In this paper, we view latent chain-of-thought as a manipulable causal process in representation space by modeling latent steps as variables in a structural causal model (SCM) and analyzing their effects through step-wise do-interventions. We study two representative paradigms (i.e., Coconut and CODI) on both mathematical and general reasoning tasks to investigate three key questions: (1) which steps are causally necessary for correctness and when answers become decidable early; (2) how does influence propagate across steps, and how does this structure compare to explicit CoT; and (3) do intermediate trajectories retain competing answer modes, and how does output-level commitment differ from representational commitment across steps. We find that latent-step budgets behave less like homogeneous extra depth and more like staged functionality with non-local routing, and we identify a persistent gap between early output bias and late representational commitment. These results motivate mode-conditional and stability-aware analyses -- and corresponding training/decoding objectives -- as more reliable tools for interpreting and improving latent reasoning systems. Code is available at https://github.com/J1mL1/causal-latent-cot.

Continuous normalizing flows (CNFs) learn an ordinary differential equation to transform prior samples into data. Flow matching (FM) has recently emerged as a simulation-free approach for training CNFs by regressing a velocity model towards the conditional velocity field. However, on constrained domains, the learned velocity model may lead to undesirable flows that result in highly unnatural samples, e.g., oversaturated images, due to both flow matching error and simulation error. To address this, we add a boundary constraint term to CNFs, which leads to reflected CNFs that keep trajectories within the constrained domains. We propose reflected flow matching (RFM) to train the velocity model in reflected CNFs by matching the conditional velocity fields in a simulation-free manner, similar to the vanilla FM. Moreover, the analytical form of conditional velocity fields in RFM avoids potentially biased approximations, making it superior to existing score-based generative models on constrained domains. We demonstrate that RFM achieves comparable or better results on standard image benchmarks and produces high-quality class-conditioned samples under high guidance weight.

Diffusion probabilistic models (DPMs) have become the state-of-the-art in high-quality image generation. However, DPMs have an arbitrary noisy latent space with no interpretable or controllable semantics. Although there has been significant research effort to improve image sample quality, there is little work on representation-controlled generation using diffusion models. Specifically, causal modeling and controllable counterfactual generation using DPMs is an underexplored area. In this work, we propose CausalDiffAE, a diffusion-based causal representation learning framework to enable counterfactual generation according to a specified causal model. Our key idea is to use an encoder to extract high-level semantically meaningful causal variables from high-dimensional data and model stochastic variation using reverse diffusion. We propose a causal encoding mechanism that maps high-dimensional data to causally related latent factors and parameterize the causal mechanisms among latent factors using neural networks. To enforce the disentanglement of causal variables, we formulate a variational objective and leverage auxiliary label information in a prior to regularize the latent space. We propose a DDIM-based counterfactual generation procedure subject to do-interventions. Finally, to address the limited label supervision scenario, we also study the application of CausalDiffAE when a part of the training data is unlabeled, which also enables granular control over the strength of interventions in generating counterfactuals during inference. We empirically show that CausalDiffAE learns a disentangled latent space and is capable of generating high-quality counterfactual images.

Real-time interactive video generation requires low-latency, streaming, and controllable rollout. Existing autoregressive (AR) diffusion distillation methods have achieved strong results in the chunk-wise 4-step regime by distilling bidirectional base models into few-step AR students, but they remain limited by coarse response granularity and non-negligible sampling latency. In this paper, we study a more aggressive setting: frame-wise autoregression with only 1--2 sampling steps. In this regime, we identify the initialization of a few-step AR student as the key bottleneck: existing strategies are either target-misaligned, incapable of few-step generation, or too costly to scale. We propose Causal Forcing++, a principled and scalable pipeline that uses causal consistency distillation (causal CD) for few-step AR initialization. The core idea is that causal CD learns the same AR-conditional flow map as causal ODE distillation, but obtains supervision from a single online teacher ODE step between adjacent timesteps, avoiding the need to precompute and store full PF-ODE trajectories. This makes the initialization both more efficient and easier to optimize. The resulting pipeline, \ours, surpasses the SOTA 4-step chunk-wise Causal Forcing under the \textbf{frame-wise 2-step setting} by 0.1 in VBench Total, 0.3 in VBench Quality, and 0.335 in VisionReward, while reducing first-frame latency by 50\% and Stage 2 training cost by sim4times. We further extend the pipeline to action-conditioned world model generation in the spirit of Genie3. Project Page: https://github.com/thu-ml/Causal-Forcing and https://github.com/shengshu-ai/minWM .

Diffusion models (DMs) have achieved remarkable success in image and video generation. However, they still struggle with (1) physical alignment and (2) out-of-distribution (OOD) instruction following. We argue that these issues stem from the models' failure to learn causal directions and to disentangle causal factors for novel recombination. We introduce the Causal Scene Graph (CSG) and the Physical Alignment Probe (PAP) dataset to enable diagnostic interventions. This analysis yields three key insights. First, DMs struggle with multi-hop reasoning for elements not explicitly determined in the prompt. Second, the prompt embedding contains disentangled representations for texture and physics. Third, visual causal structure is disproportionately established during the initial, computationally limited denoising steps. Based on these findings, we introduce LINA (Learning INterventions Adaptively), a novel framework that learns to predict prompt-specific interventions, which employs (1) targeted guidance in the prompt and visual latent spaces, and (2) a reallocated, causality-aware denoising schedule. Our approach enforces both physical alignment and OOD instruction following in image and video DMs, achieving state-of-the-art performance on challenging causal generation tasks and the Winoground dataset. Our project page is at https://opencausalab.github.io/LINA.

We introduce BCAT, a PDE foundation model designed for autoregressive prediction of solutions to two dimensional fluid dynamics problems. Our approach uses a block causal transformer architecture to model next frame predictions, leveraging previous frames as contextual priors rather than relying solely on sub-frames or pixel-based inputs commonly used in image generation methods. This block causal framework more effectively captures the spatial dependencies inherent in nonlinear spatiotemporal dynamics and physical phenomena. In an ablation study, next frame prediction demonstrated a 3.5x accuracy improvement over next token prediction. BCAT is trained on a diverse range of fluid dynamics datasets, including incompressible and compressible Navier-Stokes equations across various geometries and parameter regimes, as well as the shallow-water equations. The model's performance was evaluated on 6 distinct downstream prediction tasks and tested on about 8K trajectories to measure robustness on a variety of fluid dynamics simulations. BCAT achieved an average relative error of 1.18% across all evaluation tasks, outperforming prior approaches on standard benchmarks. With fine-tuning on a turbulence dataset, we show that the method adapts to new settings with more than 40% better accuracy over prior methods.

An essential and challenging problem in causal inference is causal effect estimation from observational data. The problem becomes more difficult with the presence of unobserved confounding variables. The front-door adjustment is a practical approach for dealing with unobserved confounding variables. However, the restriction for the standard front-door adjustment is difficult to satisfy in practice. In this paper, we relax some of the restrictions by proposing the concept of conditional front-door (CFD) adjustment and develop the theorem that guarantees the causal effect identifiability of CFD adjustment. Furthermore, as it is often impossible for a CFD variable to be given in practice, it is desirable to learn it from data. By leveraging the ability of deep generative models, we propose CFDiVAE to learn the representation of the CFD adjustment variable directly from data with the identifiable Variational AutoEncoder and formally prove the model identifiability. Extensive experiments on synthetic datasets validate the effectiveness of CFDiVAE and its superiority over existing methods. The experiments also show that the performance of CFDiVAE is less sensitive to the causal strength of unobserved confounding variables. We further apply CFDiVAE to a real-world dataset to demonstrate its potential application.

Energy markets exhibit complex causal relationships between weather patterns, generation technologies, and price formation, with regime changes occurring continuously rather than at discrete break points. Current approaches model electricity prices without explicit causal interpretation or counterfactual reasoning capabilities. We introduce Augmented Time Series Causal Models (ATSCM) for energy markets, extending counterfactual reasoning frameworks to multivariate temporal data with learned causal structure. Our approach models energy systems through interpretable factors (weather, generation mix, demand patterns), rich grid dynamics, and observable market variables. We integrate neural causal discovery to learn time-varying causal graphs without requiring ground truth DAGs. Applied to real-world electricity price data, ATSCM enables novel counterfactual queries such as "What would prices be under different renewable generation scenarios?".

Conditional generative modeling aims to learn a conditional data distribution from samples containing data-condition pairs. For this, diffusion and flow-based methods have attained compelling results. These methods use a learned (flow) model to transport an initial standard Gaussian noise that ignores the condition to the conditional data distribution. The model is hence required to learn both mass transport and conditional injection. To ease the demand on the model, we propose Condition-Aware Reparameterization for Flow Matching (CAR-Flow) -- a lightweight, learned shift that conditions the source, the target, or both distributions. By relocating these distributions, CAR-Flow shortens the probability path the model must learn, leading to faster training in practice. On low-dimensional synthetic data, we visualize and quantify the effects of CAR. On higher-dimensional natural image data (ImageNet-256), equipping SiT-XL/2 with CAR-Flow reduces FID from 2.07 to 1.68, while introducing less than 0.6% additional parameters.

Humans can look at a static scene and instantly predict what happens next -- will moving this object cause a collision? We call this ability Causal Spatial Reasoning. However, current multimodal large language models (MLLMs) cannot do this, as they remain largely restricted to static spatial perception, struggling to answer "what-if" questions in a 3D scene. We introduce CausalSpatial, a diagnostic benchmark evaluating whether models can anticipate consequences of object motions across four tasks: Collision, Compatibility, Occlusion, and Trajectory. Results expose a severe gap: humans score 84% while GPT-5 achieves only 54%. Why do MLLMs fail? Our analysis uncovers a fundamental deficiency: models over-rely on textual chain-of-thought reasoning that drifts from visual evidence, producing fluent but spatially ungrounded hallucinations. To address this, we propose the Causal Object World model (COW), a framework that externalizes the simulation process by generating videos of hypothetical dynamics. With explicit visual cues of causality, COW enables models to ground their reasoning in physical reality rather than linguistic priors. We make the dataset and code publicly available here: https://github.com/CausalSpatial/CausalSpatial

The origins of the colors of Trans-Neptunian Objects (TNOs) represent a crucial unresolved question, central to understanding the history of our Solar System. Recent observational surveys have revealed correlations between the eccentricity and inclination of TNOs and their colors. This has rekindled the long-standing debate on whether these colors reflect the conditions of TNO formation or their subsequent collisional evolution. In this study, we address this question with 98.7% certainty, using a model-agnostic, data-driven approach based on causal graphs. First, as a sanity check, we demonstrate how our model can replicate the currently accepted paradigms of TNOs' dynamical history, blindly and without any orbital modeling or physics-based assumptions. In fact, our causal model (with no knowledge of the existence of Neptune) predicts the existence of an unknown perturbing body, i.e., Neptune. We then show how this model predicts, with high certainty, that the color of TNOs is the root cause of their inclination distribution, rather than the other way around. This strongly suggests that the colors of TNOs reflect an underlying dynamical property, most likely their formation location. Moreover, our causal model excludes formation scenarios that invoke substantial color modification by subsequent irradiation. We therefore conclude that the colors of TNOs are predominantly primordial.

With the rise of Large Language Models(LLMs), it has become crucial to understand their capabilities and limitations in deciphering and explaining the complex web of causal relationships that language entails. Current methods use either explicit or implicit causal reasoning, yet there is a strong need for a unified approach combining both to tackle a wide array of causal relationships more effectively. This research proposes a novel architecture called Context Aware Reasoning Enhancement with Counterfactual Analysis(CARE CA) framework to enhance causal reasoning and explainability. The proposed framework incorporates an explicit causal detection module with ConceptNet and counterfactual statements, as well as implicit causal detection through LLMs. Our framework goes one step further with a layer of counterfactual explanations to accentuate LLMs understanding of causality. The knowledge from ConceptNet enhances the performance of multiple causal reasoning tasks such as causal discovery, causal identification and counterfactual reasoning. The counterfactual sentences add explicit knowledge of the not caused by scenarios. By combining these powerful modules, our model aims to provide a deeper understanding of causal relationships, enabling enhanced interpretability. Evaluation of benchmark datasets shows improved performance across all metrics, such as accuracy, precision, recall, and F1 scores. We also introduce CausalNet, a new dataset accompanied by our code, to facilitate further research in this domain.

We present a general causal generative modelling framework for accurate estimation of high fidelity image counterfactuals with deep structural causal models. Estimation of interventional and counterfactual queries for high-dimensional structured variables, such as images, remains a challenging task. We leverage ideas from causal mediation analysis and advances in generative modelling to design new deep causal mechanisms for structured variables in causal models. Our experiments demonstrate that our proposed mechanisms are capable of accurate abduction and estimation of direct, indirect and total effects as measured by axiomatic soundness of counterfactuals.

We study the problem of model selection in causal inference, specifically for the case of conditional average treatment effect (CATE) estimation under binary treatments. Unlike model selection in machine learning, there is no perfect analogue of cross-validation as we do not observe the counterfactual potential outcome for any data point. Towards this, there have been a variety of proxy metrics proposed in the literature, that depend on auxiliary nuisance models estimated from the observed data (propensity score model, outcome regression model). However, the effectiveness of these metrics has only been studied on synthetic datasets as we can access the counterfactual data for them. We conduct an extensive empirical analysis to judge the performance of these metrics introduced in the literature, and novel ones introduced in this work, where we utilize the latest advances in generative modeling to incorporate multiple realistic datasets. Our analysis suggests novel model selection strategies based on careful hyperparameter tuning of CATE estimators and causal ensembling.

Diffusion models are often introduced from multiple perspectives, such as VAEs, score matching, or flow matching, accompanied by dense and technically demanding mathematics that can be difficult for beginners to grasp. One classic question is: how does the reverse process invert the forward process to generate data from pure noise? This article systematically organizes the diffusion model from a fresh Langevin perspective, offering a simpler, clearer, and more intuitive answer. We also address the following questions: how can ODE-based and SDE-based diffusion models be unified under a single framework? Why are diffusion models theoretically superior to ordinary VAEs? Why is flow matching not fundamentally simpler than denoising or score matching, but equivalent under maximum-likelihood? We demonstrate that the Langevin perspective offers clear and straightforward answers to these questions, bridging existing interpretations of diffusion models, showing how different formulations can be converted into one another within a common framework, and offering pedagogical value for both learners and experienced researchers seeking deeper intuition.

Normalizing flows (NF) are a class of powerful generative models that have gained popularity in recent years due to their ability to model complex distributions with high flexibility and expressiveness. In this work, we introduce a new type of normalizing flow that is tailored for modeling positions and orientations of multiple objects in three-dimensional space, such as molecules in a crystal. Our approach is based on two key ideas: first, we define smooth and expressive flows on the group of unit quaternions, which allows us to capture the continuous rotational motion of rigid bodies; second, we use the double cover property of unit quaternions to define a proper density on the rotation group. This ensures that our model can be trained using standard likelihood-based methods or variational inference with respect to a thermodynamic target density. We evaluate the method by training Boltzmann generators for two molecular examples, namely the multi-modal density of a tetrahedral system in an external field and the ice XI phase in the TIP4P water model. Our flows can be combined with flows operating on the internal degrees of freedom of molecules and constitute an important step towards the modeling of distributions of many interacting molecules.

Faithful evaluation of language model capabilities is crucial for deriving actionable insights that can inform model development. However, rigorous causal evaluations in this domain face significant methodological challenges, including complex confounding effects and prohibitive computational costs associated with extensive retraining. To tackle these challenges, we propose a causal representation learning framework wherein observed benchmark performance is modeled as a linear transformation of a few latent capability factors. Crucially, these latent factors are identified as causally interrelated after appropriately controlling for the base model as a common confounder. Applying this approach to a comprehensive dataset encompassing over 1500 models evaluated across six benchmarks from the Open LLM Leaderboard, we identify a concise three-node linear causal structure that reliably explains the observed performance variations. Further interpretation of this causal structure provides substantial scientific insights beyond simple numerical rankings: specifically, we reveal a clear causal direction starting from general problem-solving capabilities, advancing through instruction-following proficiency, and culminating in mathematical reasoning ability. Our results underscore the essential role of carefully controlling base model variations during evaluation, a step critical to accurately uncovering the underlying causal relationships among latent model capabilities.

Inferring causal structure poses a combinatorial search problem that typically involves evaluating structures with a score or independence test. The resulting search is costly, and designing suitable scores or tests that capture prior knowledge is difficult. In this work, we propose to amortize causal structure learning. Rather than searching over structures, we train a variational inference model to directly predict the causal structure from observational or interventional data. This allows our inference model to acquire domain-specific inductive biases for causal discovery solely from data generated by a simulator, bypassing both the hand-engineering of suitable score functions and the search over graphs. The architecture of our inference model emulates permutation invariances that are crucial for statistical efficiency in structure learning, which facilitates generalization to significantly larger problem instances than seen during training. On synthetic data and semisynthetic gene expression data, our models exhibit robust generalization capabilities when subject to substantial distribution shifts and significantly outperform existing algorithms, especially in the challenging genomics domain. Our code and models are publicly available at: https://github.com/larslorch/avici.

We address causal reasoning in multivariate time series data generated by stochastic processes. Existing approaches are largely restricted to static settings, ignoring the continuity and emission of variations across time. In contrast, we propose a learning paradigm that directly establishes causation between events in the course of time. We present two key lemmas to compute causal contributions and frame them as reinforcement learning problems. Our approach offers formal and computational tools for uncovering and quantifying causal relationships in diffusion processes, subsuming various important settings such as discrete-time Markov decision processes. Finally, in fairly intricate experiments and through sheer learning, our framework reveals and quantifies causal links, which otherwise seem inexplicable.

Foundation models have brought changes to the landscape of machine learning, demonstrating sparks of human-level intelligence across a diverse array of tasks. However, a gap persists in complex tasks such as causal inference, primarily due to challenges associated with intricate reasoning steps and high numerical precision requirements. In this work, we take a first step towards building causally-aware foundation models for treatment effect estimations. We propose a novel, theoretically justified method called Causal Inference with Attention (CInA), which utilizes multiple unlabeled datasets to perform self-supervised causal learning, and subsequently enables zero-shot causal inference on unseen tasks with new data. This is based on our theoretical results that demonstrate the primal-dual connection between optimal covariate balancing and self-attention, facilitating zero-shot causal inference through the final layer of a trained transformer-type architecture. We demonstrate empirically that CInA effectively generalizes to out-of-distribution datasets and various real-world datasets, matching or even surpassing traditional per-dataset methodologies. These results provide compelling evidence that our method has the potential to serve as a stepping stone for the development of causal foundation models.