IntroductionHigh-fidelity flow fields contain richer flow information and finer-scale flow structures, which play a crucial role in understanding the flow behaviors in various physical and natural phenomena. In computational fluid dynamics (CFD), increasing the grid-resolution and employing higherorder numerical schemes can typically generate high-fidelity flow fields, but the simulations require several days to complete and incur substantial computational costs. Recently, inspired by their success in computer vision, deep-learningbased super-resolution (SR) methods have increasingly been used to reconstruct high-fidelity flow fields from low-fidelity counterparts without repeatedly solving complex partial differential equations (Fukami et al., 2023).
Numerical Accuracy Grid Resolution
PEINR Low-Fidelity
High-FidelityFigure 1. The red circles explain the error in the assumption of grid independence, showing that as the grid resolution (in the row direction) and numerical accuracy (in the column direction) increase, the attribute values at the same location differ and the high-frequency part becomes more pronounced in the corresponding frequency domain representations, which means that the structure of the flow field is more complex and refined.Existing methods of high-fidelity flow field reconstruction based on convolutional neural networks (CNNs) (Fukami et al., 2021;Yousif et al., 2021;Xu et al., 2023;Hu et al., 2024;Shen et al., 2024b;a) are clearly confined to equallyspaced Cartesian meshes because the nodes of the computational grid are reinterpreted as pixels. Nevertheless, irregular meshes are widely used in the industry because they can adequately delineate complex geometries and easily deal with localized regions that require different resolution, e.g. cells are small near walls to capture the boundary layer where flow transitions are sharp whereas they are large in the freestream where gradients are smooth. A grid can also be regarded as a graph and graph neural networks (GNNs) are reasonable candidates. However, in order to propagate information over long distances between nodes, many graph convolutional layers need to be stacked. Moreover, although pooling is a cheap operation in CNNs, it is rather challenging in GNNs (Grattarola et al., 2022;Kashefi et al., 2021).The implicit neural representation (INR) frameworks (Han & Wang, 2023;Pan et al., 2023;Tang & Wang, 2024;Jiao et al., 2024) typically learn to represent a flow field as a continuous function which can map the spatiotemporal coordinates to their corresponding values, thus are highly adaptable for different mesh types and having emerged as a new mesh-agnostic paradigm for flow reconstruction. However, these INR-based methods still face several limitations in high-fidelity flow field reconstruction: (1) In the absence of standard benchmarks datasets, existing INR-based methods assume grid independence, where attribute values at the same location in the flow field remain unchanged across different mesh resolutions. However, in the case of real simulation data, this assumption does not hold. As shown in Figure 1, the red circles highlight the same locations in flow fields with varying mesh resolutions and numerical accuracies, where the corresponding density values clearly differ. (2) Due to the I/O bottleneck (Fajardo et al., 2018), only selective time steps can be saved. There is a significant disparity between temporal and spatial complexity. Existing methods overlook this gap by simply coupling space and time, thus failing to fully simulate the complex spatiotemporal dynamics. (3) The spectral bias issue of INR method (Xu, 2018;Rahaman et al., 2019) concentrate more on the low-frequency information, which can have deficiencies in capturing fine-scale structures of flow fields. As shown in Figure 1, the corresponding frequency domain representations illustrate that high-fidelity flow fields capturing small-scale flow features more effectively can exhibit a richer presence of high-frequency information.In this paper, we first introduce HFR-Bench, a truly largescale dataset containing 24,000 unsteady flow fields of three canonical two-dimensional (2D) flow problems, 1,600 unsteady flow fields of a three-dimensional (3D) flow problem with uniform Cartesian mesh, and 8,000 2D unsteady flow fields with non-uniform mesh, amounting to a total of 5.4 TB of data. For each problem, we discretize the domain into four different grid-resolution configurations, and perform a simulation at each grid-resolution using four different numerical-precision settings. Low-and high-fidelity simulations in each pair start from identical initial conditions and share the same physical parameters. The maximum grid-resolution gap and the highest numerical-precision can reach up to 64 times and 7th-order, respectively.Compared to high-fidelity simulation, the coarse discretization of the low-fidelity simulation introduces inaccuracies and truncation errors. As shown in Figure 2(a), we calculate the correction errors between low and high-fidelity flow fields. Effectively representing the error field is the target we strive to reach with the application of the INR network. We propose a physics-enhanced INR (PEINR) which can simultaneously enhancing the numerical-precision and grid-resolution of flow fields. PEINR consists of physical encoding and transformer-based spatiotemporal fuser (TransSTF).In physical encoding, as shown in Figure 2(b), spatial coordinates ⃗ C are expanded from neighbors, taking into account the nature of stencil discretization. Temporal information is expanded by Gaussian kernel encoding to addressing the issue of disproportionate spatiotemporal dimensions. As shown in Figure 2(c), TransSTF, a temporal-aware encoder based on the multi-head attention mechanism, aims to fuse both spatial and temporal information and capture long-range temporal dependencies. We also leverage spectral block (Patro et al., 2023), to capture the high frequency components of high fidelity flow fields. We perform qualitative and quantitative analyses, to experimentally demonstrate that our approach is well-suited for high-fidelity flow field reconstruction tasks.The key contributions of our work are summarized as follows: (1) We release HFR-Bench, a truly large-scale CFD dataset including both uniform Cartesian and non-uniform meshes with 33,600 unsteady 2D and 3D vector fields, amounting to a total of 5.4 TB of data. For each simulation, data are provided at both low and high resolution, making the dataset suitable for the specific task of reconstructing high-fidelity flow fields. (2) We propose a novel physicsenhanced INR model for concurrently handling numericalprecision and grid-resolution enhancement for both uniform and non-uniform meshes. The physical encoding can alleviate the disparity between temporal and spatial complexities, capture the nonlinear characteristics of spatiotemporal dynamics and the stencil discretization of spatial dimensions.(3) We utilize the spectral block in TransSTF to alleviate the spectral bias on flow field learning. The result of ablation study can further illustrate that our method can alleviate the negative impact.
Related WorkThis section discusses the related works of implicit neural representation, and super-resolution for flow field.  tion. This function maps a given spatiotemporal position, which consists of both spatial and temporal coordinates, to its corresponding value.
Implicit neural representation.Recently, incorporating Fourier features (Mildenhall et al., 2021), periodic activations (Sitzmann et al., 2020), or multiresolution Hash tables (Müller et al., 2022) has significantly improved the performance of INR for scene reconstruction (Jiang et al., 2020;Chabra et al., 2020) and shape modeling (Genova et al., 2020;Atzmon & Lipman, 2020). Recent work by de Vito et al. (de Vito et al., 2024) in implicit neural representation for accurate CFD flow field prediction demonstrates the potential of INR architectures for steady-state CFD simulations. However, their method primarily focuses on single-resolution flow fields with stationary boundary conditions and does not address the critical challenges of reconstructing unsteady, multi-resolution flow fields with spatiotemporal coupling -a key limitation our PEINR framework resolves through physical encoding and TransSTF.PINNs (Karniadakis et al., 2021;Dong & Polak, 2024;Hosseini & Shiri, 2024) are an extended application of implicit neural representations in scientific computing. Nevertheless, our CFD data are solved based on discrete meshes, which inherently introduce numerical dissipation and truncation errors, deviating from the Navier-Stokes equations governing fluid flow. Forcing PINNs to simultaneously fit this data and satisfy the equation residuals may lead to network convergence issues or generate non-physical solutions (Farea et al., 2024).Super-resolution for flow field. In the past several years, various deep learning-based methods have been applied to tackle single image super-resolution (SISR) tasks (Lu et al., 2022;Zhang et al., 2022;Chen et al., 2023) . Inspired by their success in computer vision, SISR methods have increasingly been used to reconstruct high-fidelity flow fields by simply replacing the red, green, and blue components with physical variables (Fukami et al., 2023). Among these works, CNN-based super-resolution models (Zhenglei et al., 2017;Fukami et al., 2019;2021;Liu et al., 2020;Obiols-Sales et al., 2021;Gao et al., 2021) have been actively studied for a range of flows. To further improve the model performance, many works complicate their model by incorporating GAN (You et al., 2018;Zhiwen et al., 2019;Yousif et al., 2021;2022;Yu et al., 2022;Han & Wang, 2019;2020;Wurster et al., 2023;Hyojin et al., 2021), transformer (Wang et al., 2022;Xu et al., 2023;Hu et al., 2024;Shen et al., 2024b;a), for capturing fine-scale flow features. However, these methods are limited to flow field data on Cartesian grids.Neural operators (Kovachki et al., 2023) leverage function space mappings and efficient spectral-domain computations to overcome the fixed-grid limitations of traditional methods. However, when high-frequency energy decays rapidly, neural operators may struggle to retain fine details compared to INR. Additionally, INR supports on-demand generation of localized regions, such as boundary layers or vortex structures, without requiring full-field computation, making it more adaptable and efficient.Existing INR metods for flow field (Han & Wang, 2023;Pan et al., 2023;Tang & Wang, 2024;Jiao et al., 2024) only focuse on grid-resolution enhancement, and PEINR is the first attempt to simultaneously enhance the grid-resolution and numerical-precision by leveraging transformer and physicalenhanced INR model.
MethodWe propose a framework with an INR-based model for highfidelity flow field reconstruction. The framework consists of 4 components: inverse problem input encoding, physi- cal encoding, TransSTF, and inverse problem solutions, as shown in Figure 2. Our method allows approximating the accurate variables (density ρ, the components u, v and w of vector variable velocity V ) of high-fidelity flow field by learning its time information and the dependency of spatial information. We detail all framework components and learning schemes in the following subsections.
Inverse Problem Input Encoding and SolutionsThe inverse problem refers to reconstructing high-fidelity physical fields from sparse data (Chen et al., 2024). The CFD simulation flow field is closely related to the numerical accuracy and grid resolution, and these two factors influence each other. Finer grids can better capture flow details and reduce discretization errors, which can lead to discrepancies in flow field property values at the same location. To enhance the applicability of INR, we chose to use INR to learn the error field δF between high-fidelity and low-fidelity flow fields.The process is as follows: to establish the connection between the low-fidelity flow field F L ti at timestep t i and its high-fidelity counterpart F H ti , we need to address the inaccuracies and truncation errors introduced by the coarse discretization in low-fidelity simulations. First, we upscale the low-fidelity flow field to match the spatial resolution of the high-fidelity field using BI (bicubic interpolation) method. Although traditional upscaling methods or transferring existing mature super-resolution models could be chosen, we opted for the traditional BI method due to the diversity of flow field grid types. BI method estimates the value at each point by considering the surrounding 4 × 4 grid of known values and applying cubic interpolation along both axes, resulting in a smoother and more accurate upscaled flow field compared to simpler methods like bilinear interpolation. The resulting upscaled flow field, denoted as F B ti , provides a better approximation of the high-fidelity field but still contains residual errors.As shown in Figure 2 (a), to quantify these discrepancies, we compute the error field δF ti = F H ti -F B ti , which captures the missing fine-grained details. We obtain δF ti through the BI method, which allows us to efficiently handle the discrepancies between the low-fidelity and high-fidelity flow fields, thus laying the foundation for subsequent error learning and precise reconstruction. The goal of the PEINR network is to effectively learn and represent the error field δF ti , enabling the accurate reconstruction of high-fidelity flow fields from low-fidelity counterparts. As shown in Figure 2 (d), to obtain the inverse problems solutions, we use the error field generated by INR model along with the low-fidelity flow field to reconstruct the high-fidelity flow field.
Physical EncodingTo accurately capture the spatial and temporal dependencies in flow field representations, we introduce specialized encoding techniques for both spatial and temporal information.
SPATIAL DISCRETIZATIONSpatial discretization in computations typically involves stencil-based methods, whether using finite volume, finite difference, or finite element methods. These approaches rely on information from grid cells and their neighboring cells to construct the discretized equations. In most conventional INR methods, the network learns to represent a flow field as a continuous function with the input of dependent coordinates ⃗ C and temporal information t, ignoring the importance of spatial dependency.To address this, after randomly sampling a sufficient number of points, we expand the input spatial coordinates to include not only the original coordinates but also those of nearby neighboring points:Ψ( ⃗ C) = ⃗ C, ⃗ C neighbors (1)This localized encoding aligns with stencil-based computation paradigm of CFD, where numerical solutions inherently depend on local neighborhood interactions (e.g., finite volume discretization). By embedding this locality into our spatial localization encoding, PEINR explicitly leverages the template computation logic to resolve fine-scale flow structures. For instance, in a 2D case, we consider the nearest 4 points (up, down, left, right), transforming the input from a single coordinate tensor of shape (batchsize, 2) to an augmented tensor of shape (batchsize, 5, 2). This design explicitly encodes the local spatial correlations required for solving discretized Navier-Stokes equations, where the solution of each grid point depends on physical quantities of its neighbors.Spatial discretization allows the model to capture spatial derivatives more effectively, reflecting the underlying properties of partial differential equations (PDEs) that govern flow field computations. By considering spatial derivatives, the model can better account for the local gradients and variations in the flow field, enhancing its ability to represent fine-scale structures and accurately approximate the dynamics dictated by the governing PDEs, as shown in Figure 2 (b).
NONLINEAR TEMPORAL ENCODINGTo effectively model the nonlinear temporal dynamics in flow fields, we employ a two-step approach that combines the Gaussian Radial Basis Function (RBF) kernel and Kernel Principal Component Analysis (Kernel PCA). This method transforms the one-dimensional temporal input into a highdimensional, nonlinear feature space and then reduces the dimensionality while preserving essential temporal features.Step 1: Temporal Encoding Using RBF Kernel. The RBF kernel maps the one-dimensional temporal input t into a high-dimensional feature space where nonlinear temporal patterns can be better represented. The RBF kernel function is defined as:K (t, t ′ ) = exp - |t -t ′ | 2 2σ 2 , (2)where t and t ′ are two time instances. σ is a hyperparameter controlling the kernel's width, determining how quickly the similarity between t and t ′ decays with their distance.Given a set of temporal data points {t 1 , t 2 , . . . , t N }, we construct the RBF kernel matrix K of dimensions N × N , encoding pairwise temporal relationships. This transformation ensures translation invariance by making the kernel solely dependent on relative differences:K (t i , t j ) = K (t i -t j ) .(3)This property allows the INR model to maintain consistency across different time steps, improving long-term reconstruction accuracy.To analyze the impact of the RBF kernel on model stability and generalization, we apply Neural Tangent Kernel (NTK) theory, which describes the training evolution of infinitely wide neural networks and is essential for convergence analysis. Traditional NTKs derived from MLPs lack translation invariance, making them unsuitable for temporal modeling. By incorporating the RBF kernel, we convert the NTK into a stationary kernel that depends on relative time differences, improving stability, especially in solving time-dependent PDEs. As shown in Figure 3, the RBF-enhanced NTK exhibits structured patterns that better capture temporal dependencies, promoting smoothness and consistency over time, which enhances generalization and robustness in dynamic environments. The spectral properties of the NTK reveal that adjusting the kernel bandwidth σ allows for flexible adaptation to various time scales, balancing local and global temporal interactions.Step 2: Extracting Temporal Components with Kernel PCA. Kernel PCA is applied to extract significant temporal features from the kernel matrix by first centering the matrix to ensure zero-mean data. This is achieved using the equation:K = K -1 N K -K1 N + 1 N K1 N (4)where 1 N is an N × N matrix with all elements equal to 1/N .Next, eigenvalue decomposition of the centered kernel matrix K yields eigenvalues and eigenvectors, from which the top m eigenvectors are selected to form a low-dimensional representation. Temporal inputs are projected into this subspace to obtain their encoded representations:Φ (t i ) = V ⊤ m K (t i )(5)where Φ(t) is the nonlinear feature representation of t in the reduced space, V m is the matrix of the top m eigenvectors, and K (t i ) is the similarity vector of time point t i with all other points (i.e., the i-th column of the kernel matrix). where N is the neural network parameterized by θ. This approach enables the model to leverage the enriched temporal features for improved reconstruction accuracy.
TransSTFAs shown in Figure 2 (c), TransSTF combines spatial and temporal information to generate complex embeddings using higher-order features. It consists of multi-head attention (MHA) layers, ResuMLPs, and spectral blocks. The MHA layer with query (Q), key (K), and value (V ) inputs aims to refine feature maps, where Q, K, and V come from a mixture of spatial-temporal data and ResuMLP-extracted features. The output matrix of MHA (M) is shown as follows:M(Q, K, V ) = Concat ( att 1 , . . . , att h ) W Owhere att i = softmaxQ i W Q i K T i W K i √ d k V i W V i (7)ResuMLP. The ResuMLP (Han & Wang, 2022) improves gradient propagation by adding depth and complexity to the network. It includes residual blocks and a SIREN (Sitzmann et al., 2020) activation function (sin(ωx)), with ω set to 30, as recommended (Sitzmann et al., 2020). The output is scaled to the [-1, 1] range, suitable for sinusoidal activations.Spectral Block. The spectral block (Patro et al., 2023) captures different frequency components and reduces spectral bias. It includes a spectral gating network with FFT for converting to spectral space, weighted gating to adjust frequency component weights, and an inverse FFT (IFFT) to return the signal to physical space, enhancing the capture of high-frequency features like edges.
ExperimentDataset. We generate a truly large-scale dataset HFR-Bench, containing 2D and 3D unsteady flow fields including both uniform Cartesian and non-uniform meshes in different high-order weighted essentially non-oscillatory (WENO) schemes, amounting to a total of 5.4 TB of data. In uniform Cartesian meshes, we choose the Rayleigh-Taylor instability (RT), Riemann (RM), and Forward Facing Step (FFS) problems for the 2D flow fields, and the shock-longitudinal vortex interaction (SV) case in 3D. In non-uniform structured meshes, we simulate the problem of a flow past a cylinder (Cylinder). The discrepancy in grid-resolution and numerical-precision between F L and F H data can be described by the upscaling factor α and the improvement factorβ: α = ST HF /ST LF , β = SP HF /SP LF ,(8)where ST denotes the number of grid points and SP denotes the number of points in the stencil.Baselines. We compare our method with three classic methods: (1) Bicubic interpolation (BI) is a common traditional image processing algorithm based on interpolation. ( 2) NIF (23'JMLR) (Pan et al., 2023)   1.Our method and all the baseline methods are trained with the MSE loss in 2000 epochs and every method including ours can converge within 1000 epochs. Learning ratio is set to 1e -5 and decrease after 20 epochs if there is no loss degradation and we adopt the AdamW optimizer for optimization. In spatial discretization, for 2D cases, we consider the nearest 4 points, and for 3D cases, the nearest 9 points. In temporal nonlinear encoding, we set the σ as 10 with time steps normalized to [0,1]. During design, we set the number of residual layers of a ResuMLP to 10 and the max number of neutrons in a ResuMLP is 64. Experiments are conducted within a range of ±10% around the optimal hyperparameters, using the inference results as the standard to confirm that the hyperparameters are indeed optimal. For the other models compared in the table, we utilized their original code and conducted experiments by only varying the input length.Experiments are conducted on a single channel for constructing ρ in flow fields, and two (three) channels for constructing velocity fields in 2D (3D). For uniform Cartesian meshes, flow fields from timesteps 460 to 480 (out of 500) are used for training, with the final 20 steps as test samples. Results for uniform meshes (FFS, RM, RT, SV) are shown for step 500. For non-uniform meshes, training samples are taken from steps 400 to 500 at intervals of 5, with the remaining non-multiples of 5 used for testing. Results for non-uniform meshes are presented for step 494.
Quantitative ComparisonWe take MSE (Mean Squared Error) as the loss function and report PSNR (Peak Signal-to-Noise Ratio), SSIM (Structural Similarity Index), CORR (Correlation Coefficient) and DD (Dissipation difference) results as the evaluation metrics.The dissipation operator is used to assess the model performance in terms of capturing the discontinuities and rotating features with lower values indicating better performance with lower values indicating better performance.The quantitative results are shown in Table 2, which quantitatively compares PEINR with different grid-resolution and numerical-precision against the results generated by baselines using data-level metrics (higher values are better except for DD). In Table 2, all experiments are onechannel, conducted to reconstruct the variable ρ of flow fields. PEINR can generally outperform the baselines, except for the slightly inferior PSNR with respect to Coord-Net (Han & Wang, 2023) in the RM dataset. The observed PSNR difference stems from PEINR's design prioritizing physical accuracy over numerical optimization for discontinuous flows. Riemann problems involve strong discontinuities (shocks, contact surfaces) where PEINR's localized spatial encoding intentionally preserves sharp gradients, making it more sensitive to errors in these regions. While this approach leads to slightly lower PSNR (a mean-squarederror metric favoring smoothness), it avoids the physical distortions visible in CoordNet's results (Figure 4(c)(d)) where large errors near discontinuities appear.As shown in 5, we also present the four evaluation met-rics for Table 2 (a) across all extrapolated timesteps. The results clearly demonstrate that PEINR exhibits an absolute advantage in generalization performance.Ablation Study To evaluate the effectiveness of importance sampling and spectral block of PEINR, we compare with two baseline methods PEINR 1 and PEINR 2 . PEINR 1 removes both spectral block and the physical encoding, while PEINR 2 only removes the physical encoding. As shown in Table 2, PEINR outperforms PEINR 2 in the metric DD in all cases, which indicates that our physical encoding method can preserve more physical peculiarity. The superiority of PEINR 2 compared with PEINR 1 highlights that the added spectral block can also help the model to perform better.
Qualitative ComparisonThe qualitative evaluation provides several contour plots and streamline rendering results. To facilitate analysis, we normalized all results to the interval [-1, 1] for ease of observation. The areas of interest to domain experts are marked with red squares in the first column, and we zoom in on these areas in the following columns. BI can not capture complex fluids  4 (g) (h). NIF tends to blur the turbulence structures, resulting in poor reconstructions of the smallscale ground-truth features, regardless of α or β. PEINR can efficiently handle the simultaneous enhancement of the numerical-precision and grid-resolution.In Figure 6, we compare streamline rendering results of the synthesized vector fields generated by baselines. Figure 6 (a) and (b) display 48 streamlines generated from different seeds in the SV data set. While NIF and CoordNet can capture more details compared with BI but the tail of vortex obtainted are diverging and they fail to produce more complex flow details. In contrast, our approach not only maintains a high level of detail but also ensures that the complex behaviors of the flow, including vortex dynamics and the underlying fluid structure, are consistently preserved throughout the simulation. Figure 7 displays the comparison results of Cylinder dataset in non-uniform structured meshes, and it can illustrate that our method achieves better results on both uniform Cartesian grid data and non-uniform structured data compared to other baselines.
ConclusionWe introduce a large-scale comprehensive flow simulation dataset covering 5 canonical flow problems of 2D and 3D in both uniform Cartesian and non-uniform meshes. For each problem, results were obtained using four grid-resolution and four numerical-precision settings. In total, the dataset contains 33,600 vector fields, resulting in approximately 5.4 TB of data. Using this dataset, we creatively propose the coordinate-based INR method with the attention mechanism and physical encoding, which achieves satisfactory results  in the field of high-fidelity flow fields reconstruction, so that domain experts can observe high-fidelity flow fields with less storage cost. However, the PEINR model is, thus far, tailored and appraised for particular flow scenarios. In the future, we would like to explore meta-learning approaches to enhance the performance on out-of-distribution samples by considering the relationship among different flow fields.
Impact StatementThis work introduces HFR-Bench, the first large-scale benchmark dataset for high-fidelity flow field reconstruction. By integrating physics-based encoding with transformer architectures, our proposed PEINR framework significantly improves the accuracy and robustness of implicit neural representations. These contributions pave the way for more generalizable and physically consistent AI solutions in fluid simulation tasks.  PSNR. This represents the ratio of the maximum possible power of the signal and the destructive noise power that affects its representation accuracy. PSNR is computed using the mean squared error by the following formula: 
AcknowledgmentFigure 2 .2Figure 2. Overview of our method PEINR. (a) We first calculate the difference field between the high-fidelity flow field and low-fidelity counterpart. (b) In physical encoding, we leverage Gaussian coordinate encoding and localized encoding to handle temporal information and spatial coordinates. (c) The TransSTF block fuses temporal and spatial information and consists of ResuMLP, multi-head attention and spectral block. (d) The generated high-fidelity flow fileds are obtained by combining diffenrence fields and low-fidelity counterparts.
Figure 3 .3Figure 3. (a) The NTK without mapping exhibits irregular patterns, indicating dependency on absolute time positions and leading to inconsistent predictions. (b) Applying the Gaussian kernel mapping results in a structured diagonal pattern, ensuring translation invariance and improved generalization. (c) The NTK spatial decay plot shows how kernel values diminish across input space, with smaller σ emphasizing local interactions and larger σ capturing broader dependencies. (d) The NTK Fourier spectrum demonstrates that larger σ values allow high-frequency variations, while smaller σ favor smooth, low-frequency behaviors.
Figure 4 .4Figure 4. Comparison results of the variable ρ flow field of uniform Cartesian meshes. given by NIF, CoordNet and our method in the 2D (upscaling α = 16) and 3D datasets (upscaling α = 64). Results of FFS datasets in (a) WENO3 (β = 1), (b) from WENO3 to WENO7 with β = 2.6. Results of RM datasets in (c) WENO3 (β = 1), (d) from WENO3 to WENO5 with β = 1.4. Results of RT datasets in (e) WENO3 (β = 1), (f) from WENO3 to WENO5 with β = 1.4. (g) (h) Results of SV datasets from WENO3 to WENO5 with β = 1.4 from different viewpoints.
Figure 4 displays synthesized flow fields of density ρ of FFS datasets in (a) (b), RM datasets in (c) (d), RT datasets in (e) (f) and 3D SV datasets in (g) (h) with different upscaling factors α and improvement factors β.
Figure 5 .5Figure 5. Comparison of model generalization performance on unseen timesteps beyond the training range. The PEINR method consistently achieves higher PSNR, SSIM and CORR values while maintaining lower DD compared to other methods, demonstrating its superior generalization capability in long-term flow field predictions.
Figure 6 .6Figure 6. Comparison of streamline rendering results of threechannel experiments of SV dataset. Results given by BI, NIF and CoordNet are the inferred results (i.e., the networks do not see these vector fields during training). (a) and (b) display 48 streamlines from different seeds.
This work was supported by the the National Key Research and Development Program of China (2023YFA1011704), and the National Key Research and Development Program of China (2021YFB0300101).
Figure 7 .Figure 9 .79Figure 7. Comparison results of the variable u flow field with 25 uniformly distributed contours (upscaling α = 4) given by BI, NIF, CoordNet and our method in the Cylinder dataset of non-uniform structured meshes. Results in (a) WENO3 with β = 1, and (b) from WENO3 to WENO5 (β = 1.4).
PFigure 1010Figure 10. x-direction velocity component u of dataset cylinder contours with 25 equally spaced isosurfaces in diffenrent grids and computation accuracy.
Table 1 .1The details of each data set. GR, NP, BS and MM denote grid resolution, numerical precision, batch size and memory, respectively.datasetVariableGRInputNPGROutputNPBSTrain Time(s) MM(GB)Inference Time(s)RTρ,u,v120×480WENO3480×1920WENO3 WENO580001022.6816.71RMρ,u,v200×200WENO3800×800WENO3 WENO580001110.5313.59FFSρ,u,v36×12 180×48WENO3144×48 720×192WENO3 WENO780001117.2915.4Cylinderu,v51×381WENO3101×761WENO3 WENO58000195.119.59SVρ,u,v,w 96×32 × 32 WENO3 384×128 × 128 WENO5 360002368.4575.31
Table 2 .2PSNR, SSIM×10 2 , CORR×10 2 and DD×10 5 values with different upscaling factors α and improvement factors β.The best