TITLE: How to solve the equations of linear combination of sigmoid functions?
QUESTION [5 upvotes]: Let $\sigma(x)=\frac{1}{1+e^{-x}}$ be the sigmoid function.
How to solve such kind of equations?
\begin{align*}
\sigma(x+y)+\sigma(x-y)=a\\
\sigma(2x+y)+3\sigma(3x-y)=b\\
\end{align*}
I guess this kind of equations are related to neural networks. I don't know how to solve it.
Thank you very much.

REPLY [2 votes]: If $z$ is small, you can write $$\sigma(z)=\frac{1}{1+e^{-z}}=\frac{1}{2}+\frac{z}{4}+O\left(z^2\right)$$ and then the solutions of the equations write $$x=2 (a-1)$$ $$y=11 a-2 b-7$$ If this is not the case, Robert Israel gave the rigorous answer.