1. The Original Question (informal)
I have a function (vector field) $\mathbf{F}:\mathbb{R}^N\rightarrow\mathbb{R}^N$ with expression $\mathbf{F}(\mathbf{x})=\sigma(\mathbf{Wx+b})$, where $\mathbf{W}\in\mathbb{R}^{N\times N}$ is a matrix, $\mathbf{b}\in\mathbb{R}^{N}$ is a vector, $\sigma(\cdot)$ is the so-called "Sigmoid" function (an element-wise non-linear mapping) with $\sigma(x)=1/(1+e^{-x})$.
Does there exists a scalar-valued function $f:\mathbb{R}^N\rightarrow\mathbb{R}$ satisfying $\nabla f\equiv \mathbf{F}$?
2. Some of My Efforts
I may know from Wikipedia and MathInsight that if $\mathbf{F}$ is path-independent, then $f$ exists.
In addition, in my case, I may not want to get the explicit form of $f$. I may just want to explore the existence of $f$. And I do not care if it is elementary or not.
However, after a long struggle, I still do not know that how to prove or disprove the original question. Specifically, it is difficult for me to determine if $\mathbf{F}$ is path-independent or not.
Is there a way to figure it out?
I am finding, searching, and thinking ...
You should work with one component at a time; your problem is then : $$ \frac{\partial f}{\partial x_i} = F_i = \sigma\left(\sum_{j=1}^NW_{ij}x_j+b_i\right) = \frac{1}{1-\exp\left(-\sum_{j=1}^NW_{ij}x_j+b_i\right)} $$ hence $$ f(x_i) = \int\frac{\exp\left(\sum_{j=1}^NW_{ij}x_j+b_i\right)}{\exp\left(\sum_{j=1}^NW_{ij}x_j+b_i\right)-1}\mathrm{d}x_i = \frac{1}{W_{ii}}\ln\left(\exp\left(\sum_{j=1}^NW_{ij}x_j+b_i\right)-1\right) + C(\{x_k\}_{k\neq i}) $$ As the functions $C$ cannot mix all the components of $\mathbf{x}$ at the same time $-$ what the expression $\sum_{j=1}^NW_{ij}x_j+b_i$ does for each $x_i$ $-$, one concludes that there is no solution.