Suppose you have a Turing-type reaction-diffusion system $$ \begin{cases} \partial_t \phi = & f(\phi, \psi) + D_\phi \nabla^2\phi \\ \partial_t \psi = & g(\phi, \psi) + D_\psi \nabla^2\psi \end{cases} $$ where
- $\phi(r, t)$ is a real number representing the concentration of a chemical at point $r \in \mathbb{R}^2$ and time $t \in \mathbb{R}$;
- $\psi(r, t)$ is a real number representing the concentration of another chemical at point $r \in \mathbb{R}^2$ and time $t \in \mathbb{R}$;
- $D_{\phi}$ and $D_{\psi}$ denote the diffusion coefficients of the chemicals.
Suppose that the system has a solution $(\phi^*, \psi^*)$ such that:
- there exists a time $t_0$ such that $\phi^*(r,t) = \phi^*(r,t_0)$ and $\psi^*(r,t) = \psi^*(r,t_0)$ for every $t \geq t_0$ and every $r \in \mathbb{R}^2$;
- there exists a closed interval $[a,b]$ that contains the range of both $\phi^*$ and $\psi^*$.
Now we describe how to associate a 24 bit per pixel RGB image of dimension $N \times M$ to each stable solution $(\phi^*, \psi^*)$ restricted to a domain $[0,N \cdot h] \times [0,M \cdot h] \subset \mathbb{R}^2$, where $h$ is a positive real number.
- the image is laid over a grid of $N \times M$ pixels;
- for every pair of discrete coordinates $(i,j) \in \{1,\ldots, N\} \times \{1,\ldots, M\}$ we form the pair $(x_{ij}, y_{ij})$ where $x_{ij}$ is the average of $\phi^*(r,t)$ and $y_{ij}$ is the average value of $\psi^*(r,t)$ for $r \in [i\cdot h,(i+1) \cdot h] \times [j \cdot h,(j+1) \cdot h]$ and $t \geq t_0$;
- the interval $[a,b]$ is divided into $256$ equal parts $C_k = [\frac{(b-a)k}{256}, \frac{(b-a)(k+1)}{256}]$ for each $k = 0, \ldots, 254$. The cell $(i,j)$ is given the RGB color having 255 in the blue component and the $u$-th degree of RED and the $v$-th degree of GREEN iff $(x_{ij}, y_{ij}) \in C_u \times C_v$.
Let $(\phi_1, \psi_1), \ldots, (\phi_k, \psi_k), \ldots$ be an infinite sequence of approximations to the solutions $(\phi^*, \psi^*)$ given by a numerical method, like Euler's.
For a fixed (large) natural number $K$ we can create $K$ RGB images of $N \times M$ pixels, each one corresponding to a pair $(\phi_k, \psi_k)$ (where $k \in \{1,\ldots, K\}$).
Now we can use those $K$ images as a training set for a neural network.
Is it possible to extract a strong correspondence between the computation steps of an algorithm that numerically solves the system of PDEs and the back-propagation algorithm that trains the neural network whose precision increases as $K$ increases?
