Let's consider the classic heat equation and ignore the boundary and initial conditions for simplicity,
$$u_t(t,x) = u_{xx}(t,x).$$
I have a neural network that attempts to solve this by inputting a random uniform distribution (which we call $p$) for sampling $(t,x)$ from the domain and minimizes the loss ($L^2$ norm) of $\hat u_t - \hat u_{xx}$, where $\hat u$ is the predicted output from the neural network.
Now I want to find the optimal sampling distribution, $p^{*}$, for which we get the most accurate and fastest solution. I thought of using another network that takes the $\hat u$ as input and outputs a $\hat p$, which can again be passed on to the original network to obtain another $\hat{\hat u}$, etc, and this sequence might converge? Is this similar to a GAN? But for a GAN, you need real training data and here we don't know anything about the real solution. Maybe it's a reinforcement learning problem?
