The Poisson equation for \( g(x) \) in one dimension is
$$ \begin{equation} \tag{13} -g''(x) = f(x) \end{equation} $$where \( f(x) \) is a given function for \( x \in (0,1) \).
The conditions that \( g(x) \) is chosen to fulfill, are
$$ \begin{align*} g(0) &= 0 \\ g(1) &= 0 \end{align*} $$This equation can be solved numerically using programs where e.g Autograd and TensorFlow are used. The results from the networks can then be compared to the analytical solution. In addition, it could be interesting to see how a typical method for numerically solving second order ODEs compares to the neural networks.