On quadrature rules for solving Partial Differential Equations using Neural Networks

Neural Networks have been widely used to solve Partial Differential Equations. These methods require approximate definite integrals using quadrature rules. Here, we illustrate via 1D numerical examples the problems that may arise in these applications and propose different alternatives overcome them, namely: Monte Carlo methods, adaptive integration, polynomial approximations of Network output, inclusion regularization terms loss. We also discuss advantages limitations each proposed alternative. advocate use for high dimensions (above 3 or 4), integration low (3 below). The is a mathematically elegant alternative valid any spacial dimension, however, it requires certain regularity assumptions on solution complex mathematical analysis when dealing with sophisticated Networks.