Semi-autonomous neural networks differential equation solver

The finite element method frequently needs complex grids to solve partial differential equations. This becomes more serious in highly dimensional problems and complicated geometries. In this article we present an improved gridless solver, which trains a neural network to fit the differential equation solution. The advantage of a gridless method is its easier scalability to problems with a high number of dimensions. A smart stopping criterion, based on statistical learning theory concepts, makes the method more autonomous than preceding algorithms. The proposed method uses a simple rule to include the boundary conditions in the error measure of the network. For validation, we show the results of solving some simple first and second order equations and one from a classical application problem.