Neural Networks to solve Partial Differential Equations: a Comparison with Finite Elements

Abstract: We compare the Finite Element Method (FEM) simulation of a standard Partial Differential Equation thermal problem of a plate with a hole with a Neural Network (NN) simulation. The largest deviation from the true solution obtained from FEM ($0.015$ for a solution on the order of unity) is easily achieved with NN too without much tuning of the hyperparameters. Accuracies below $0.01$ instead require refinement with an alternative optimizer to reach a similar performance with NN. A rough comparison between the Floating Point Operations values, as a machine-independent quantification of the computational performance, suggests a significant difference between FEM and NN in favour of the former. This also strongly holds for computation time: for an accuracy on the order of $10^{-5}$, FEM and NN require $54$ and $1100$ seconds, respectively. A detailed analysis of the effect of varying different hyperparameters shows that accuracy and computational time only weakly depend on the major part of them. Accuracies below $0.01$ cannot be achieved with the "adam'' optimizers and it looks as though accuracies below $10^{-5}$ cannot be achieved at all. In conclusion, the present work shows that for the concrete case of solving a steady-state 2D heat equation, the performance of a FEM algorithm is significantly better than the solution via networks.