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