Evaluating Error Bound for Physics-Informed Neural Networks on Linear Dynamical SystemsDownload PDF

Published: 21 Oct 2022, Last Modified: 05 May 2023DLDE 2022 PosterReaders: Everyone
Keywords: Error Estimation, Error Bound, Physics-Informed Neural Network, Linear Dynamical Systems
TL;DR: Given a linear ODE system, the algorithm evaluates error of any PINN using only the residual, without any assumptions on network size/finetuning, within milliseconds.
Abstract: There have been extensive studies on solving differential equations using physics-informed neural networks. While this method has proven advantageous in many cases, a major criticism lies in its lack of analytical error bounds. Therefore, it is less credible than its traditional counterparts, such as the finite difference method. This paper shows that one can mathematically derive explicit error bounds for physics-informed neural networks trained on a class of linear dynamical systems using only the network's residuals (point-wise loss) over the domain. Our work shows a link between network residuals and the absolute error of solution. Our approach is semi-phenomonological and independent of knowledge of the actual solution or the complexity or architecture of the network. Using the method of manufactured solution on linear ODEs and system of linear ODEs, we empirically verify the error evaluation algorithm and demonstrate that the actual error strictly lies within our derived bound.
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