A certified wavelet-based physics-informed neural network for the solution of parameterized partial differential equations

TL;DR

This paper introduces a wavelet-based, certified physics-informed neural network that provides a computable error bound for solving parameterized PDEs, enhancing model reduction techniques with reliable error estimates.

Contribution

It develops a wavelet-based approach for PINNs that includes a computable error bound, applicable to elliptic PPDEs with improved reliability.

Findings

Wavelet-based error bounds are effective for elliptic PPDEs.

The approach improves the reliability of PINNs in model reduction.

Numerical examples confirm the quantitative effectiveness of the error bounds.

Abstract

Physics Informed Neural Networks (PINNs) have frequently been used for the numerical approximation of Partial Differential Equations (PDEs). The goal of this paper is to construct PINNs along with a computable upper bound of the error, which is particularly relevant for model reduction of Parameterized PDEs (PPDEs). To this end, we suggest to use a weighted sum of expansion coefficients of the residual in terms of an adaptive wavelet expansion both for the loss function and an error bound. This approach is shown here for elliptic PPDEs using both the standard variational and an optimally stable ultra-weak formulation. Numerical examples show a very good quantitative effectivity of the wavelet-based error bound.