Physics-informed neural networks (PINNs) for numerical model error approximation and superresolution

TL;DR

This paper introduces physics-informed neural networks (PINNs) to explicitly quantify and improve the accuracy of finite element models by approximating errors and achieving superresolution, demonstrated on a 2D elastic plate.

Contribution

The paper presents a novel application of PINNs for simultaneous model error approximation and superresolution in finite element analysis, outperforming purely data-driven methods.

Findings

PINNs accurately predict displacement errors with small deviations from ground truth.

Physics-informed loss functions enhance neural network performance over data-only approaches.

Method successfully applied to 2D elastic plate with different mesh resolutions.

Abstract

Numerical modeling errors are unavoidable in finite element analysis. The presence of model errors inherently reflects both model accuracy and uncertainty. To date there have been few methods for explicitly quantifying errors at points of interest (e.g. at finite element nodes). The lack of explicit model error approximators has been addressed recently with the emergence of machine learning (ML), which closes the loop between numerical model features/solutions and explicit model error approximations. In this paper, we propose physics-informed neural networks (PINNs) for simultaneous numerical model error approximation and superresolution. To test our approach, numerical data was generated using finite element simulations on a two-dimensional elastic plate with a central opening. Four- and eight-node quadrilateral elements were used in the discretization to represent the reduced-order and higher-order models, respectively. It was found that the developed PINNs effectively predict model errors in both x and y displacement fields with small differences between predictions and ground truth. Our findings demonstrate that the integration of physics-informed loss functions enables neural networks (NNs) to surpass a purely data-driven approach for approximating model errors.