Energy-based error bound of physics-informed neural network solutions in elasticity

TL;DR

This paper introduces an energy-based a posteriori error bound for physics-informed neural network solutions in elasticity, providing a reliable measure of the global discretization error.

Contribution

It proposes a novel energy-based error estimator for physics-informed neural networks applied to elasticity problems, with proven bounding properties.

Findings

The error bound effectively estimates the global discretization error.

The proposed method demonstrates the asymptotic behavior of neural network solutions.

The error estimator is validated through a demonstrating example.

Abstract

An energy-based a posteriori error bound is proposed for the physics-informed neural network solutions of elasticity problems. An admissible displacement-stress solution pair is obtained from a mixed form of physics-informed neural networks, and the proposed error bound is formulated as the constitutive relation error defined by the solution pair. Such an error estimator provides an upper bound of the global error of neural network discretization. The bounding property, as well as the asymptotic behavior of the physics-informed neural network solutions, are studied in a demonstrating example.