NH-PINN: Neural homogenization based physics-informed neural network for multiscale problems

TL;DR

This paper introduces NH-PINN, a novel approach combining homogenization techniques with physics-informed neural networks to effectively solve multiscale equations, significantly improving accuracy over traditional PINN methods.

Contribution

The paper proposes a 3-step homogenization-based method that enhances PINN accuracy for multiscale problems, including an oversampling strategy for periodic cell problems.

Findings

Significant accuracy improvement in solving multiscale equations with NH-PINN.

Oversampling strategy effectively addresses periodic cell problem challenges.

PINN-based homogenization can outperform traditional numerical homogenization methods.

Abstract

Physics-informed neural network (PINN) is a data-driven approach to solve equations. It is successful in many applications; however, the accuracy of the PINN is not satisfactory when it is used to solve multiscale equations. Homogenization is a way of approximating a multiscale equation by a homogenized equation without multiscale property; it includes solving cell problems and the homogenized equation. The cell problems are periodic; and we propose an oversampling strategy which greatly improves the PINN accuracy on periodic problems. The homogenized equation has constant or slow dependency coefficient and can also be solved by PINN accurately. We hence proposed a 3-step method to improve the PINN accuracy for solving multiscale problems with the help of the homogenization. We apply our method to solve three equations which represent three different homogenization. The results show that the proposed method greatly improves the PINN accuracy. Besides, we also find that the PINN aided homogenization may achieve better accuracy than the numerical methods driven homogenization; PINN hence is a potential alternative to implementing the homogenization.