Meshfree Variational Physics Informed Neural Networks (MF-VPINN): an adaptive training strategy

TL;DR

This paper presents MF-VPINN, a meshfree, adaptive training method for physics-informed neural networks that improves accuracy by selectively adding test functions based on error indicators, eliminating the need for domain triangulation.

Contribution

Introduction of a meshfree, adaptive training strategy for VPINNs that enhances accuracy without domain triangulation.

Findings

Higher accuracy than traditional VPINNs with same test functions

Effective adaptive test function addition based on error indicators

Comparison of four training strategies demonstrates improved performance

Abstract

In this paper, we introduce a Meshfree Variational-Physics-Informed Neural Network. It is a Variational-Physics-Informed Neural Network that does not require the generation of the triangulation of the entire domain and that can be trained with an adaptive set of test functions. In order to generate the test space, we exploit an a posteriori error indicator and add test functions only where the error is higher. Four training strategies are proposed and compared. Numerical results show that the accuracy is higher than the one of a Variational-Physics-Informed Neural Network trained with the same number of test functions but defined on a quasi-uniform mesh.