Variational Physics Informed Neural Networks: the role of quadratures and test functions

TL;DR

This paper investigates how quadrature precision and test function degree influence the convergence of Variational Physics Informed Neural Networks in solving elliptic boundary-value problems, revealing counterintuitive strategies for optimal error decay.

Contribution

It provides a theoretical analysis of VPINN convergence based on quadrature and test functions, supported by numerical experiments, within a Petrov-Galerkin framework.

Findings

Lower-degree test functions with high-precision quadratures improve convergence for smooth solutions.

The inf-sup condition is crucial for error estimates and convergence.

Numerical results confirm theoretical predictions about optimal test function and quadrature choices.

Abstract

In this work we analyze how quadrature rules of different precisions and piecewise polynomial test functions of different degrees affect the convergence rate of Variational Physics Informed Neural Networks (VPINN) with respect to mesh refinement, while solving elliptic boundary-value problems. Using a Petrov-Galerkin framework relying on an inf-sup condition, we derive an a priori error estimate in the energy norm between the exact solution and a suitable high-order piecewise interpolant of a computed neural network. Numerical experiments confirm the theoretical predictions and highlight the importance of the inf-sup condition. Our results suggest, somehow counterintuitively, that for smooth solutions the best strategy to achieve a high decay rate of the error consists in choosing test functions of the lowest polynomial degree, while using quadrature formulas of suitably high precision.