Adaptive anisotropic composite quadratures for residual minimisation in neural PDE approximations

TL;DR

This paper introduces an adaptive quadrature method for neural PDE approximation that improves accuracy and efficiency by controlling quadrature errors and selectively refining quadratures during training.

Contribution

It proposes a novel anisotropic adaptive composite quadrature strategy combined with a refresh-based training approach for neural PDE solvers.

Findings

The adaptive method narrows the gap between training and reference losses.

It uses quadrature points more efficiently than non-adaptive strategies.

The approach achieves strong approximation accuracy on benchmark problems.

Abstract

We study the role of numerical quadrature in residual-minimisation methods for neural network approximation of partial differential equations. We first present an abstract error framework that separates approximation, quadrature and optimisation errors, and derive a nonlinear Strang-type estimate quantifying how inaccuracies in the discrete loss affect the final approximation. Motivated by this analysis, we propose an anisotropic adaptive composite quadrature strategy that controls the relative quadrature error of the residual loss using richer reference quadratures and bisection-based refinement. We then introduce a refresh-based training methodology that rebuilds the quadrature only when an online error indicator exceeds a prescribed threshold, balancing accuracy and computational cost. Numerical experiments on a range of benchmark problems show that the proposed approach narrows the gap between training and reference losses, uses quadrature points more efficiently and delivers strong approximation accuracy relative to non-adaptive quadrature strategies.