Adaptive anisotropic composite quadratures for residual minimisation in neural PDE approximations
For researchers using neural networks to solve PDEs, this work provides a principled way to control quadrature errors, improving the reliability and efficiency of residual-based training.
The paper addresses the impact of numerical quadrature errors on residual-minimization methods for neural PDE solvers. It proposes an adaptive anisotropic composite quadrature strategy that reduces the gap between training and reference losses, achieving more efficient use of quadrature points and strong approximation accuracy compared to non-adaptive methods.
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.