M\'ethode de quadrature pour les PINNs fond\'ee th\'eoriquement sur la hessienne des r\'esiduels

TL;DR

This paper introduces a theoretically grounded quadrature method based on the Hessian of the residuals to improve the selection of collocation points in Physics-informed Neural Networks, enhancing their efficiency in solving PDEs.

Contribution

The paper proposes a novel quadrature approach leveraging the Hessian of residuals to adaptively select collocation points in PINNs, grounded in theoretical analysis.

Findings

Improved accuracy in PINNs through Hessian-based sampling

Enhanced convergence rates with adaptive collocation

Theoretical validation of the quadrature method

Abstract

Physics-informed Neural Networks (PINNs) have emerged as an efficient way to learn surrogate neural solvers of PDEs by embedding the physical model in the loss function and minimizing its residuals using automatic differentiation at so-called collocation points. Originally uniformly sampled, the choice of the latter has been the subject of recent advances leading to adaptive sampling refinements. In this paper, we propose a new quadrature method for approximating definite integrals based on the hessian of the considered function, and that we leverage to guide the selection of the collocation points during the training process of PINNs.