Provably accurate adaptive sampling for collocation points in physics-informed neural networks

• URL: http://arxiv.org/abs/2504.00910v1
• Date: Tue, 01 Apr 2025 15:45:08 GMT
• Title: Provably accurate adaptive sampling for collocation points in physics-informed neural networks
• Authors: Antoine Caradot, Rémi Emonet, Amaury Habrard, Abdel-Rahim Mezidi, Marc Sebban,
• Abstract summary: Physics-informed Neural Networks (PINN) have emerged as an efficient way to learn surrogate solvers.<n>We introduce a provably accurate sampling method for collocation points based on the Hessian of the PDE residuals.
• Score: 11.912466054588327
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Despite considerable scientific advances in numerical simulation, efficiently solving PDEs remains a complex and often expensive problem. Physics-informed Neural Networks (PINN) have emerged as an efficient way to learn surrogate solvers by embedding the PDE 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 for PINNs. In this paper, leveraging a new quadrature method for approximating definite integrals, we introduce a provably accurate sampling method for collocation points based on the Hessian of the PDE residuals. Comparative experiments conducted on a set of 1D and 2D PDEs demonstrate the benefits of our method.

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