Provably Accurate Adaptive Sampling for Collocation Points in Physics-informed Neural Networks
This work addresses the challenge of expensive PDE simulations for researchers in computational science, offering an incremental improvement in adaptive sampling for PINNs.
The paper tackles the problem of efficiently solving PDEs by introducing a provably accurate adaptive sampling method for collocation points in Physics-informed Neural Networks, based on the Hessian of PDE residuals, and demonstrates benefits through comparative experiments on 1D and 2D PDEs.
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.