Flow-Matching Sampling in Physics-Informed Neural Networks for PDEs with Sharp Source Terms

Yana Khassan Nibal, Dmitry Efremenko, Fedor Buzaev, Denis Derkach

Keywords: pinn, pde, sampling for pinn

Abstract:

Singularities in the source functions of partial differential equations (PDEs) pose
significant challenges for physics-informed neural networks (PINNs), often lead-
ing to numerical instability and requiring a large number of sampling points to
achieve accurate solutions, which increases computational costs. In this paper, we
propose a novel sampling strategy that use diffusion models for generative sam-
pling based on the distribution of PDE residuals. Using the optimal transport cou-
pling flow-matching technique, our method adaptively generates additional sam-
pling points in regions with high residuals, enhancing both solution accuracy and
efficiency. Unlike existing approaches, which explicitly model probability densi-
ties proportional to residuals, our technique uses flow matching to directly sample
from complex residual distributions, improving PINN performance for problems
with sharply localized source terms. We validate our method on the Poisson equa-
tion with singular source functions and the linear elasticity equation in materials
with complex geometries, achieving up to 10× lower MSE compared to baseline
methods and outperforming normalizing flow-based sampling.

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Submission Number: 53