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Flow-Matching Sampling in Physics-Informed Neural Networks for PDEs with Sharp Source Terms
Singularities in the source functions of partial differential equations (PDEs) pose significant challenges for physics-informed neural networks (PINNs), often leading 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 uses diffusion models for generative sampling based on the distribution of PDE residuals. Using the optimal transport coupling flow-matching technique, our method adaptively generates additional sampling points in regions with high residuals, enhancing both solution accuracy and efficiency. Unlike existing approaches, which explicitly model probability densities 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 equation with singular source functions and the linear elasticity equation in materials with complex geometries, achieving up to 10x lower MSE compared to baseline methods and outperforming normalizing flow-based sampling.