PINN-ffusion: Physics-Informed Neural Network via Diffusion-based Adaptive Sampling and Anisotropic Domain Decomposition

Physics-Informed Neural Networks (PINNs) provide a flexible framework for solving partial differential equations (PDEs) by incorporating physical constraints into neural network training. However, their performance is often limited by the non-uniform distribution of PDE residuals, which are typically concentrated in localized regions such as boundary layers and sharp gradients. Existing approaches address this issue through either adaptive sampling as a minimax optimization problem or domain decomposition based on fixed partitioning schemes, which lead to instability and limited flexibility. In this study, we propose PINN-ffusion, a unified framework that reformulates adaptive sampling and domain decomposition as a coupled problem of probability evolution and geometry learning. Instead of adversarial optimization, we construct the sampling distribution via a diffusion process, enabling stable adaptation while preserving the original PINN objective. In addition, we introduce a geometry-aware domain decomposition based on diversity and uncertainty metrics, generalizing isotropic partitions into anisotropic structures aligned with the residual landscape. The proposed framework allows sampling and domain geometry to co-evolve during training, improving the allocation of computational resources to complex regions. We provide a theoretical justification of consistency with the PINN objective and demonstrate through experiments that PINN-ffusion achieves improved accuracy and stability, particularly in problems with localized and anisotropic error structures.