Moving sample method for solving time-dependent partial differential equations
Beining Xu, Haijun Yu, Jiayu Zhai, Kejun Tang, Xiaoliang Wan
TL;DR
This work tackles the challenge of solving time-dependent PDEs with sharp local features using physics-informed neural networks by introducing Moving Sample Method (MSM), an adaptive sampling framework where collocation points migrate toward high-residual regions. The method combines a residual-driven sampling density with a velocity field learned from residual dynamics, linking sampling updates to transport equations and equidistribution principles. A pair of neural networks—one for the PDE solution $u_{\boldsymbol{\theta}}$ and one for the sampling potential $\boldsymbol{\phi}_{\boldsymbol{\eta}}$— jointly drive the solution and the adaptive mesh via $\boldsymbol{v}_{\boldsymbol{\eta}}=\nabla \boldsymbol{\phi}_{\boldsymbol{\eta}}$, with losses $\text{Loss}_u$ and $\text{Loss}_{\boldsymbol{v}}$ incorporating the residual $r_t$ and its time derivative $R_t$. Across Allen–Cahn, rotation, Burgers’, Fokker–Planck, and high-dimensional advection benchmarks, MSM-PINNs achieve higher accuracy with fewer sampling points than vanilla PINNs, particularly in regions exhibiting singular behavior or moving sharp features; the approach improves stability and efficiency but introduces extra training cost for the velocity field and requires careful handling of domain-boundary dynamics.
Abstract
Solving time-dependent partial differential equations (PDEs) that exhibit sharp gradients or local singularities is computationally demanding, as traditional physics-informed neural networks (PINNs) often suffer from inefficient point allocation that wastes resources on regions already well-resolved. This paper presents an adaptive sampling framework for PINNs aimed at efficiently solving time-dependent partial differential equations with pronounced local singularities. The method employs a residual-driven strategy, where the spatial-temporal distribution of training points is iteratively updated according to the error field from the previous iteration. This targeted allocation enables the network to concentrate computational effort on regions with significant residuals, achieving higher accuracy with fewer sampling points compared to uniform sampling. Numerical experiments on representative PDE benchmarks demonstrate that the proposed approach improves solution quality.