PINN-MG: A Multigrid-Inspired Hybrid Framework Combining Iterative Method and Physics-Informed Neural Networks (2410.05744v1)

Abstract: Iterative methods are widely used for solving partial differential equations (PDEs). However, the difficulty in eliminating global low-frequency errors significantly limits their convergence speed. In recent years, neural networks have emerged as a novel approach for solving PDEs, with studies revealing that they exhibit faster convergence for low-frequency components. Building on this complementary frequency convergence characteristics of iterative methods and neural networks, we draw inspiration from multigrid methods and propose a hybrid solving framework that combining iterative methods and neural network-based solvers, termed PINN-MG (PMG). In this framework, the iterative method is responsible for eliminating local high-frequency oscillation errors, while Physics-Informed Neural Networks (PINNs) are employed to correct global low-frequency errors. Throughout the solving process, high- and low-frequency components alternately dominate the error, with each being addressed by the iterative method and PINNs respectively, thereby accelerating the convergence. We tested the proposed PMG framework on the linear Poisson equation and the nonlinear Helmholtz equation, and the results demonstrated significant acceleration of the PMG when built on Gauss-Seidel, pseudo-time, and GMRES methods. Furthermore, detailed analysis of the convergence process further validates the rationality of the framework. We proposed that the PMG framework is a hybrid solving approach that does not rely on training data, achieving an organic integration of neural network methods with iterative methods.