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 (NNs) have emerged as a novel approach for solving PDEs, with studies showing that they exhibit faster convergence for low-frequency components. Building on these complementary frequency-convergence characteristics of iterative methods and NNs, and drawing inspiration from multigrid methods, we propose a hybrid solving framework consisting of a combination of iterative methods and NN-based solvers, termed physics-informed neural network multigrid (PINN-MG, abbreviated as PMG). In this framework, the iterative methods eliminate local high-frequency oscillation errors, while PINNs 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 methods and PINNs, respectively, thereby accelerating the convergence. We validate the proposed PMG framework on the linear Poisson equations and nonlinear Helmholtz equations. The results show significant acceleration of the PMG built on the Gauss-Seidel (GS), pseudo-time, and generalized minimal residual (GMRES) methods. A detailed analysis of the convergence process validates the rationality of the framework. To further evaluate the advantages of PMG, we apply it to indefinite Helmholtz equations, a class of problems in which traditional solvers often diverge due to the divergence of the low-frequency components. The PMG framework effectively overcomes this divergence, improving both the stability and the convergence speed. We propose the PMG framework as a data-free hybrid solver that does not rely on any pretraining and, more importantly, provides a unified mechanism to tightly couple the NN methods with classical iterative solvers, achieving an organic and iterative integration of the two paradigms.

Key words: physics-informed neural network (PINN), iterative method, hybrid computing architecture, spectral bias