A Label-Free Hybrid Iterative Numerical Transferable Solver for Partial Differential Equations

Abstract

In scientific computing, traditional numerical methods for partial differential equations (PDEs), such as finite difference method and finite element method, often need to solve (large-scale) linear systems of equations. It is known that classical iterative solvers, such as Jacobi iteration and Gauss-Seidel iteration, have the smoothing property, i.e. the high-frequency part of the solution can be efficiently captured while the low-frequency part cannot. Multigrid offers a general methodology that utilizes the smoothing property of iterative solvers in a hierarchical manner. Meanwhile, machine learning-based methods for PDEs, such as deep operator network and Fourier neural operator, show the spectral bias, i.e. the low-frequency part of the solution can be efficiently captured while the high-frequency part cannot. The recently developed hybrid iterative numerical transferable solver (HINTS) offers an alternative choice that combines the advantages of classical iterative solvers on fine grids and operator learning methods on coarse grids. In this work, we propose a label-free HINTS for PDEs with the following features: (1) the training of the operator learning component is totally label-free, i.e. we do not need solutions to a given problem, which are typically obtained by classical solvers, (2) the resolution of the operator learning component is far coarser than that of the linear system of equations to be solved, (3) the success of label-free HINTS depends on whether the high-frequency component of the solution is captured on fine grids or not. Numerical experiments, including Possion equation in two and three dimensions, Hemholtz equation in two and three dimensions, anisotropic diffusion equation in two dimensions, are conducted to demonstrate the features of the proposed method. Based on these results, we conclude that the label-free HINTS provides a valuable addition for solving linear systems of equations arising from numerical PDEs.