Nodal Hybrid Neural Solvers for Parametric PDE Systems

TL;DR

This paper introduces advanced neural network-based solvers for complex PDE systems on unstructured meshes, achieving mesh-independent convergence and robustness, thus extending the applicability of learning-based PDE solvers.

Contribution

It develops the G-FNS, AG-FNS, and ML-AG-FNS methods, extending Fourier neural solvers to unstructured meshes and coupled PDE systems with multilevel and adaptive features.

Findings

Achieves mesh-independent convergence rates.

Effectively captures multiscale error modes.

Demonstrates robustness on complex anisotropic and elasticity problems.

Abstract

The numerical solution of partial differential equations (PDEs) is fundamental to scientific and engineering computing. In the presence of strong anisotropy, material heterogeneity, and complex geometries, however, classical iterative solvers often suffer from reduced efficiency and require substantial problem-dependent tuning. The Fourier neural solver (FNS) is a learning-based hybrid iterative solver for such problems without extensive manual parameter tuning, but its original design is primarily effective for scalar PDEs on structured meshes and is difficult to extend directly to unstructured meshes or strongly coupled PDE systems. Building on the FNS framework, we introduce block smoothing operators and graph neural networks to construct a solver for unstructured systems, termed the graph Fourier neural solver (G-FNS). We further incorporate a coordinate transformation network to develop the adaptive graph Fourier neural solver (AG-FNS), and then extend this formulation to a frequency-domain multilevel variant, ML-AG-FNS. Rigorous analysis shows that, under suitable mathematical assumptions, the proposed method achieves mesh-independent convergence rate. Error-spectrum visualizations further indicate that AG-FNS can capture complex multiscale error modes. Extensive experiments on two-dimensional anisotropic diffusion and on two- and three-dimensional isotropic/anisotropic linear elasticity problems over unstructured meshes demonstrate strong robustness and efficiency. The proposed framework can be used either as a solver or as a preconditioner for Krylov subspace methods. Overall, it substantially extends the original FNS methodology and broadens the applicability of this class of neural solvers.