Nodal Hybrid Neural Solvers for Parametric PDE Systems
This work addresses the problem of efficient PDE solving for scientific and engineering applications, offering a novel extension to existing neural solvers but is incremental in building upon the FNS framework.
The authors tackled the challenge of solving parametric PDE systems on unstructured meshes with strong anisotropy and heterogeneity, where classical solvers are inefficient, by developing a neural solver framework (ML-AG-FNS) that achieves mesh-independent convergence and demonstrates strong robustness and efficiency in experiments on diffusion and elasticity problems.
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.