Hybrid Iterative Solvers with Geometry-Aware Neural Preconditioners for Parametric PDEs

TL;DR

This paper introduces Geo-DeepONet, a geometry-aware neural network that improves the robustness and efficiency of hybrid iterative solvers for parametric PDEs across diverse unstructured domains without retraining.

Contribution

We develop Geo-DeepONet, a novel neural operator that incorporates domain geometry, enabling effective hybrid solvers for parametric PDEs on arbitrary meshes.

Findings

Enhanced robustness of solvers across diverse geometries

Improved convergence rates in numerical experiments

Effective application to real-world PDE problems

Abstract

The convergence behavior of classical iterative solvers for parametric partial differential equations (PDEs) is often highly sensitive to the domain and specific discretization of PDEs. Previously, we introduced hybrid solvers by combining the classical solvers with neural operators for a specific geometry 1, but they tend to under-perform in geometries not encountered during training. To address this challenge, we introduce Geo-DeepONet, a geometry-aware deep operator network that incorporates domain information extracted from finite element discretizations. Geo-DeepONet enables accurate operator learning across arbitrary unstructured meshes without requiring retraining. Building on this, we develop a class of geometry-aware hybrid preconditioned iterative solvers by coupling Geo-DeepONet with traditional methods such as relaxation schemes and Krylov subspace algorithms. Through numerical experiments on parametric PDEs posed over diverse unstructured domains, we demonstrate the enhanced robustness and efficiency of the proposed hybrid solvers for multiple real-world applications.