Hybrid Iterative Solvers with Geometry-Aware Neural Preconditioners for Parametric PDEs
Youngkyu Lee, Francesc Levrero Florencio, Jay Pathak, George Em Karniadakis
cs.LG, cs.NA, math.NA
Submitted: 2025-12-16
Updated: 2026-08-20
Comments: 19 pages, 10 figures, 3 tables
Journal ref: International Journal for Numerical Methods in Engineering 127 no. 15 (2026) e70393
DOI: 10.1002/nme.70393
License: http://creativecommons.org/licenses/by-nc-nd/4.0/
The gist: 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.
Terminology
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.
Sources
- Automatic discovery of optimal meta-solvers via multi-objective optimization
- Automatic discovery of optimal meta-solvers for time-dependent nonlinear PDEs
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