NSPOD: acceleratingthe convergence ofKrylov-based iterative linearsolvers via approximated PODs

2026-05-08

AI Summary

This work addresses the high sensitivity of Krylov iterative solvers to geometry, boundary conditions, and material parameters when solving parametric partial differential equations, as well as the limited generalization and acceleration capabilities of existing neural operator-based preconditioners. The authors propose NSPOD, a multigrid-like deep operator network preconditioner that, for the first time, integrates neural operators with Proper Orthogonal Decomposition (POD) subspaces. By approximating solutions within a low-dimensional POD subspace, NSPOD effectively accelerates Krylov solvers without requiring retraining, even on unstructured meshes derived from complex CAD geometries. Demonstrated on linear PDEs in solid mechanics, the method significantly reduces iteration counts and outperforms state-of-the-art preconditioners such as algebraic multigrid, thereby overcoming key performance bottlenecks in current approaches.

Abstract

The convergence of Krylov-based linear iterative solvers applied to parametric partial differential equations (PDEs) is often highly sensitive to the domain, its discretization, the location/values of the applied Dirichlet/Neumann boundary conditions, body forces and material properties, among others. We have previously introduced hybridization of classical linear iterative solvers with neural operators for specific geometries, but they tend to not perform well on geometries not previously seen during training. We partially addressed this challenge by introducing the deep operator network Geo-DeepONet and hybridizing it with Krylov-based iterative linear solvers, which, despite learning effectively across arbitrary unstructured meshes without requiring retraining, led to only modest reductions in iterations compared to state-of-the-art preconditioners. In this study we introduce Neural Subspace Proper Orthogonal Decomposition (NSPOD), a multigrid-like deep operator network-based preconditioner which can dramatically reduce the number of iterations needed for convergence in Krylov-based linear iterative solvers, even when compared to state-of-the-art methods such as algebraic multigrid preconditioners. We demonstrate its efficiency via numerical experiments on a linearized version of solid mechanics PDEs applied to unstructured domains obtained from complex CAD geometries. We expect that the findings in this study lead to more efficient hybrid preconditioners that can match, or possibly even surpass, the convergence properties of the current gold standard preconditioning methods for solid mechanics PDEs.

Problem

• Krylov-based iterative solvers
• parametric PDEs
• preconditioning
• convergence acceleration
• unstructured domains

Innovation

• Neural Subspace POD
• Krylov solvers
• deep operator networks
• preconditioning
• unstructured meshes