NSPOD: Accelerating the convergence of Krylov-based iterative linear solvers via approximated PODs

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 (Zhang et al., 2024). We partially addressed this challenge by introducing the deep operator network Geo-DeepONet and hybridizing it with Krylov-based iterative linear solvers (Lee et al., 2025a), 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 reduce the number of iterations needed for convergence in Krylov-based linear iterative solvers on complex curved domains, performing comparably to state-of-the-art algebraic multigrid preconditioners, and substantially improving on other widely used standard preconditioners, such as Jacobi or SOR, in both number of iterations and computational time. We expect that the findings in this study lead to more efficient hybrid preconditioners that complement the convergence properties of current gold-standard preconditioning methods for solid mechanics PDEs in three dimensions. 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 when applied to solid mechanics PDEs.