NOWS: Neural Operator Warm Starts for Accelerating Iterative Solvers

TL;DR

NOWS introduces a hybrid approach using neural operators to provide high-quality initial guesses, significantly accelerating classical PDE solvers while maintaining their stability and convergence guarantees.

Contribution

It presents a novel hybrid method that integrates neural operators with traditional iterative solvers, reducing computational time by up to 90% without altering existing discretizations.

Findings

Consistently reduces iteration counts and runtime across benchmarks.

Achieves up to 90% reduction in computational time.

Maintains stability and convergence guarantees of classical solvers.

Abstract

Partial differential equations (PDEs) underpin quantitative descriptions across the physical sciences and engineering, yet high-fidelity simulation remains a major computational bottleneck for many-query, real-time, and design tasks. Data-driven surrogates can be strikingly fast but are often unreliable when applied outside their training distribution. Here we introduce Neural Operator Warm Starts (NOWS), a hybrid strategy that harnesses learned solution operators to accelerate classical iterative solvers by producing high-quality initial guesses for Krylov methods such as conjugate gradient and GMRES. NOWS leaves existing discretizations and solver infrastructures intact, integrating seamlessly with finite-difference, finite-element, isogeometric analysis, finite volume method, etc. Across our benchmarks, the learned initialization consistently reduces iteration counts and end-to-end runtime, resulting in a reduction of the computational time of up to 90 %, while preserving the stability and convergence guarantees of the underlying numerical algorithms. By combining the rapid inference of neural operators with the rigor of traditional solvers, NOWS provides a practical and trustworthy approach to accelerate high-fidelity PDE simulations.