NOWS: Neural Operator Warm Starts for Accelerating Iterative Solvers

2025-11-04

AI Summary

High-fidelity partial differential equation (PDE) simulations remain computationally prohibitive for querying, real-time control, and design optimization; while data-driven surrogate models offer speed, they suffer from poor generalizability and limited reliability. This paper introduces Neural Operator Warm-Starting (NOWS), a method that embeds a learned solution operator into classical Krylov subspace iterative solvers—such as conjugate gradient and GMRES—to provide high-quality initial guesses. NOWS preserves the original solver’s architecture and is agnostic to discretization schemes, supporting finite differences, finite elements, and isogeometric analysis while rigorously maintaining numerical stability and convergence guarantees. Evaluated on multiple benchmark PDE problems, NOWS reduces iteration counts substantially and achieves up to 90% end-to-end runtime reduction. It establishes a critical trade-off between speed and reliability—marking the first systematic integration of neural operators as warm-starting components within traditional iterative solvers, rather than as solver replacements.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Solvers and Tools
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Data management and stream processing for Web, mobile and wireless applications
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• Accelerating iterative PDE solvers with neural operators
• Reducing computational bottlenecks in high-fidelity simulations
• Generating quality initial guesses for Krylov methods

Innovation

• Neural operators accelerate iterative solvers
• Generates initial guesses for Krylov methods
• Integrates with existing discretizations and infrastructures