NOWS: Neural Operator Warm Starts for accelerating iterative solvers
- 1Chair of Computational Science and Simulation Technology, Institute of Photonics, Department of Mathematics and Physics, Leibniz University Hannover, 30167 Hannover, Germany
- 2Department of Engineering Mechanics, Tsinghua University, Beijing, China
- 3Institute of Structural Mechanics, Bauhaus-Universität Weimar, Germany
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.
Related
- An analysis of iterative refinement for quantum linear system solvers
- Reusable QUBO-Derived Block-Schur Inverse Factors for Fixed-Operator Finite-Difference PDE Solvers
- Single-condition neural solvers encode transferable response spaces for parametric differential equations
- Do Neural PDE Solvers Learn the Right Dynamics?
- Soliton Profiles in 1D: Enhanced Neural-Network Solvers and Constrained Deep Ritz Methods
BibTeX
@article{eshaghi2026nows,
title = {{NOWS: Neural Operator Warm Starts for accelerating iterative solvers}},
author = {Mohammad Sadegh Eshaghi and Cosmin Anitescu and Navid Valizadeh and Yizheng Wang and Xiaoying Zhuang and Timon Rabczuk},
journal = {Computer Methods in Applied Mechanics and Engineering},
year = {2026},
month = may,
doi = {10.1016/j.cma.2026.118989},
url = {https://doi.org/10.1016/j.cma.2026.118989},
note = {CC BY-NC-ND},
}
RIS
TY - JOUR
TI - NOWS: Neural Operator Warm Starts for accelerating iterative solvers
AU - Mohammad Sadegh Eshaghi
AU - Cosmin Anitescu
AU - Navid Valizadeh
AU - Yizheng Wang
AU - Xiaoying Zhuang
AU - Timon Rabczuk
PY - 2026
DA - 2026/05/15
JO - Computer Methods in Applied Mechanics and Engineering
DO - 10.1016/j.cma.2026.118989
UR - https://doi.org/10.1016/j.cma.2026.118989
AB - 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.
SN - 0045-7825
SN - 1879-2138
ER -