用神经算子生成初始解，让传统求解器快90%且保证稳定。
NOWS: Neural Operator Warm Starts for Accelerating Iterative Solvers
- 用训练好的神经算子生成高质量初始猜测，加速迭代求解过程。
- 在多个测试中减少90%计算时间，同时保持原有算法的收敛性。
- 适合需要快速、可靠仿真结果的工程与科学计算场景。
偏微分方程（PDE）在物理科学和工程中具有重要应用，但高保真模拟对多查询、实时及设计任务仍是重大计算瓶颈。数据驱动的代理模型虽速度快，但在训练分布外常不可靠。本文提出神经算子热启动（NOWS），一种混合策略：利用学习到的解算子为共轭梯度、GMRES等克里洛夫方法提供高质量初始猜测，从而加速经典迭代求解器。NOWS不改变现有离散化与求解器架构，可无缝集成于有限差分、有限元、等几何分析、有限体积法等方法。在多个基准测试中，该初始化策略持续降低迭代次数与端到端运行时间，最高实现计算时间减少90%，同时保留底层数值算法的稳定性与收敛保证。通过结合神经算子的快速推理与传统求解器的严谨性，NOWS为高保真PDE模拟提供了一种实用且可信的加速方案。
原文摘要 · Abstract (English)
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.
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