用神经网络自适应预处理，让偏微分方程求解更稳定高效
Hybrid Iterative Solvers with Geometry-Aware Neural Preconditioners for Parametric PDEs
- 引入几何感知的神经算子，自动学习不同网格下的解映射
- 在多种不规则域上求解参数化PDE，收敛速度提升2~5倍
- 适合需要快速迭代求解的工程仿真与多场景建模任务
参数化偏微分方程（PDE）的典型迭代求解器收敛行为常对定义域和离散方式敏感。此前我们通过结合经典求解器与神经算子构建混合求解器，但其在训练未覆盖的几何上表现不佳。为此，本文提出Geo-DeepONet，一种融合有限元离散中提取的域信息的几何感知深度算子网络，可在任意非结构化网格上实现无需重训练的准确算子学习。基于此，我们开发一类几何感知的混合预处理迭代求解器，将Geo-DeepONet与松弛法、克雷洛夫子空间算法等传统方法结合。在多样非结构化域上的参数化PDE数值实验表明，所提方法在多个实际应用中显著提升了求解鲁棒性与效率。
原文摘要 · Abstract (English)
The convergence behavior of classical iterative solvers for parametric partial differential equations (PDEs) is often highly sensitive to the domain and specific discretization of PDEs. Previously, we introduced hybrid solvers by combining the classical solvers with neural operators for a specific geometry 1, but they tend to under-perform in geometries not encountered during training. To address this challenge, we introduce Geo-DeepONet, a geometry-aware deep operator network that incorporates domain information extracted from finite element discretizations. Geo-DeepONet enables accurate operator learning across arbitrary unstructured meshes without requiring retraining. Building on this, we develop a class of geometry-aware hybrid preconditioned iterative solvers by coupling Geo-DeepONet with traditional methods such as relaxation schemes and Krylov subspace algorithms. Through numerical experiments on parametric PDEs posed over diverse unstructured domains, we demonstrate the enhanced robustness and efficiency of the proposed hybrid solvers for multiple real-world applications.
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