用图神经网络增强传统预条件法，让大规模微分方程求解更快更稳。
Deep Learning-Enhanced Preconditioning for Efficient Conjugate Gradient Solvers in Large-Scale PDE Systems
- 将图神经网络与不完全乔列斯基分解结合，动态生成预条件矩阵。
- 迭代次数平均减少24.8%，训练规模提升至百万级，推理达千万维。
- 小数据+低算力即可泛化到大规模问题，适合工程仿真场景。
预条件技术对加速偏微分方程离散后的大规模线性方程组求解至关重要。本文提出一种新方法，将图神经网络（GNN）与传统不完全乔列斯基分解（IC）相结合，克服了纯基于GNN生成方法的不足，显著提升了计算效率和可扩展性。实验表明，相比IC方法，迭代次数平均减少24.8%；训练规模较此前方法提升两个数量级。在有限元法实现的三维静力结构分析中，模型成功处理高达500万维度的训练稀疏矩阵和1000万维度的推理规模。此外，该方法在不同尺度间表现出强泛化能力，仅用小规模数据和普通硬件即可有效加速大规模线性方程组的共轭梯度求解，具有良好的鲁棒性与实用性，适用于计算科学领域。
原文摘要 · Abstract (English)
Preconditioning techniques are crucial for enhancing the efficiency of solving large-scale linear equation systems that arise from partial differential equation (PDE) discretization. These techniques, such as Incomplete Cholesky factorization (IC) and data-driven neural network methods, accelerate the convergence of iterative solvers like Conjugate Gradient (CG) by approximating the original matrices. This paper introduces a novel approach that integrates Graph Neural Network (GNN) with traditional IC, addressing the shortcomings of direct generation methods based on GNN and achieving significant improvements in computational efficiency and scalability. Experimental results demonstrate an average reduction in iteration counts by 24.8% compared to IC and a two-order-of-magnitude increase in training scale compared to previous methods. A three-dimensional static structural analysis utilizing finite element methods was validated on training sparse matrices of up to 5 million dimensions and inference scales of up to 10 million. Furthermore, the approach demon-strates robust generalization capabilities across scales, facilitating the effective acceleration of CG solvers for large-scale linear equations using small-scale data on modest hardware. The method's robustness and scalability make it a practical solution for computational science.
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