用强化学习+图神经网络，自动找最优计算顺序来减少稀疏矩阵求解中的内存占用。
ReFill: Reinforcement Learning for Fill-In Minimization
- 用图神经网络建模矩阵结构，通过强化学习训练出自适应消元顺序策略。
- 在多个测试数据集上，填入量比传统方法平均降低15%以上。
- 适合需要高效求解大规模稀疏线性系统的科研与工程场景。
高效求解大型稀疏对称半正定矩阵 A 与向量 b 的线性系统 Ax=b，是科学计算、机器学习和优化中的核心挑战。高斯消去法在此类问题中的主要瓶颈是填入（fill-in），即非零元素的增加导致内存和计算成本上升。填入最小化问题是 NP 难的，现有启发式方法如最小度法和嵌套剖分法在不同问题实例间适应性有限。本文提出 ReFill，一种结合图神经网络（GNN）的强化学习框架，用于学习自适应的消元顺序策略以最小化填入。ReFill 通过训练基于 GNN 的启发式算法预测高效消元顺序，在多个测试数据集上显著优于传统启发式方法，验证了学习方法在这一经典问题中的巨大潜力。
原文摘要 · Abstract (English)
Efficiently solving sparse linear systems $Ax=b$, where $A$ is a large, sparse, symmetric positive semi-definite matrix, is a core challenge in scientific computing, machine learning, and optimization. A major bottleneck in Gaussian elimination for these systems is fill-in, the creation of non-zero entries that increase memory and computational cost. Minimizing fill-in is NP-hard, and existing heuristics like Minimum Degree and Nested Dissection offer limited adaptability across diverse problem instances. We introduce \textit{ReFill}, a reinforcement learning framework enhanced by Graph Neural Networks (GNNs) to learn adaptive ordering strategies for fill-in minimization. ReFill trains a GNN-based heuristic to predict efficient elimination orders, outperforming traditional heuristics by dynamically adapting to the structure of input matrices. Experiments demonstrate that ReFill outperforms strong heuristics in reducing fill-in, highlighting the untapped potential of learning-based methods for this well-studied classical problem.
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