发表机构
University of Macau（澳门大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究提出将图卷积融入多重网格的可学习框架，以提升偏微分方程求解性能，其参数量少、训练成本低，收敛速率优于传统方法，参数泛化性良好。
AI 中文摘要
本文提出了一种新颖的框架，将可学习图卷积与几何多重网格方法相结合，用于求解偏微分方程（PDEs）。首先将偏微分方程的离散化表示为图结构，从而可应用图卷积提升多重网格的性能。通过将图卷积融入多重网格的各个组件，如平滑算子和网格间转移算子，我们开发了一种可学习多重网格，能根据底层问题特性自适应优化其性能。在该框架中，图卷积被直接嵌入多重网格循环内，将整个多重网格求解器有效转化为专用神经网络架构，而非将经典求解器与代理模型组合。该可学习多重网格框架的参数量少，仅需极少训练即可获得良好性能。数值实验验证了该方法在求解部分挑战性问题时的有效性，与传统多重网格方法相比收敛速率得到提升。同时，研究了所学习参数在不同问题设置间的泛化性，包括源项、系数、几何形状和网格尺寸的变化，且采用了恰当的权重共享和迁移学习策略。
英文摘要
This paper presents a novel framework that integrates learnable graph convolutions with the geometric multigrid method for solving partial differential equations (PDEs). The discretization of PDEs is first represented as a graph structure, enabling the application of graph convolutions to enhance the multigrid performance. By incorporating graph convolutions into the multigrid components such as smoothing and inter-grid transfer operators, we develop a learnable multigrid that can adaptively optimize its performance based on the underlying problem characteristics. In this framework, the graph convolutions are embedded directly within the multigrid cycle, effectively transforming the entire multigrid solver into a specialized neural network architecture, rather than combining a classical solver with surrogate models. The learnable multigrid framework is lightweight in terms of parameter count and requires minimal training effort to achieve good performance. Numerical experiments demonstrate the effectiveness of the proposed approach in solving some challenging problems, showing improved convergence rates compared to traditional multigrid methods. The generalizability of the learned parameters across different problem settings, including varying source terms, coefficients, geometries, and mesh sizes, is also investigated with proper weight-sharing and transfer-learning strategies.
