用神经网络模拟快速多极算法，高效学习椭圆型方程的格林函数。
Learning Greens Operators through Hierarchical Neural Networks Inspired by the Fast Multipole Method
- 基于快速多极法的分层计算结构，分离近场与远场交互
- 通过分层表示，实现对格林算子的高效学习
- 适合需处理长程相互作用的物理仿真与机器学习任务
快速多极法（FMM）是一种高效的数值算法，用于计算引力和静电场中N体问题的长程力。该方法利用了底层动力系统中格林函数的多极展开特性。尽管在物理和工程领域广泛应用，但将FMM与现代机器学习架构结合的研究仍不充分。本文提出一种新型神经网络架构——神经快速多极法（Neural FMM），将FMM的信息流融入分层机器学习框架，用于学习椭圆型偏微分方程的格林算子。该架构利用FMM的分层计算流程，分离局部与远场相互作用，并高效学习其各自表示。
原文摘要 · Abstract (English)
The Fast Multipole Method (FMM) is an efficient numerical algorithm for computation of long-ranged forces in $N$-body problems within gravitational and electrostatic fields. This method utilizes multipole expansions of the Green's function inherent to the underlying dynamical systems. Despite its widespread application in physics and engineering, the integration of FMM with modern machine learning architectures remains underexplored. In this work, we propose a novel neural network architecture, the Neural FMM, that integrates the information flow of the FMM into a hierarchical machine learning framework for learning the Green's operator of an Elliptic PDE. Our Neural FMM architecture leverages a hierarchical computation flow of the FMM method to split up the local and far-field interactions and efficiently learn their respective representations.
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