离散化
计算机科学
人工神经网络
快速多极方法
偏微分方程
操作员(生物学)
理论计算机科学
算法
应用数学
数学优化
数学
人工智能
多极展开
数学分析
生物化学
量子力学
转录因子
基因
物理
抑制因子
化学
作者
Zongyi Li,Nikola B. Kovachki,Kamyar Azizzadenesheli,Burigede Liu,Kaushik Bhattacharya,Andrew M. Stuart,Anima Anandkumar
出处
期刊:Cornell University - arXiv
日期:2020-01-01
被引量:109
标识
DOI:10.48550/arxiv.2006.09535
摘要
One of the main challenges in using deep learning-based methods for simulating physical systems and solving partial differential equations (PDEs) is formulating physics-based data in the desired structure for neural networks. Graph neural networks (GNNs) have gained popularity in this area since graphs offer a natural way of modeling particle interactions and provide a clear way of discretizing the continuum models. However, the graphs constructed for approximating such tasks usually ignore long-range interactions due to unfavorable scaling of the computational complexity with respect to the number of nodes. The errors due to these approximations scale with the discretization of the system, thereby not allowing for generalization under mesh-refinement. Inspired by the classical multipole methods, we propose a novel multi-level graph neural network framework that captures interaction at all ranges with only linear complexity. Our multi-level formulation is equivalent to recursively adding inducing points to the kernel matrix, unifying GNNs with multi-resolution matrix factorization of the kernel. Experiments confirm our multi-graph network learns discretization-invariant solution operators to PDEs and can be evaluated in linear time.
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