PDE dataset (Parametric Partial Differential Equation dataset)

使用基于深度学习的方法来模拟物理系统和求解偏微分方程 (PDE) 的主要挑战之一是在神经网络所需的结构中制定基于物理的数据。图神经网络 (GNN) 在这一领域得到了普及，因为图提供了一种模拟粒子相互作用的自然方式，并提供了一种离散化连续模型的清晰方法。然而，由于计算复杂度相对于节点数量的不利缩放，为逼近此类任务而构建的图通常会忽略远程交互。由于这些近似值导致的误差随着系统的离散化而缩放，因此不允许在网格细化下进行泛化。受经典多极方法的启发，我们提出了一种新颖的多级图神经网络框架，该框架仅以线性复杂度捕获所有范围内的交互。我们的多级公式相当于递归地向核矩阵添加诱导点，将 GNN 与核的多分辨率矩阵分解统一起来。实验证实，我们的多图网络学习了 PDE 的离散化不变解算子，并且可以在线性时间内进行评估。

One of the main challenges in using deep learning-based methods to simulate physical systems and solve partial differential equations (PDEs) is formulating physics-based data within the structure required by neural networks. Graph neural networks (GNNs) have gained popularity in this field, as graphs provide a natural way to model particle interactions and a clear approach to discretizing continuous models. However, due to the unfavorable scaling of computational complexity with respect to the number of nodes, the graphs constructed for such approximation tasks often overlook long-range interactions. The errors resulting from these approximations scale with the discretization of the system, thus failing to generalize under grid refinement. Motivated by classical multipole methods, we propose a novel multilevel graph neural network framework that captures interactions across all ranges with only linear complexity. Our multilevel formulation is equivalent to recursively adding inducing points to the kernel matrix, unifying GNNs with the multiresolution matrix decomposition of kernels. Experiments confirm that our multi-graph network learns discretization-invariant solution operators for PDEs and can be evaluated in linear time.

搜集汇总

背景与挑战

背景概述

该数据集旨在支持基于深度学习的物理系统模拟和偏微分方程求解，通过图神经网络解决远程交互捕获的挑战。它提出了一种多级图神经网络框架，以线性复杂度实现离散化不变解算子，并在实验中验证了其有效性。