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.
翻译:在使用深层次的学习方法模拟物理系统和解决部分差异方程式(PDEs)方面,使用深层次的模拟物理系统和解决部分差异方程式(PDEs)的主要挑战之一是在理想的神经网络结构中制定基于物理的数据。图形神经网络(GNNS)在这一领域越来越受欢迎,因为图形提供了模拟粒子相互作用的自然方式,并提供了将连续模型分解的清晰方式。但是,为类似任务而构造的图表通常忽略了远程互动,因为对节点数量而言,计算复杂性的大小不易降低。这些近似尺度导致系统离散,从而不允许在网状精化下进行泛化。在经典的多极方法的启发下,我们提出了一个新的多层次的图形神经网络框架,将所有范围的互动都包含线性复杂度。我们的多层次配方形配方相当于在内核矩阵中反复添加导引点,使GNNS与多分辨率矩阵因子化。实验证实了我们的多谱网络在光线性PDE公司中学习离式的内溶性内溶性解决方案。