SNN-PDE: Learning Dynamic PDEs from Data with Simplicial Neural Networks

Dynamics of many complex systems, from weather and climate to spread of infectious diseases, can be described by partial differential equations (PDEs). Such PDEs involve unknown function(s), partial derivatives, and typically multiple independent variables. The traditional numerical methods for solving PDEs assume that the data are observed on a regular grid. However, in many applications, for example, weather and air pollution monitoring delivered by the arbitrary located weather stations of the National Weather Services, data records are irregularly spaced. Furthermore, in problems involving prediction analytics such as forecasting wildfire smoke plumes, the primary focus may be on a set of irregular locations associated with urban development. In recent years, deep learning (DL) methods and, in particular, graph neural networks (GNNs) have emerged as a new promising tool that can complement traditional PDE solvers in scenarios of the irregular spaced data, contributing to the newest research trend of physics informed machine learning (PIML). However, most existing PIML methods tend to be limited in their ability to describe higher dimensional structural properties exhibited by real world phenomena, especially, ones that live on manifolds. To address this fundamental challenge, we bring the elements of the Hodge theory and, in particular, simplicial convolution defined on the Hodge Laplacian to the emerging nexus of DL and PDEs. In contrast to conventional Laplacian and the associated convolution operation, the simplicial convolution allows us to rigorously describe diffusion across higher order structures and to better approximate the complex underlying topology and geometry of the data. The new approach, Simplicial Neural Networks for Partial Differential Equations (SNN PDE) offers a computationally efficient yet effective solution for time dependent PDEs. Our studies of a broad range of synthetic data and wildfire processes demonstrate that SNN PDE improves upon state of the art baselines in handling unstructured grids and irregular time intervals of complex physical systems and offers competitive forecasting capabilities for weather and air quality forecasting.

从天气、气候到传染病传播的诸多复杂系统，其动力学行为均可通过偏微分方程（Partial Differential Equations, PDEs）进行描述。此类偏微分方程包含未知函数、偏导数项，且通常涉及多个独立变量。传统的偏微分方程数值求解方法均假设数据采集自规则网格。然而在诸多实际应用场景中，例如由国家气象局（National Weather Services）任意布设的气象站所开展的天气与空气污染监测数据，其记录的空间分布往往呈现不规则性。此外，在野火烟雾羽流预报等预测分析任务中，研究的核心关注点往往集中在与城市开发相关的一系列不规则点位上。近年来，深度学习（Deep Learning, DL）方法，尤其是图神经网络（Graph Neural Networks, GNNs），已成为极具前景的新型工具，可在不规则分布数据场景下弥补传统偏微分方程求解器的不足，助力物理知情机器学习（Physics Informed Machine Learning, PIML）这一前沿研究方向的发展。然而，当前多数已有的物理知情机器学习方法，在刻画真实世界现象所展现的高维结构特性方面存在局限，尤其是针对定义于流形之上的相关问题。为解决这一基础性挑战，我们将霍奇理论（Hodge theory）的相关元素——尤其是定义于霍奇拉普拉斯算子（Hodge Laplacian）上的单纯形卷积——引入深度学习与偏微分方程的新兴交叉研究领域。与传统拉普拉斯算子及其配套的卷积操作不同，单纯形卷积能够严格刻画高阶结构间的扩散过程，更精准地逼近数据复杂的底层拓扑与几何特性。这一全新方法——面向偏微分方程的单纯形神经网络（Simplicial Neural Networks for Partial Differential Equations, SNN PDE）——为时变偏微分方程提供了一种计算高效且效果优异的解决方案。我们针对大量合成数据集与野火演化过程开展的研究表明，SNN PDE在处理复杂物理系统的非结构化网格与不规则时间间隔任务上，性能优于现有最优基准模型，同时在天气与空气质量预报任务中展现出极具竞争力的预测能力。