论文标题
通过图神经网络从稀疏数据中学习连续的时间PDE
Learning continuous-time PDEs from sparse data with graph neural networks
论文作者
论文摘要
许多动态系统的行为遵循复杂但仍未知的偏微分方程（PDE）。虽然已经提出了几种机器学习方法直接从数据中学习PDE，但以前的方法仅限于离散的时间近似或对定期网格的观测值的限制假设。我们为动态系统提出了一个通用的连续时间差异模型，该系统通过消息传递图形神经网络对其管理方程进行参数化。该模型接受任意空间和时间离散化，从而消除了观察点和时间间隔的位置的约束。该模型通过连续的伴随方法训练，从而实现有效的神经PDE推断。我们证明了该模型与非结构化网格，任意时间步骤和嘈杂观察的能力。我们将我们的方法与几种涉及以最先进的预测性能的第一和高阶PDE的著名物理系统的现有方法进行了比较。
The behavior of many dynamical systems follow complex, yet still unknown partial differential equations (PDEs). While several machine learning methods have been proposed to learn PDEs directly from data, previous methods are limited to discrete-time approximations or make the limiting assumption of the observations arriving at regular grids. We propose a general continuous-time differential model for dynamical systems whose governing equations are parameterized by message passing graph neural networks. The model admits arbitrary space and time discretizations, which removes constraints on the locations of observation points and time intervals between the observations. The model is trained with continuous-time adjoint method enabling efficient neural PDE inference. We demonstrate the model's ability to work with unstructured grids, arbitrary time steps, and noisy observations. We compare our method with existing approaches on several well-known physical systems that involve first and higher-order PDEs with state-of-the-art predictive performance.
