Rapidly developing machine learning methods has stimulated research interest in computationally reconstructing differential equations (DEs) from observational data which may provide additional insight into underlying causative mechanisms. In this paper, we propose a novel neural-ODE based method that uses spectral expansions in space to learn spatiotemporal DEs. The major advantage of our spectral neural DE learning approach is that it does not rely on spatial discretization, thus allowing the target spatiotemporal equations to contain long range, nonlocal spatial interactions that act on unbounded spatial domains. Our spectral approach is shown to be as accurate as some of the latest machine learning approaches for learning PDEs operating on bounded domains. By developing a spectral framework for learning both PDEs and integro-differential equations, we extend machine learning methods to apply to unbounded DEs and a larger class of problems.

翻译：快速发展的机器学习方法激发了从观测数据中计算重构微分方程的研究兴趣，这能为揭示潜在因果机制提供新的见解。本文提出一种基于神经常微分方程的新方法，该方法利用空间上的谱展开来学习时空微分方程。我们提出的谱神经微分方程学习方法的显著优势在于无需依赖空间离散化，因此允许目标时空方程包含作用于无界空间域的长程非局部空间相互作用。研究表明，我们的光谱方法在学习有界域上的偏微分方程时，与最新机器学习方法具有同等精度。通过建立学习偏微分方程和积分微分方程的谱框架，我们将机器学习方法扩展至无界微分方程及更广泛的问题类别。