A Spectral Approach for Learning Spatiotemporal Neural Differential
  Equations
Mingtao Xia Xiangting Li Qijing Shen Tom Chou
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.