A Deep Learning-based Collocation Method for Modeling Unknown PDEs from Sparse Observation

Deep learning-based modeling of dynamical systems driven by partial differential equations (PDEs) has become quite popular in recent years. However, most of the existing deep learning-based methods either assume strong physics prior, or depend on specific initial and boundary conditions, or require data in dense regular grid making them inapt for modeling unknown PDEs from sparsely-observed data. This paper presents a deep learning-based collocation method for modeling dynamical systems driven by unknown PDEs when data sites are sparsely distributed. The proposed method is spatial dimension-independent, geometrically flexible, learns from sparsely-available data and the learned model does not depend on any specific initial and boundary conditions. We demonstrate our method in the forecasting task for two-dimensional wave equation and Burgers-Fisher equation in multiple geometries with different boundary conditions.