  In this paper, we consider the problem of learning prediction models for
spatiotemporal physical processes driven by unknown partial differential
equations (PDEs). We propose a deep learning framework that learns the
underlying dynamics and predicts its evolution using sparsely distributed data
sites. Deep learning has shown promising results in modeling physical dynamics
in recent years. However, most of the existing deep learning methods for
modeling physical dynamics either focus on solving known PDEs or require data
in a dense grid when the governing PDEs are unknown. In contrast, our method
focuses on learning prediction models for unknown PDE-driven dynamics only from
sparsely observed data. The proposed method is spatial dimension-independent
and geometrically flexible. We demonstrate our method in the forecasting task
for the two-dimensional wave equation and the Burgers-Fisher equation in
multiple geometries with different boundary conditions, and the ten-dimensional
heat equation.
