A Deep Learning Approach for Predicting Spatiotemporal Dynamics From Sparsely Observed Data

TL;DR

This paper introduces a deep learning framework capable of predicting the evolution of unknown PDE-driven spatiotemporal processes from sparse data, demonstrating flexibility across various geometries and dimensions.

Contribution

The proposed method uniquely learns dynamics from sparse, unstructured data without prior knowledge of PDEs, and is spatially dimension-independent and geometrically flexible.

Findings

Successfully forecasted 2D wave and Burgers-Fisher equations in various geometries.

Accurately predicted 10D heat equation dynamics from sparse data.

Outperformed existing methods in handling unknown PDEs with sparse observations.

Abstract

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.