Machine discovery of partial differential equations from spatiotemporal data: A sparse Bayesian learning framework

Abstract

This study presents a general framework, namely, Sparse Spatiotemporal System Discovery (S3d), for discovering dynamical models given by Partial Differential Equations (PDEs) from spatiotemporal data. S3d is built on the recent development of sparse Bayesian learning, which enforces sparsity in the estimated PDEs. This approach enables a balance between model complexity and fitting error with theoretical guarantees. The proposed framework integrates Bayesian inference and a sparse priori distribution with the sparse regression method. It also introduces a principled iterative re-weighted algorithm to select dominant features in PDEs and solve for the sparse coefficients. We have demonstrated the discovery of the complex Ginzburg–Landau equation from a traveling-wave convection experiment, as well as several other PDEs, including the important cases of Navier–Stokes and sine-Gordon equations, from simulated data.