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

Area of Science:

• Dynamical systems theory
• Computational physics
• Machine learning

Background:

• Discovering governing Partial Differential Equations (PDEs) from spatiotemporal data is crucial for scientific modeling.
• Existing methods often struggle with model complexity and feature selection.
• Sparse Bayesian learning offers a promising approach for parsimonious model identification.

Purpose of the Study:

• To present a general framework, Sparse Spatiotemporal System Discovery (S3d), for discovering dynamical models from spatiotemporal data.
• To leverage sparse Bayesian learning for identifying sparse Partial Differential Equations (PDEs).
• To balance model complexity and fitting error with theoretical guarantees.

Main Methods:

• Integration of Bayesian inference with a sparse prior distribution and sparse regression.
• Development of a principled iterative re-weighted algorithm for dominant feature selection.
• Application of the framework to both experimental and simulated spatiotemporal data.

Main Results:

• Successful discovery of the complex Ginzburg-Landau equation from experimental traveling-wave convection data.
• Accurate identification of other significant PDEs, including Navier-Stokes and sine-Gordon equations, from simulated data.
• Demonstration of S3d's capability in uncovering complex dynamical systems.

Conclusions:

• The Sparse Spatiotemporal System Discovery (S3d) framework effectively identifies dynamical models represented by Partial Differential Equations (PDEs).
• The integration of sparse Bayesian learning provides a robust method for feature selection and model parsimony.
• S3d offers a powerful tool for scientific discovery from spatiotemporal data.