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

Ye Yuan; Xiuting Li; Liang Li; Frank J Jiang; Xiuchuan Tang; Fumin Zhang; Jorge Goncalves; Henning U Voss; Han Ding; Jürgen Kurths

doi:10.1063/5.0160900

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

Chaos. 2023 Nov 1;33(11):113122. doi: 10.1063/5.0160900.

Authors

Ye Yuan¹, Xiuting Li², Liang Li³, Frank J Jiang⁴, Xiuchuan Tang⁵, Fumin Zhang⁶, Jorge Goncalves⁷, Henning U Voss⁸, Han Ding⁹, Jürgen Kurths¹⁰

Affiliations

¹ School of Artificial Intelligence and Automation, State Key Laboratory of Digital Manufacturing Equipments and Technology, Huazhong University of Science and Technology, Wuhan 430074, People's Republic of China.
² College of Informatics, Huazhong Agricultural University, Wuhan 430070, People's Republic of China.
³ School of Artificial Intelligence and Automation, Key Laboratory of Image Processing and Intelligent Control, Huazhong University of Science and Technology, Wuhan 430074, People's Republic of China.
⁴ Division of Decision and Control Systems, KTH Royal Institute of Technology, 10044 Stockholm, Sweden.
⁵ School of Automation, Tsinghua University, Beijing 100084, People's Republic of China.
⁶ School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, Georgia 30309, USA.
⁷ Department of Engineering, University of Cambridge, Cambridge, United Kingdom and the Luxembourg Centre for Systems Biomedicine, University of Luxembourg, L-4362 Belvaux, Esch-sur-Alzette, Luxembourg.
⁸ Cornell MRI Facility, College of Human Ecology, Cornell University, Ithaca, New York 10065, USA.
⁹ School of Mechanical Science and Engineering, State Key Laboratory of Digital Manufacturing Equipments and Technology, Huazhong University of Science and Technology, Wuhan 430074, People's Republic of China.
¹⁰ Research Domain IV-Transdisciplinary Concepts & Methods, Potsdam Institute for Climate Impact Research, Potsdam D-14415, Germany.

PMID: 37967264
DOI: 10.1063/5.0160900

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.