Physics-informed AI and ML-based sparse system identification algorithm for discovery of PDE's representing nonlinear dynamic systems

Abstract

Sparse system identification of nonlinear dynamic systems is still challenging, especially for stiff and high-order differential equations for noisy measurement data. The use of highly correlated functions makes distinguishing between true and false functions difficult, which limits the choice of functions. In this study, an equation discovery method has been proposed to tackle these problems. The key elements include a) use of B-splines for data fitting to get analytical derivatives superior to numerical derivatives, b) sequentially regularized derivatives for denoising (SRDD) algorithm, highly effective in removing noise from signal without system information loss, c) uncorrelated component analysis (UCA) algorithm that identifies and eliminates highly correlated functions while retaining the true functions, and d) physics-informed spline fitting (PISF) where the spline fitting is updated gradually while satisfying the governing equation with a dictionary of candidate functions to converge to the correct equation sequentially. The complete framework is built on a unified deep-learning architecture that eases the optimization process. The proposed method is demonstrated to discover various differential equations at various noise levels, including three-dimensional, fourth-order, and stiff equations. The parameter estimation converges accurately to the true values with a small coefficient of variation, suggesting robustness to the noise.

References in corpus (10)

• Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations
• Integrating Machine Learning and Multiscale Modeling: Perspectives, Challenges, and Opportunities in the Biological, Biomedical, and Behavioral Sciences
• PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Network
• Data-driven approximation of the Koopman generator: Model reduction, system identification, and control
• A Hybrid Science-Guided Machine Learning Approach for Modeling and Optimizing Chemical Processes
• DeepMoD: Deep learning for Model Discovery in noisy data
• Robust learning from noisy, incomplete, high-dimensional experimental data via physically constrained symbolic regression
• Sparse Identification of Nonlinear Dynamical Systems via Reweighted ${\ell }_1$-regularized Least Squares
• Parsimony-Enhanced Sparse Bayesian Learning for Robust Discovery of Partial Differential Equations
• Discovering Nonlinear PDEs from Scarce Data with Physics-encoded Learning