Sparse parametric data-driven model identification for intermittently and latently forced dynamical systems

Abstract

This study explores data-driven, sparse Ordinary Differential Equation (ODE) – based modelling techniques for mechanical machinery, aiming to enhance diagnostic and prognostic capabilities. Effectively using current methods, such as Discrete Fourier Transforms and Kurtosis analysis, requires extensive domain knowledge, whereas interpretable data-driven ODE models offer a more intuitive approach. The Sparse Identification of Nonlinear Dynamics (SINDy) framework is employed to identify sparse parametric ODEs. However, SINDy faces challenges with normalised datasets and non-autonomous systems, such as those with intermittent and latent forces. Two novel heuristics for SINDy are proposed: the first heuristic is a new regression strategy, Sequential Thresholding of the Coefficient of Variation (STCV), which uses a magnitude-free regularisation approach, making it robust against data normalisation. In comparison, STCV represents an advancement over the Ensemble SINDy (E-SINDy) approach. While E-SINDy uses bootstrap aggregating to mitigate overfitting, STCV applies Bayesian Linear Regression for computational efficiency and employs a magnitude-free regularisation approach, enhancing its robustness to data normalisation. The second heuristic allows SINDy to model non-autonomous systems by filtering out significant, temporally sparse errors. This method, named Non-Autonomous SINDy (NAut-SINDy), employs model state prediction error and kinetic energy normalisation, enhancing its application to a broader range of systems. The effectiveness of STCV and NAut-SINDy is showcased through various examples, including simulated Lorenz, Rossler, Van der Poll, and Duffing oscillators and real-world scenarios involving damaged bearings and half-car suspension models. NAut-SINDy's utility particularly emphasises diagnostics, making it possible to identify internal crack shapes in bearings and updating vehicle suspension models online to detect damage. These advancements in SINDy enhance its practicality and accuracy in modelling complex engineering systems.