An improved sparse identification of nonlinear dynamics with Akaike information criterion and group sparsity.

A crucial challenge encountered in diverse areas of engineering applications involves speculating the governing equations based upon partial observations. On this basis, a variant of the sparse identification of nonlinear dynamics (SINDy) algorithm is developed. First, the Akaike information criterion (AIC) is integrated to enforce model selection by hierarchically ranking the most informative model from several manageable candidate models. This integration avoids restricting the number of candidate models, which is a disadvantage of the traditional methods for model selection. The subsequent procedure expands the structure of dynamics from ordinary differential equations (ODEs) to partial differential equations (PDEs), while group sparsity is employed to identify the nonconstant coefficients of partial differential equations. Of practical consideration within an integrated frame is data processing, which tends to treat noise separate from signals and tends to parametrize the noise probability distribution. In particular, the coefficients of a species of canonical ODEs and PDEs, such as the Van der Pol, Rössler, Burgers' and Kuramoto-Sivashinsky equations, can be identified correctly with the introduction of noise. Furthermore, except for normal noise, the proposed approach is able to capture the distribution of uniform noise. In accordance with the results of the experiments, the computational speed is markedly advanced and possesses robustness.

SINDy-SA framework: enhancing nonlinear system identification with sensitivity analysis.

Machine learning methods have revolutionized studies in several areas of knowledge, helping to understand and extract information from experimental data. Recently, these data-driven methods have also been used to discover structures of mathematical models. The sparse identification of nonlinear dynamics (SINDy) method has been proposed with the aim of identifying nonlinear dynamical systems, assuming that the equations have only a few important terms that govern the dynamics. By defining a library of possible terms, the SINDy approach solves a sparse regression problem by eliminating terms whose coefficients are smaller than a threshold. However, the choice of this threshold is decisive for the correct identification of the model structure. In this work, we build on the SINDy method by integrating it with a global sensitivity analysis (SA) technique that allows to hierarchize terms according to their importance in relation to the desired quantity of interest, thus circumventing the need to define the SINDy threshold. The proposed SINDy-SA framework also includes the formulation of different experimental settings, recalibration of each identified model, and the use of model selection techniques to select the best and most parsimonious model. We investigate the use of the proposed SINDy-SA framework in a variety of applications. We also compare the results against the original SINDy method. The results demonstrate that the SINDy-SA framework is a promising methodology to accurately identify interpretable data-driven models. Supplementary Information: The online version contains supplementary material available at 10.1007/s11071-022-07755-2.

Low-Computational-Cost Hybrid FEM-Analytical Induction Machine Model for the Diagnosis of Rotor Eccentricity, Based on Sparse Identification Techniques and Trigonometric Interpolation.

Since it is not efficient to physically study many machine failures, models of faulty induction machines (IMs) have attracted a rising interest. These models must be accurate enough to include fault effects and must be computed with relatively low resources to reproduce different fault scenarios. Moreover, they should run in real time to develop online condition-monitoring (CM) systems. Hybrid finite element method (FEM)-analytical models have been recently proposed for fault diagnosis purposes since they keep good accuracy, which is widely accepted, and they can run in real-time simulators. However, these models still require the full simulation of the FEM model to compute the parameters of the analytical model for each faulty scenario with its corresponding computing needs. To address these drawbacks (large computing power and memory resources requirements) this paper proposes sparse identification techniques in combination with the trigonometric interpolation polynomial for the computation of IM model parameters. The proposed model keeps accuracy similar to a FEM model at a much lower computational effort, which could contribute to the development and to the testing of condition-monitoring systems. This approach has been applied to develop an IM model under static eccentricity conditions, but this may extend to other fault types.

Nonlinear parametric models of viscoelastic fluid flows.

Reduced-order models (ROMs) have been widely adopted in fluid mechanics, particularly in the context of Newtonian fluid flows. These models offer the ability to predict complex dynamics, such as instabilities and oscillations, at a considerably reduced computational cost. In contrast, the reduced-order modelling of non-Newtonian viscoelastic fluid flows remains relatively unexplored. This work leverages the sparse identification of nonlinear dynamics (SINDy) algorithm to develop interpretable ROMs for viscoelastic flows. In particular, we explore a benchmark oscillatory viscoelastic flow on the four-roll mill geometry using the classical Oldroyd-B fluid. This flow exemplifies many canonical challenges associated with non-Newtonian flows, including transitions, asymmetries, instabilities, and bifurcations arising from the interplay of viscous and elastic forces, all of which require expensive computations in order to resolve the fast timescales and long transients characteristic of such flows. First, we demonstrate the effectiveness of our data-driven surrogate model to predict the transient evolution and accurately reconstruct the spatial flow field for fixed flow parameters. We then develop a fully parametric, nonlinear model capable of capturing the dynamic variations as a function of the Weissenberg number. While the training data are predominantly concentrated on a limit cycle regime for moderate W i , we show that the parametrized model can be used to extrapolate, accurately predicting the dominant dynamics in the case of high Weissenberg numbers. The proposed methodology represents an initial step in applying machine learning and reduced-order modelling techniques to viscoelastic flows.

Discovering dynamic models of COVID-19 transmission.

Existing models about the dynamics of COVID-19 transmission often assume the mechanism of virus transmission and the form of the differential equations. These assumptions are hard to verify. Due to the biases of country-level data, it is inaccurate to construct the global dynamic of COVID-19. This research aims to provide a robust data-driven global model of the transmission dynamics. We apply sparse identification of nonlinear dynamics (SINDy) to model the dynamics of COVID-19 global transmission. One advantage is that we can discover the nonlinear dynamics from data without assumptions in the form of the governing equations. To overcome the problem of biased country-level data on the number of reported cases, we propose a robust global model of the dynamics by using maximin aggregation. Real data analysis shows the efficiency of our model.

Sparse nonlinear models of chaotic electroconvection.

Convection is a fundamental fluid transport phenomenon, where the large-scale motion of a fluid is driven, for example, by a thermal gradient or an electric potential. Modelling convection has given rise to the development of chaos theory and the reduced-order modelling of multiphysics systems; however, these models have been limited to relatively simple thermal convection phenomena. In this work, we develop a reduced-order model for chaotic electroconvection at high electric Rayleigh number. The chaos in this system is related to the standard Lorenz model obtained from Rayleigh-Benard convection, although our system is driven by a more complex three-way coupling between the fluid, the charge density, and the electric field. Coherent structures are extracted from temporally and spatially resolved charge density fields via proper orthogonal decomposition (POD). A nonlinear model is then developed for the chaotic time evolution of these coherent structures using the sparse identification of nonlinear dynamics (SINDy) algorithm, constrained to preserve the symmetries observed in the original system. The resulting model exhibits the dominant chaotic dynamics of the original high-dimensional system, capturing the essential nonlinear interactions with a simple reduced-order model.

Sparse identification of nonlinear dynamics for model predictive control in the low-data limit.

Data-driven discovery of dynamics via machine learning is pushing the frontiers of modelling and control efforts, providing a tremendous opportunity to extend the reach of model predictive control (MPC). However, many leading methods in machine learning, such as neural networks (NN), require large volumes of training data, may not be interpretable, do not easily include known constraints and symmetries, and may not generalize beyond the attractor where models are trained. These factors limit their use for the online identification of a model in the low-data limit, for example following an abrupt change to the system dynamics. In this work, we extend the recent sparse identification of nonlinear dynamics (SINDY) modelling procedure to include the effects of actuation and demonstrate the ability of these models to enhance the performance of MPC, based on limited, noisy data. SINDY models are parsimonious, identifying the fewest terms in the model needed to explain the data, making them interpretable and generalizable. We show that the resulting SINDY-MPC framework has higher performance, requires significantly less data, and is more computationally efficient and robust to noise than NN models, making it viable for online training and execution in response to rapid system changes. SINDY-MPC also shows improved performance over linear data-driven models, although linear models may provide a stopgap until enough data is available for SINDY. SINDY-MPC is demonstrated on a variety of dynamical systems with different challenges, including the chaotic Lorenz system, a simple model for flight control of an F8 aircraft, and an HIV model incorporating drug treatment.