Uncertainty Analysis and Experimental Validation of Identifying the Governing Equation of an Oscillator Using Sparse Regression

In recent years, the rapid growth of computing technology has enabled identifying mathematical models for vibration systems using measurement data instead of domain knowledge. Within this category, the method Sparse Identification of Nonlinear Dynamical Systems (SINDy) shows potential for interpretable identification. Therefore, in this work, a procedure of system identification based on the SINDy framework is developed and validated on a single-mass oscillator. To estimate the parameters in the SINDy model, two sparse regression methods are discussed. Compared with the Least Squares method with Sequential Threshold (LSST), which is the original estimation method from SINDy, the Least Squares method Post-LASSO (LSPL) shows better performance in numerical Monte Carlo Simulations (MCSs) of a single-mass oscillator in terms of sparseness, convergence, identified eigenfrequency, and coefficient of determination. Furthermore, the developed method SINDy-LSPL was successfully implemented with real measurement data of a single-mass oscillator with known theoretical parameters. The identified parameters using a sweep signal as excitation are more consistent and accurate than those identified using impulse excitation. In both cases, there exists a dependency of the identified parameter on the excitation amplitude that should be investigated in further research.

Identification of time series models using sparse Takagi–Sugeno fuzzy systems with reduced structure

Simplifying fuzzy models, including those for predicting time series, is an important issue in terms of their interpretation and implementation. This simplification can involve both the number of inference rules (i.e., structure) and the number of parameters. This paper proposes novel hybrid methods for time series prediction that utilize Takagi–Sugeno fuzzy systems with reduced structure. The fuzzy sets are obtained using a global optimization algorithm (particle swarm optimization, simulated annealing, genetic algorithm, or pattern search). The polynomials are determined by elastic net regression, which is a sparse regression. The simplification is based on reducing the number of polynomial parameters in the then-part by using sparse regression and removing unnecessary rules by using labels. A new quality criterion is proposed to express a compromise between the model accuracy and its simplification. The experimental results show that the proposed methods can improve a fuzzy model while simplifying its structure.

Boosting the Model Discovery of Hybrid Dynamical Systems in An Informed Sparse Regression Approach

We present an efficient data-driven sparse identification of dynamical systems. The work aims at reconstructing the different sets of governing equations and identifying discontinuity surfaces in hybrid systems when the number of discontinuities is known a priori. In a two-stages approach, we first locate the switches between separate vector fields. Then, the dynamics among the manifolds are regressed, in this case by making use of the existing algorithm of Brunton et al. [1]. The reconstruction of the discontinuity surfaces comes as the outcome of a statistical analysis implemented via symbolic regression with small clusters (micro-clusters) and a rigid library of models. These allow to classify all the feasible discontinuities that are clustered and to reduce them into the actual discontinuity surfaces. The performances of the sparse regression hybrid model discovery are tested on two numerical examples, namely, a canonical spring-mass hopper and a free/impact electromagnetic energy harvester, engineering archetypes characterized by the presence of a single and double discontinuity, respectively. Results show that a supervised approach, i.e. where the number of discontinuities is preassigned, is computationally efficient and it determines accurately both discontinuities and set of governing equations. A large improvement in the time of computation is found with the maximum achievable reliability. Informed regression-based identification offers the prospect to outperform existing data-driven identification approaches for hybrid systems at the expense of instructing the algorithm for expected discontinuities.

Physics-informed learning of governing equations from scarce data

Harnessing data to discover the underlying governing laws or equations that describe the behavior of complex physical systems can significantly advance our modeling, simulation and understanding of such systems in various science and engineering disciplines. This work introduces a novel approach called physics-informed neural network with sparse regression to discover governing partial differential equations from scarce and noisy data for nonlinear spatiotemporal systems. In particular, this discovery approach seamlessly integrates the strengths of deep neural networks for rich representation learning, physics embedding, automatic differentiation and sparse regression to approximate the solution of system variables, compute essential derivatives, as well as identify the key derivative terms and parameters that form the structure and explicit expression of the equations. The efficacy and robustness of this method are demonstrated, both numerically and experimentally, on discovering a variety of partial differential equation systems with different levels of data scarcity and noise accounting for different initial/boundary conditions. The resulting computational framework shows the potential for closed-form model discovery in practical applications where large and accurate datasets are intractable to capture.