Recovery of Differential Equations from Impulse Response Time Series Data for Model Identification and Feature Extraction

Abstract

Time recordings of impulse-type oscillation responses are short and highly transient. These characteristics may complicate the usage of classical spectral signal processing techniques for a) describing the dynamics and b) deriving discriminative features from the data. However, common model identification and validation techniques mostly rely on steady-state recordings, characteristic spectral properties and non-transient behavior. In this work, a recent method, which allows reconstructing differential equations from time series data, is extended for higher degrees of automation. With special focus on short and strongly damped oscillations, an optimization procedure is proposed that fine-tunes the reconstructed dynamical models with respect to model simplicity and error reduction. This framework is analyzed with particular focus on the amount of information available to the reconstruction, noise contamination and non-linearities contained in the time series input. Using the example of a mechanical oscillator, we illustrate how the optimized reconstruction method can be used to identify a suitable model and to extract features from uni-variate and multivariate time series recordings in an engineering-compliant environment. Moreover, the determined minimal models allow for identifying the qualitative nature of the underlying dynamical systems as well as testing for the degree and strength of non-linearity. The reconstructed differential equations would then be potentially available for classical numerical studies, such as bifurcation analysis. These results represent a physically interpretable enhancement of data-driven modeling approaches in structural dynamics.

Reduced Order Dynamical Models For Complex Dynamics in Manufacturing and Natural Systems Using Machine Learning

“…To improve the performance of our SINDy-recovered models, we further optimize the associated sparsity matrix, Ξ, by applying a constrained nonlinear optimization scheme. This optimization beyond the SINDy method is based on the work of [5], which suggests that while SINDy is capable of finding the location of nonzero elements of Ξ, it cannot necessarily find optimal values for each since Ξ is discontinuous over λ [5]. We apply the method outlined in [5] of sequential quadratic programming (SQP) implemented using MATLAB's fmincon.…”

“…This optimization beyond the SINDy method is based on the work of [5], which suggests that while SINDy is capable of finding the location of nonzero elements of Ξ, it cannot necessarily find optimal values for each since Ξ is discontinuous over λ [5]. We apply the method outlined in [5] of sequential quadratic programming (SQP) implemented using MATLAB's fmincon. Here we set upper and lower bounds for each nonzero element of Ξ as the given constraints to fmincon and construct an optimization function using the mean absolute error (MAE) across all state variables between the training data and the integrated model.…”

“…In [3], the SINDy algorithm outlined in [4] is expanded to include methods for handling control and forcing inputs to a system.The authors show the performance of this algorithm on both a Lotka-Volterra predator-prey model and the Lorenz System with external inputs to simultaneously demonstrate the failure of naive SINDy, with no consideration of input, and the success of the modified SINDy algorithm when these inputs are taken into account. The authors of [5] suggest that the SINDy algorithm described in [4] and [3] may be further improved by employing bounded nonlinear optimization of the model coefficients. We apply both the input and optimization modification in our implementation of SINDy.…”

On spike-and-slab priors for Bayesian equation discovery of nonlinear dynamical systems via sparse linear regression

“…Spurred on by the rapid increase in computational power and growing rates of data collection, recent years have seen a booming interest in discovering governing differential equations of motion of nonlinear dynamical systems from time-series data [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. Governing differential equations of motion are ordinary or partial differential equations that characterise system behaviour and provide an understanding of the physics of the underlying phenomena.…”

“…Since then, the sparse regression approach for datadriven equation discovery of differential equations has been further developed in many studies. Examples include sparse identification of biological networks with rational basis functions [4], model selection using an integral formulation of the differential equation to reduce noise effects [5], model selection for dynamical selection combining sparse regression and information criteria [6], extension of sparse identification to nonlinear systems with control [7], discovery of coordinates for sparse representation of governing equations [8], extracting structured differential equations with under-sampled data [9], sparse learning of stochastic dynamical equations [10], model selection for nonlinear dynamical systems with switching behaviour [11], recovery of differential equations from short impulse response time-series data [12], identification of parametric partial differential equations [13][14][15]. There are also studies that proposed black-box approaches using deep neural networks [16][17][18] for equation discovery of differential equations; however, they are mostly useful for forecasting and do not provide explicit equations for interpretation.…”