All Times ET
Keywords: Dynamical Systems, System Identification, Statistical Learning, Sparse Regression, Optimization, Adaptive Lasso
As data-centric engineering continues to expand and we proceed to collect, store, and manipulate data in more prolific quantities, the ability to automatically extract governing equations from this information remains a crucial challenge in the fields of data science and engineering. Although data surrounds us, developing models to describe nonlinear dynamical systems such as climate and stock market trends, fluid flow dynamics, and epidemiology is still a strenuous task. Currently, engineers construct generalized linear models manually to identify, expand, and forecast these systems over time; a process better known as system identification. With this process, the only assumption is that just a few essential terms regulate the dynamics of the model's underlying structure, which holds for many physical systems (Brunton et al., 2016). Therefore, we represent the equations using a sparse functional basis. Here, we aim to develop a method to determine equations automatically since these data sets are often otherwise intractable. We thus employ several optimization techniques to perform system identification by extracting a sparse solution for these dynamic equations to visualize and interpret the data accurately.
In this work, we outline our Automatic Sparse Regression (TAPER) algorithm, which provides an iterative process that implements various statistical learning methods to identify nonlinear dynamical systems and extract their governing equations from data. We then demonstrate the TAPER algorithm's ability to yield the correct dynamic representation of the Lorenz chaotic system, a model used to describe atmospheric convection, with a minimal number of observations necessary. Based on these results, our algorithm develops an automated process for the identification of the nonlinear Lorenz equations, which offers advancement to state-of-the-art semi-automated methods by at least one order of magnitude (Brunton et al., 2016).