Automated symbolic model identification for nonlinear dynamical systems from time-series data with limited sampling frequency and precision

The advancement in data science has influenced many fields in science and engineering. One revolutionizing application is the use of data-driven modeling methods to extract the governing equations of unknown systems from measurement data. Such modeling approaches require little a priori knowledge and have the potential to automate the scientific discovery process. Different from the traditional black-box modeling in system identification, these methods seek concise mathematical expressions to describe the system, which provide better explainability of the underlying process. One of the implementations of data-driven model discovery in the engineering field is the Sparse Identification of Nonlinear Dynamics (SINDy) method. This method utilizes sparse regression and compressed sensing techniques to find models that are linear in the parameters. We studied the behavior of the SINDy method under nonideal data, where both the sampling frequency and the measurement precision are limited, and proposed two improvements to make the method more robust in real-world applications. Under low sampling frequencies, the derivative approximated by numerical methods suffers from large systematic error. We solved this problem using neural-network-based system identification, utilizing adjacent samples in the phase space to improve the derivative estimation for time-invariant systems. Low measurement precision under low sampling frequencies makes noise hard to filter. We mitigated this problem by designing a sparse regression method with high noise tolerance, using random dropout and adaptive thresholding on ridge coefficient (DATRidge). The DATRidge method can find multiple candidate models of different sparsity and fitness. We use a Pareto front to present the results for making trade-offs. We tested our method on four model systems: the Van der Pol oscillator, the Lorenz system, the boiling water reactor (BWR) model by March-Leuba et al., and the glycolytic oscillator in bioengineering. Our method successfully recovered all the nonlinear polynomial equations in these systems, using data with limited sampling frequency and precision. In all the tests, our method shows a much higher tolerance to imperfect data than the SINDy method. As an extension, we studied the possibility of estimating variable relevance by applying layer-wise relevance propagation (LRP) on the neural network model. Since a dependent variable may not be affected by all other variables, excluding the irrelevant variable helps reduce the dimension of the search space. We show that due to the curse of dimensionality, the data required for making such estimations grows exponentially with the number of variables in the system. Since the trajectory of a dynamical system usually does not provide dense coverage of the phase space, the relevance estimation is generally not reliable.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Nuclear, Plasma, Radiolgc Engr

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2022

Author dc:creator

• Zhou, Long

Contributors dc:contributor

• Zhang, Yang
• Huff, Kathryn D.
• Bretl, Timothy Wolfe
• Uddin, Rizwan

• System Identification
• Black-box Modeling
• Nonlinear Systems

Statement dc:rights

• Copyright 2022 Long Zhou

Language dc:language

en, eng

Handle dc:identifier

https://hdl.handle.net/2142/117539

OAI record oai:identifier

https://www.ideals.illinois.edu/oai-pmh?verb=GetRecord&metadataPrefix=etdms&identifier=oai%3Awww.ideals.illinois.edu%3A2142%2F117539