Discovering Governing Equations of Dynamical Systems

Abstract

Data-driven discovery methods are gradually changing the study of dynamical systems in many fields, such as engineering, physics, economics, biology, and chemistry. However, traditional methods of finding governing equations are based on mathematical derivation and require a lot of data, expertise, and computing time. This thesis explores automating the discovery of governing equations from data, especially using the Automatic Regression for Governing Equations (ARGOS) framework, which marks an advancement in the data-driven identification of dynamical systems. The research begins with an overview of data-centric engineering, underlining the essential statistical and machine learning methodologies. It then focuses on the Sparse Identification of Nonlinear Dynamical Systems (SINDy) framework to demonstrate the requirements of automating the identification process. Subsequent chapters present two innovative methods for automatically calculating derivatives and identifying partial differential equations (PDEs) from data with limited prior knowledge. These methods are developed to expand the usage of the ARGOS framework and rigorously benchmarked against other state-of-the-art algorithms using success rates with varying signal-to-noise ratios (SNRs) and sample sizes, demonstrating the accuracy of the proposed methods in model discovery. The application of the ARGOS framework is demonstrated through the case study of COVID-19 data in mainland China using Bayesian ARGOS, which is a new ARGOS extension and provides interpretable and meaningful results. The thesis also tackles the computational challenges and outlines future directions in automating the discovery of dynamical systems, including stochastic differential equations. Overall, this thesis has three contributions. First, it proposes an automatic framework to reduce manual parameter tuning for dynamical system identification. Second, it offers systematic testing templates, facilitating advancements across diverse scientific disciplines. Third, analysing COVID-19 data illustrates that the ARGOS framework is a candidate method for dealing with real-world problems.