Data-driven discovery and model reduction of complex systems
de Silva, Brian M
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Dynamical systems play an integral role in the continued success of scientific theories in describing and predicting the world around us. They are at the heart of countless scientific models, including electromagnetic theory (Maxwell's equations and Lorenz's force law), general relativity (the Einstein field equations), and quantum mechanics (the Schrodinger equation). This thesis considers three problems related to the practical use of dynamical systems in scientific inquiry: identifying a system, leveraging predictions from dynamical systems in applications, and efficiently solving a class of dynamical systems. We begin by applying the sparse identification of nonlinear dynamical systems (SINDy) technique for system identification to the canonical problem of falling bodies. Using an experimental dataset consisting of noisy measurements of sports balls dropped from a tall bridge, we highlight challenges faced by practitioners attempting to perform system identification when working with real-world datasets and provide strategies for overcoming them. We briefly present PySINDy, an open source software package in Python designed for the sparse identification of nonlinear dynamical systems. Next we turn to the problem of detecting sensor faults in airplane flight data. We propose a simple physics-based approach which learns a model for the underlying relationships between sensors. Online, the method predicts future sensor readings from current ones, flagging potential sensor failures when incoming measurements disagree with predictions. The overall model is a hybrid of ideas and methods from dynamic mode decomposition, Kalman filters, and machine learning. Its performance is demonstrated on two artificial, but realistic, flight datasets, and one real-world one. Finally, we introduce a numerical method for the efficient solution of parametrized elliptic partial differential equations (PDEs) on component-based domains. The method takes advantage of analytical properties of the differential operator defining the PDE, the component-based nature of the domain, and empirically-discovered structure in solutions of the PDE. The result is a compact reduced-order model which can be efficiently evaluated to approximate solutions to the PDE in a many-query setting.
- Applied mathematics