Machine learning and data decompositions for complex networked dynamical systems
Lusch, Bethany Ann
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Machine learning has become part of our daily lives. Its applications include personalized advertisements, stock price predictions, and self-driving cars. The goal of this thesis is to study ways to apply machine learning to complex dynamical systems in science. Each of the methods we discuss involves an optimization problem to fit a model to data. First, we study pairwise-conditional Granger causality, a popular statistical method in fields such as economics and neuroscience for inferring causal connections from time series data. We systematically test this method on data generated by a nonlinear model with known network structure. We find significant discrepancies between the original and inferred networks, unless the true structure is extremely sparse or dense. This work illustrates that network inference is a fundamentally challenging task which needs further innovative developments to be accurate. Second, we develop a specialized tensor decomposition to extract important spatial modes from a data set and sparsely fit time dynamics from an over-complete library to each spatial mode. This decomposition is more readily interpretable than others because the output includes the analytic forms of the time dimension. It is especially intended for data sets that other methods struggle with due to transient and intermittent phenomena. We demonstrate its usage on real crime and climate data. Finally, we simulate damage from traumatic brain injury and neurodegenerative disease on artificial neural networks used in deep learning. We use well-established biophysical data on focal axonal swellings to quantitatively study the progress of impairments on our model of cognition. Our model provides intuitively appealing results about the manner in which cognitive impairments arise. Together, these methods demonstrate ways to use the frameworks of machine learning and optimization to study complex systems and advance science.
- Applied mathematics