UC San Diego

Complex systems are found at the heart of the most pressing intellectual challenges of the 21st century. Weather and climate, ecological and neurobiological networks, many body quantum systems, and even the operation of human societies can be described as the interaction of simple components from which collectively emerge distinct phenomena. In this thesis, key concepts from dynamical systems theory and machine learning are employed to analyze complex systems in both neurobiological contexts and weather prediction. The first of these concepts is synchronization, from which data assimilation is derived in order to describe the optimal interpolation between an imperfect physical model and sparse/noisy data. Data assimilation is applied to biophysical problems such as the design of VLSI circuits for neuromorphic computing, as well as the estimation of a biological neuron's physical characteristics from voltage and calcium fluorescence measurements. Generalized synchronization is then deployed as the framework to examine the operation of recurrent neural networks, particularly the reservoir computing (RC) architecture. RC operates through the synchronization of a high dimensional artificial network together with chaotic input data, and is able to produce state of the art forecasting performance on a number of benchmark systems in numerical weather prediction. Invariant quantities such as the Lyapunov exponent spectrum and the fractal dimension of the RC are calculated and shown to be directly related to the input dynamics of the data. The concluding remarks unify both RC and data assimilation together into a machine learning forecasting cycle that produces forecasts from sparse and noisy data with no physical model.