Data-driven predictions of the future often suffer from error propagation and overconfidence. In many scenarios where forecasting is practical, the data follows a quasi-periodic pattern, which means that data from one point in time relays useful information about the data one period later. Through a Koopman theoretic approach, we make use of this feature to make long-term probabilistic forecasts that do not suffer from error propagation. Overwhelmingly, data-driven forecasts attempt to predict the exact value of a quantity into the future; however, such point-forecasts fail to describe the uncertainty in that quantity. Even when models are designed to make probabilistic forecasts, they are often overconfident and rely on the model to supply all sources of uncertainty. By forecasting the parameters of a probability distribution describing a quantity, rather than the quantity itself, we are also able to overcome the problem of overconfidence. Furthermore, our model relays novel and useful information about temporal patterns in the uncertainty of a dataset. We apply our model to electric load data from the 2017 Global Energy Forecasting Competition and show significant improvements compared to competing forecasts. I implemented the approach and ran experiments to explore the ways in which Koopman theory can be leveraged to robustly mitigate error and overconfidence in electric load forecasting. Our contributions to probabilistic forecasting of energy demand have the potential to lessen global warming, and the approach can be used in forecasting problems more broadly.

Systems of non-linear differential equations are often difficult to analytically solve, control, and analyze, whereas systems of linear differential equations are relatively straightforward to solve, control, and analyze because we possess tools to study systems of linear differential equations. Koopman analysis allows us to transform a system of non-linear differential equations into a linear system. The caveat is that sometimes, the resulting linear system is infinite-dimensional, meaning that the mapping between the original space and the Koopman space is infinite-dimensional. This poses a computational challenge because infinite dimensional vector spaces are difficult to computationally work with. Koopman analysis has traditionally been done on one fixed point (equilibrium). Previously, work has been done to study specific examples of systems that have closure i.e. systems with a finite-dimensional Koopman operator. In this project, we looked at how to apply Koopman operators to systems with multiple fixed points. We found effective eigenfunctions that linearize low-dimensional non-linear dynamical systems analytically if possible, computationally otherwise. Under the assumption that the right-hand side of the differential equations are polynomials, we identified appropriate eigenfunctions that linearize the Koopman space with a possible invertible mapping. Previously, we have been able to find a closed-form solution that generates eigenfunctions for one-dimensional systems that have a polynomial form. However, in practice, the resulting integral equation can be difficult to computationally solve with current methods, and edge cases such as singularities and asymptotes are not well understood. Using implicit SINDy (an algorithm that approximates dynamical systems given data), we attempted to find polynomial decompositions that allowed us to describe the eigenfunctions with rational polynomial functions. This work is significant because better understanding dynamical systems allows us to better understand dynamic fields such as natural disaster detection, the firing of neurons, and the spread of pandemics.

Neurons in the brain are dynamical in nature, maintaining constantly changing states. Neurons modulate voltage based on input currents and produce spikes when the voltage exceeds a certain threshold. Additional dynamics after spiking, called evoked after-spike currents, are important for computation and memory over time scales. The diversity of neuronal dynamics and the variability in parameters underlying them give rise to rich and varied dynamics across networks. We hypothesize that the complexity and diversity of biological dynamics in the brain play a critical role in predictive coding of temporally complex systems, and that diverse forms of after-spike currents enable computation over variable timescales. Current artificial neural networks (ANNs), that emulate the structure of biological neural networks, successfully learn relationships between static patterns but have difficulty learning dynamic patterns that change over time. We aim to incorporate complex biological dynamics and diversity in ANNs and thereby systematically explore the function of such dynamics in network computation and learning. To this end, we construct ANNs that express biologically realistic dynamics, developing methods to learn dynamics-generating parameters, such as membrane capacitance and threshold, in individual neurons. Theoretically, diverse dynamics of individual neurons will enable even more complex dynamics when combined in networks and may improve performance on tasks requiring computation over complex timescales, such as determining actions based on temporal patterns of cues. Hence, we hypothesize that when trained on temporally challenging tasks, our networks will learn diverse dynamics across neurons. We present the diversity of parameters learned and the resulting distribution of firing patterns and compare performance between our neural networks and traditional networks that only learn connection weights. This research will inform learning methods for training novel biologically inspired neural networks and will also shed light on the physiological role of diversity in the brain.