Next week AIMdyn’s team of scientists will present their research at the SIAM Conference on Applications of Dynamical Systems (DS23) that will be held in Portland, OR. See below for brief descriptions of our team’s talks.
Monday, May 15. 9:00-9:25
Koopman Reduced Order Modeling with Confidence Bounds
Abstract. This talk introduces a reduced order modeling technique based on Koopman operator theory that gives confidence bounds on the model’s predictions. It is based on a data-driven spectral decomposition of said operator. The reduced order model is constructed using a finite number of Koopman eigenvalues and modes while the rest of spectrum is treated as a noise process. This noise process is used to extract the confidence bounds. Additionally, we propose a heuristic algorithm to choose the number of deterministic modes to keep in the model. We assume Gaussian observational noise in our models. As the number of modes used for the reduced order model increases, we approach a deterministic plus Gaussian noise model. The Gaussian-ity of the noise can be measured via a Shapiro-Wilk test. As the number of modes increase, the modal noise better approximates a Gaussian distribution. As the number of modes increases past the threshold, the standard deviation of the modal distribution decreases rapidly. This allows us to propose a heuristic algorithm for choosing the number of deterministic modes to keep for the model.
Monday, May 15. 4:45-5:10
A Koopman Operator Theoretic Approach for Studying Deep Learning
William T. Redman, Maria Fonoberova, Ryan Mohr, Ioannis G. Kevrekidis, Igor Mezic
Abstract. Deep learning exists as something of a contradiction. On the one hand, there has been an immense amount of practical development on methods that are able to perform highly complex tasks with remarkable success. On the other hand, there is a great lack of theoretically grounded understanding of all but the simplest machine learning paradigms. An outstanding problem then is to bridge this gap and provide meaningful, quantitative analysis of state-of-the-art deep learning models that can be connected to theory. In this talk, I will argue that Koopman operator theory, by its nature of being a data-driven dynamical systems framework that is intimately connected to the geometrical state-space theory that emerged over the past century, is a natural choice for achieving this bridging. I will highlight the success recent work that has used Koopman operator theory to analyze, optimize, and design deep neural networks has found and discuss open questions in this growing subfield.
Monday, May 15. 5:45-6:10
Koopman Operator Methods for Analysis and Prediction of Arctic Sea Ice Dynamics
Alan Cao, James Hogg, Maria Fonoberova, Qinghua Ding, Igor Mezic
Abstract. Recent reports have noted the rapid decay of sea ice concentration (SIC) in the Barents Sea region of the Arctic, indicative of a potential “tipping point” with the expected increases in global temperature. We present results based on Koopman mode analysis (KMA) of NSIDC SIC data showing Koopman eigenvalues corresponding to exponential decrease in SIC in the Barents Sea. Specifically, we show that a Koopman mode exhibiting exponential decay is consistently detected from KMA applied to observational satellite data from both the entire Arctic region and to only the Barents Sea. This mode is not found when the Barents Sea SIC is excluded from the training data, indicating that the identified decaying dynamics are spatially localized to the Barents Sea. The geographic distribution of Koopman modes corresponding to the identified exponential decay eigenvalues is also found to have a spatial locus in the Barents Sea region. We conclude that the Barents Sea is undergoing a nonlinear decrease in SIC, which may be a signature of a potential tipping point.
Tuesday, May 16. 8:30-8:55
Theory and Computation of the Spectral Properties of Pullback Operators in Dynamical Systems
Abstract. Koopman operator methods along with the associated numerical algorithms have provided a powerful methodology for the data-driven study of nonlinear dynamical systems. In this talk, we will give a brief outline of how the Koopman group of operators can be generalized beyond function spaces to the space of sections of various vector bundles over the state space. We describe their relationship with the standard Koopman operator on functions as well as describe the new spectral invariants produced by these generalized operators. We then demonstrate how the recently developed spectral exterior calculus framework can be utilized to compute the spectral properties of the generator of the induced operator on sections of the cotangent bundle. We conclude with some applications of the algorithm to some well-known dynamical systems.