Transport- and Measure-Theoretic Approaches for Modeling, Identifying, and Forecasting Dynamical Systems

Explore advanced mathematical approaches for analyzing dynamical systems through this 46-minute conference talk that introduces the Distributional Koopman Operator (DKO) as a novel framework for performing Koopman analysis on random dynamical systems. Learn how this innovative method enables analysis using only aggregate distribution data, eliminating the need for particle tracking or detailed trajectory information. Discover how the DKO generalizes the stochastic Koopman operator to work with observables of probability distributions, utilizing transfer operators to propagate probability distributions forward in time. Understand the linear properties and semigroup characteristics of the DKO, and examine how dynamical mode decomposition approximations can converge to the DKO in large data scenarios. Gain insights into applications for random dynamical systems where trajectory information is unavailable, and see how this framework elevates analysis from atomistic Lagrangian particles to continuum probability distributions. The presentation covers transport and measure-theoretic approaches for modeling, identifying, and forecasting complex systems, providing valuable perspectives for researchers working in computational mathematics, dynamical systems theory, and electrochemical system modeling.