Measure Transport for Data-Driven Dynamical Systems

Data Science Seminar

Jonah Botvinick-Greenhouse
Yale University

Abstract

Learning dynamical systems in the presence of noisy, sparse, and uncertain observations is a central challenge in data-driven science and engineering. In this talk, I will present methods based on measure transport that target long-term statistical behavior rather than pointwise time-series reconstruction, making them well suited to regimes where standard trajectory-based methods struggle. We begin by reframing system identification as a distribution-matching problem in which invariant measures extracted from data are compared with model-simulated invariant measures. To guarantee identifiability of the underlying dynamics from distributional data, we introduce data-driven coordinate transformations inspired by Takens’ embedding theorem. While classical embedding methods assume noise-free dynamics, we leverage tools from measure transport to extend these concepts to a probabilistic setting. Our formulation recasts state reconstruction as a transformation between probability distributions and yields a practical computational framework for recovering high-dimensional systems from partial observations. Throughout, we demonstrate the effectiveness of our methods through numerical studies on fluid flows, Hall-effect thrusters, and large-scale geophysical datasets.