Transport- and Measure-Theoretic Approaches for Modeling, Identifying, and Forecasting Dynamical Systems
Abstract
Measures provide valuable insights into long-term and global behaviors across a broad range of dynamical systems. In this talk, we present our recent research efforts that employ measure theory and optimal transport to tackle core challenges in system identification, parameter recovery, and predictive modeling. First, we adopt a PDE-constrained optimization perspective to learn ODEs and SDEs from slowly sampled trajectories, enabling stable forward models and uncertainty quantification. We then use optimal transportation to align physical measures for parameter estimation, even when time-derivative data is unavailable. Our second result extends the celebrated Takens’ time-delay embedding, a foundational result in dynamical systems, from state space to probability distributions. It establishes a robust theoretical and computational framework for state reconstruction that remains effective under noisy and partial observations. Finally, we show that by comparing invariant measures in time-delay coordinates, one can overcome identifiability challenges and achieve unique recovery of the underlying dynamics even though it is not generally possible to uniquely reconstruct dynamics using invariant statistics alone. Collectively, these works demonstrate how measure-theoretic and transport-based methods can robustly identify, analyze, and forecast real-world dynamical systems and the great research potential of measure-theoretic approaches for dynamical systems.
