Learning Linear Evolution Operators: Theory and Applications

Dynamical systems are central to science and engineering, with applications in climate modeling, molecular dynamics, robotics, neuroscience, finance, and beyond. Remarkably, despite their diversity, many such systems can be understood within the unifying framework of linear evolution operators. The key idea is to study how functions of the state evolve over time, instead of tracking the full state. This transforms a nonlinear problem into a linear one and enables spectral analysis to uncover global system dynamics. This perspective has a long history, rooted in foundational work by Markov, Koopman, and von Neumann. However, while conceptually powerful, linear evolution operators are often computationally intractable in high-dimensional settings. Consequently, over the past two decades, significant effort has focused on data-driven methods, yet their theoretical guarantees remain poorly understood. This talk presents a framework for placing data-driven approaches to dynamical systems on a firm statistical foundation. By formulating the problem of learning linear evolution operators from a statistical perspective, we develop novel learning algorithms backed by finite-sample learning guarantees, which show promising results in challenging real-world applications in molecular dynamics, climate modeling, and robotics. The algorithms provably learn evolution operators and their spectra, and leverage modern deep learning architectures to learn representations of dynamics efficiently and reliably. Finally, if time permits, I will discuss how linear operators play a central role in reinforcement learning and statistical inference, including uncertainty quantification and causality.

MASSIMILIANO PONTIL

Massimiliano Pontil is a Principal Investigator at the Italian Institute of Technology, where he leads the Computational Statistics and Machine Learning (CSML) unit, and serves as co-director of the ELLIS Unit Genoa, a joint effort of the IIT and the University of Genoa. He is also a Professor at University College London and a member of the UCL Centre for Artificial Intelligence. His research concerns the theory and algorithms of machine learning, including kernel methods, meta-learning, multi-task and transfer learning, sparse estimation, and statistical learning theory. More information on Massimiliano’s research interests and accomplishments can be found here.

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