Massimiliano Pontil
[intermediate/advanced] Operator Learning for Dynamical Systems
Summary
The course gives an introduction to data-driven approaches to learn dynamical systems, with a focus on stochastic systems and efficient and reliable learning algorithms. A main goal is to learn the evolution operators associated with the system, in particular the family of transfer operators. For continuous systems, we also investigate learning the closely related infinitesimal generator, essential for characterizing solutions to stochastic differential equations. We review recent operator estimators, whose domain is either a reproducing kernel Hilbert space or the span of basis functions learned via a neural network. The framework of statistical learning theory is used to study the algorithms. In particular we present spectral learning bounds for the operator estimators, highlighting the important role of controlling the estimator’s rank. Examples of dynamical systems will be discussed during the lecture.
Syllabus
Part 1: Introduction to dynamical systems and data-driven approaches:
- What is a dynamical system
- Deterministic vs. stochastic systems
- Discrete vs. Continuous. SDEs
- Example of continuous systems: Brownian motion and Langevin equation
- Ergodic decomposition
- Transfer operator and spectral decomposition
Part 2: Statistical learning approach to learning the transfer operator:
- Reproducing kernel Hilbert spaces
- Risk formulation of transfer operator regression
- Risk formulation of infinitesimal generator regression
- Empirical risk minimization and reduced rank estimator
Part 3: Advanced topics:
- Spectral learning bounds
- Large scale kernel algorithms
- Deep representation learning
- Long term forecasting
References
V. Kostic, P. Novelli, A. Maurer, C. Ciliberto, L. Rosasco, M. Pontil. Learning dynamical systems via Koopman operator regression in reproducing kernel hilbert spaces. NeurIPS 2022.
V. Kostic, K. Lounici, P. Novelli, M. Pontil. Koopman operator learning: sharp spectral rates and spurious eigenvalues. NeurIPS 2023.
G. Meanti, A. Chatalic, V. Kostic, P. Novelli, M. Pontil, L. Rosasco. Estimating Koopman operators with sketching to provably learn large scale dynamical systems. NeurIPS 2023.
V. Kostic, P. Novelli, R. Grazzi, K. Lounici, M. Pontil. Learning invariant representations of time-homogeneous stochastic dynamical systems. ICLR 2024.
V. Kostic, K. Lounici, P. Inzerilli, P. Novelli., M. Pontil. Consistent long-term forecasting of ergodic dynamical systems. ICML 2024.
Pre-requisites
Fundamentals of linear algebra and probability theory. Some familiarity with Hilbert spaces and kernel methods is a plus but not strictly required.
Short bio
Massimiliano Pontil is Senior Researcher at the Italian Institute of Technology, where he leads the CSML research unit, and co-director of ELLIS unit Genoa. He is also Professor at University College London and member of the UCL Centre for Artificial Intelligence. He has made significant contributions to machine learning theory and algorithms, including the areas of kernel methods, meta-learning, multitask and transfer learning, sparse estimation, and statistical learning theory. More information on Massimiliano’s research interests and accomplishments can be found here: https://www.iit.it/people-details/-/people/massimiliano-pontil.
