Spectral Properties of Pullback Operators on Vector Bundles of a Dynamical System

Speaker: Allan Avila, AIMdyn Inc.

DateMay 20 and June 10, 2022, 9:00 am PT

Abstract:

The spectrum of the Koopman operator has been shown to encode many important statistical and geometric properties of a dynamical system. In this talk, we consider induced linear operators acting on the space of sections of the tangent, cotangent, and tensor bundles of the state space.
We begin by first demonstrating how these operators are indeed natural generalizations of Koopman operators acting on functions.  We then draw connections between the various operators' spectra and characterize the algebraic and differential topological properties of their spectrum. We describe the discrete spectrum of these operators for linear dynamical systems and derive spectral expansions for linear vector fields.  We define the notion of an "eigendistribution", provide conditions for an eigendistribution to be integrable and demonstrate how to recover the foliations arising from their integral manifolds via the level sets of certain Koopman eigenfunctions. Lastly, we demonstrate that the characteristic Lyapunov exponents of a uniformly hyperbolic dynamical system are in the spectrum of the induced operators on sections of the tangent or cotangent bundle.
We then apply our results to generalize the well-known fact that the flows of commuting vector fields commute and recover the original statement as a particular case of our result. We also apply our results to recover the Lyapunov exponents and the stable/unstable foliations of Arnold's Cat map via the spectrum of the induced operator on sections of the tangent bundle. If time permits, we will describe a numerical method for computing the spectrum of the induced operators on sections of the cotangent bundle.

Short Bio:

Allan received his B.S. in Mechanical Engineering from the University of California Riverside in 2015, and an M.S. and PhD in Mechanical Engineering from the University of California Santa Barbara in 2020. He is a member of the Society of Industrial and Applied Mathematics, and a 2018 Hispanic Scholarship recipient. Allan is now a researcher at AIMdyn Inc. and his general research is on data-driven analysis and forecasting of dynamical systems via spectral operator methods.

 

 

A Convex Optimization Approach to Learning Koopman Operators from Data

Speaker: Mario Sznaier, Northeastern University, USA
DateApril 8 2022, 9:00 am PT
Abstract:

Koopman operators provide tractable means of learning linear approximations of  non-linear dynamics. Many approaches have been proposed to find these operators,  typically  based upon approximations using an a-priori fixed class of models. However, choosing appropriate models and bounding the approximation error is far from trivial.  Motivated by these difficulties, in this talk we propose  an optimization based approach to  learning  Koopman operators from data. Our results show that the Koopman operator, the associated Hilbert space of observables and a suitable dictionary can be obtained by solving two rank-constrained semi-definite programs (SDP). While in principle these problems are NP-hard, the use of standard relaxations of rank leads to  convex SDPs.  In the second portion of the talk we will briefly discuss some new architectures motivated by Koopman operator theory. . However, as shown with a simple example, these results cannot be extended to actuated models. The talk will conclude by exploring the connection between LPV identification, bilinear identification and actuated Koopman models.

Short Bio:

Mario Sznaier is currently the Dennis Picard Chaired Professor at the Electrical and Computer Engineering Department, Northeastern University, Boston. Prior to joining Northeastern University, Dr. Sznaier was a Professor of Electrical Engineering at the Pennsylvania State University and also held visiting positions at the California Institute of Technology. His research interest include robust identification and control of hybrid systems, robust optimization, and dynamical vision. Dr. Sznaier is currently serving as chair of the IFAC Technical Committee on Robust Control and Founding Editor in Chief of the section on AI and Control of the Journal Frontiers in Control. Engineering . Past recent service include Associate Editor for Automatica (2005-2021), Program Chair of the 2017 IEEE Conf. on Decision and Control, General Chair of the 2016 IEEE Multi Systems Conference, Chair of the  IEEE Control Systems Society Technical Committee on Computational Aspects of Control Systems Design (2013-2017),  He is a distinguished member of the IEEE Control Systems Society  and a Fellow of the IEEE for his contributions to robust control, identification and dynamic vision. A list of publications and current research projects can be found at http://robustsystems.coe.neu.edu.

Koopman-based Learning in Continuous-time Optimization

Speaker: Eduardo Mojica-Nava, Universidad Nacional de Colombia, Colombia.

Date: March 10, 2022

Abstract:

The operator-theoretic framework has emerged as a successful tool for data-driven learning of nonlinear dynamical systems. Koopman operator theory has demonstrated a strong connection between algorithms and dynamical systems. In contrast to the traditional dynamics of states, the Koopman operator is a linear infinite-dimensional representation on the space of observables. This new approach has brought capabilities for data-driven detecting of essential elements in global geometric analysis such as stable/unstable manifolds and invariant sets by means of spectral properties. On the other hand, saddle-point dynamics have become an important strategy in the design of optimal feedback controllers. In this talk, the Koopman operator framework is used to obtain a data-driven continuous-time optimization algorithm for solving constrained optimization problems using its connection with dynamical systems for numerical algorithms. Stability and convergence results are demonstrated for three different optimization problem cases: strictly convex functions, quadratic objective function, and nonconvex function with multiple saddle-points. Several numerical examples are simulated for illustrating the proposed approach.

Short Bio:

Eduardo Mojica-Nava received the B.S. degree in electronics engineering from the Universidad Industrial de Santander, Bucaramanga, Colombia, in 2002; the M.Sc. degree in electronics engineering from the Universidad de Los Andes, Bogotá, Colombia; and the Ph.D. degree in Automatique et Informatique Industrielle from the École des Mines de Nantes, Nantes, France in co-tutelle with UAndes in 2010. He is currently a full professor with the Department of Electrical and Electronics Engineering, Universidad Nacional de Colombia, Bogotá, Colombia. He has been visiting professor at the University of Mons, Belgium in 2017 and Politecnico de Milano, Italy, in 2021. His current research interests include optimization and control of complex networked systems, operator theory method, switched and hybrid systems, and control in smart grids applications.

KoopmanizingFlows: Diffeomorphically Learning Stable Koopman Operators

Speaker: Petar Bevanda, Technical University of Munich, Germany

Date: February 3, 2022

Abstract:

Global linearization methods for nonlinear systems inspired by the infinite-dimensional, linear Koopman operator have received increased attention for data-driven modeling of nonlinear dynamics in recent years. By lifting a finite-dimensional nonlinear system to a higher-dimensional linear operator representation, superior complexity-accuracy balance compared to conventional nonlinear modeling is possible through the use of efficient linear techniques for prediction, analysis and control. Learning meaningful finite-dimensional representations of Koopman operators presents a challenging problem, as one needs to learn linear time-invariant (LTI) features that are both Koopman-invariant (evolve linearly under the dynamics) as well as relevant (spanning the original state) - a generally unsupervised learning task. For a structured solution to this unsupervised problem, we propose learning Koopman-invariant coordinates by composing a lifted aggregate system of a latent linear model with a diffeomorphic learner based on Normalizing Flows. Using an unconstrained parameterization of stable matrices along with the aforementioned feature construction, we learn the Koopman operator features without assuming a predefined library of functions or knowing the spectrum, while ensuring stability regardless of the operator approximation accuracy – resulting in the KoopmanizingFlow Stable Dynamical Systems (KF-SDS) framework. We demonstrate the superior efficacy of the proposed method in comparison to a state-of-the-art method on the well-known LASA handwriting dataset. This talk is based on a preprint of ours found here.

Short bio:

Petar Bevanda is PhD student at the Technical University of Munich (TUM). He received his M.Sc. degree in Electrical and Computer Engineering at the Technical University of Munich in 2020. Beginning of 2021, he began pursuing his PhD degree at the Chair of Information-oriented Control in the Department of Electrical and Computer Engineering at Technical University of Munich. His research interests include data-driven modeling based on linear evolution operators, model predictive control and safe learning.

ResDMD: Rigorous data-driven computation of spectral properties of Koopman operators for dynamical systems

Speaker: Matt Colbrook, University of Cambridge, United Kingdom 

DateDecember 15, 2021

Abstract:

Koopman operators are infinite-dimensional operators that globally linearize nonlinear dynamical systems, making their spectral information valuable for understanding dynamics. However, Koopman operators can have continuous spectra and infinite-dimensional invariant subspaces, making computing their spectral information a considerable challenge. This paper describes data-driven algorithms with rigorous convergence guarantees for computing spectral information of Koopman operators from trajectory data. We introduce residual dynamic mode decomposition (ResDMD), which provides the first scheme for computing the spectra and pseudospectra of general Koopman operators from snapshot data without spectral pollution. Using the resolvent operator and ResDMD, we compute smoothed approximations of spectral measures associated with measure-preserving dynamical systems. We prove explicit convergence theorems for our algorithms, which can achieve high-order convergence even for chaotic systems, when computing the density of the continuous spectrum and discrete spectrum. We demonstrate our algorithms on the tent map, Gauss iterated map, nonlinear pendulum, double pendulum, Lorenz system, and an 11-dimensional extended Lorenz system. Finally, we provide kernelized variants of our algorithms for dynamical systems with a high-dimensional state-space. This allows us to compute the spectral measure associated with the dynamics of a protein molecule that has a 20,046-dimensional state-space and compute nonlinear Koopman modes with error bounds for turbulent flow past aerofoils with Reynolds number greater than 10^5 that has a 295,122-dimensional state-space. This talk is based on joint work with Alex Townsend and our preprint can be found here.

Short Bio:

Matthew Colbrook is a Junior Research Fellow at Trinity College Cambridge and a Fondation Sciences Mathématiques de Paris Postdoctoral Fellow at École Normale Supérieure. He holds a PhD from the Department of Applied Mathematics and Theoretical Physics, University of Cambridge (2020). His research is centred on numerical analysis and foundations of computation in infinite-dimensional spectral problems, PDEs, and deep learning/neural networks for scientific computation, as well as a framework for determining the boundaries of what is and what is not computationally possible. He is a recipient of the IMA Lighthill-Thwaites Prize, the Smith-Knight prize and the Mayhew prize.