Koopman Operator and Machine Learning (DesCartes – WP3)

The DesCartes programme is developing a hybrid AI, combining Learning, Knowledge and Reasoning, which has good properties (need for less resources and data, security, robustness, fairness, respect for privacy, ethics), and demonstrated on industrial applications of the smart city (digital energy, monitoring of structures, air traffic control).

The program brings together 80 permanent researchers (half from France, half from Singapore), with the support of large industrial groups (Thales SG, EDF SG, ESI group, CETIM Matcor, ARIA etc.).

The research will take place mainly in Singapore, at the premises of CNRS@CREATE, with a competitive salary and generous funding for missions.

Read more about the DesCartes program here.

WP3 aims at supporting the whole Descartes program in order to develop advanced optimization-based solutions in the context of hybrid AI. Any AI system or machine learning algorithm ultimately involves a formulation with an objective or loss function to be minimized. The modelling of the problem as well as the chosen objective function optimization algorithm is crucial to the success of the overall AI task. This is all the more crucial in the context of hybrid AI, which seeks to integrate physics-inspired models with machine learning algorithms. We will address this problem from two complementary angles, namely optimization-based methods and machine learning-based methods.

This sub-project focuses on developing data-driven modelling and control methodologies for dynamical systems, with particular emphasis on the interaction of Koopman operator analysis and machine learning. The Koopman operator approach is a leading framework for data-driven analysis of complex nonlinear dynamical systems. In this framework, a nonlinear dynamical system is represented by an infinite dimensional linear operator that governs the evolution of functions defined on the state-space. This operator provides an equivalent linear representation of the underlying system with all information (e.g., stability, ergodicity, invariant sets, stable/unstable manifolds) being encoded within the operator and its spectrum. Finite-dimensional approximations of this operator then allow one to recover this information numerically, leading to methods for analysis, model reduction, state-estimation and control of high-dimensional systems, relying solely on mature and efficient tools of linear algebra and convex optimization. However, applying Koopman operator analysis to practical problems of high dimensions remains a significant challenge, and modern machine learning offers a promising approach to overcome these issues.

In this project, we will investigate the coupling of machine learning and Koopman operator analysis from several angles, including:

1. Coupling of dictionary learning type of methods to Koopman operator analysis, particular for dynamical systems with temporal inputs (control systems) and parametric dynamical systems. The main challenge here is to develop learning-based methods for invariant subspace identification for such systems.
2. Koopman operator approach for optimization-based control of nonlinear dynamical systems. The main challenge is to preserve convexity of the resulting optimization problem to be solved (e.g., within model predictive control) when applying a Koopman embedding to the control system.

Competences in some of the domains listed below will be highly considered:

• Applied Dynamical Systems
• Optimization and Control
• Machine Learning (Deep Learning)

Keywords:

1. Koopman Operator
2. Machine Learning
3. Deep Learning
4. Convex and Non-convex Optimization
5. Control
6. Programming (python, matlab)