News

In this recent preprint, we investigate the estimation error for kernel-based estimates for the Koopman Operator. Kernel methods are highly versatile and widely used in machine learning. In the context of dynamical systems, kernel-based approaches to model the associated Koopman Operator have been introduced in recent years, eliminating the need to explicitly choose a finite-dimensional approximation subspace. Here, we derive rigorous error bounds for the estimation error due to the use of finite data. more

We are happy that Lei Guo has joined as our second Ph.D. student. Lei will be working on efficient control of stochastic systems using Koopman operators. Welcome Lei! more

In this new manuscript, Thomas Berger and I investigate a new conceptual approach to the control of stochastic dynamics governed by Langevin equations. We derive conditions for which the funnel controller, a very simple and essentially model-free feedback controller, is feasible and can be guaranteed to achieve a prescribed control objective. more

We welcome Heejin Woo from Chonnam National University in South Korea to our group. Heejin will be with us as a student intern for half a year, working on simulation-based methods for the analysis of quantum systems.

I’m very pleased that this new paper (with Stefan Klus and Sebastian Peitz) has just appeared in Journal of Physics A. Koopman operator theory has so far been applied successfully to classical problems from fluid dynamics, molecular dynamics, climate science, engineering, and many others. However, there is also an intricate connection between dynamical systems driven by stochastic differential equations and quantum mechanics. In the paper, we compare the ground-state transformation and Nelson's stochastic mechanics and demonstrate how data-driven methods developed for the approximation of the Koopman operator can be used to analyze quantum physics problems. We also exploit the relationship between Schrödinger operators and stochastic control problems to show that modern data-driven methods for stochastic control can be used to solve the stationary or imaginary-time Schrödinger equation. Our findings open up a new avenue towards solving Schrödinger's equation using recently developed tools from data science. more

We are happy to welcome our first group member, Vahid Nateghi, who joined the group June 1, 2022 working on “Data-driven model reduction for stochastic systems”. more

Low-rank tensor formats, especially the tensor train (TT) format, have emerged as a powerful tool for the solution of large scale problems. In the context of modeling dynamical systems using Koopman operators, tensor formats arise when a product basis is employed as Galerkin subspace. The use of low-rank formats is basically inevitable in this case due to the combinatorial explosion of the basis set size. In this work, we present a data-driven method to efficiently approximate the Koopman generator using the TT format. The centerpiece of the method is a TT representation of the tensor of generator evaluations at all data sites. We analyze consistency and complexity of the approach, and present extensions to two practically relevant settings, namely the use of importance sampling data, and estimation of a projected generator on coarse grained coordinates. We illustrate our findings using two examples: the first is a low-dimensional diffusion process using a model potential, the second is a coarse grained representation of the deca alanine peptide. more

As part and restart of the MPI Colloquia Series I will give my Inaugural Lecture as a hybrid presentation on "Data-driven Modeling of Dynamical Systems - Analysis and Model Reduction".

Time:    Thursday, May 12, 2022 at 2:00 pm
Place:   Seminar room Prigogine (up to 40 persons, only with face masks) more

The DMP group has officially started  on April 1, 2022!