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Perspektivenwechsel. Bildinformationen anzeigen


Foto: Universität Paderborn

Dr. Feliks Nüske

Dr. Feliks Nüske

Lehrstuhl für Angewandte Mathematik

Mitglied - Wissenschaftlicher Mitarbeiter

+49 5251 60-3774
Technologiepark 21
33100 Paderborn
1. Research Topics
  1. Data-driven Modeling of Dynamical Systems: Spectral Analysis, Prediction, Control and Model Reduction of Complex Dynamical Systems based on data.
  2. Data-driven Approximation of Linear Operators: Approximation of linear operators on reproducing kernel Hilbert spaces and in low-rank tensor formats.
  3. Molecular Dynamics Simulations: Data-driven methods for metastability analysis, coarse graining, enhanced sampling, and control.
2. Further Information

Google Scholar


Dr. Feliks Nüske
01.09.2019 - heute

Postdoctoral Researcher

Paderborn University (Germany), Institute of Mathematics

16.03.2017 - 31.08.2019

Postdoctoral Research Associate

Rice University (Houston, TX, U.S.), Department of Chemistry and Center for Theoretical Biological Physics


Ph. D. Mathematics

Freie Universität Berlin (Germany), Title of the Thesis: The Variational Approach to Conformational Dynamics


M. Sc. Mathematics

Freie Universität Berlin (Germany), Title of the Thesis: A variational approach for conformation dynamics


B. Sc. Mathematics

Freie Universität Berlin (Germany), Title of the Thesis: Die Mehler-Formel und der Satz von Egorov (in German)

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Finite-data error bounds for Koopman-based prediction and control

F. Nüske, S. Peitz, F. Philipp, M. Schaller, K. Worthmann, in: arXiv:2108.07102, 2021

The Koopman operator has become an essential tool for data-driven approximation of dynamical (control) systems in recent years, e.g., via extended dynamic mode decomposition. Despite its popularity, convergence results and, in particular, error bounds are still quite scarce. In this paper, we derive probabilistic bounds for the approximation error and the prediction error depending on the number of training data points; for both ordinary and stochastic differential equations. Moreover, we extend our analysis to nonlinear control-affine systems using either ergodic trajectories or i.i.d. samples. Here, we exploit the linearity of the Koopman generator to obtain a bilinear system and, thus, circumvent the curse of dimensionality since we do not autonomize the system by augmenting the state by the control inputs. To the best of our knowledge, this is the first finite-data error analysis in the stochastic and/or control setting. Finally, we demonstrate the effectiveness of the proposed approach by comparing it with state-of-the-art techniques showing its superiority whenever state and control are coupled.

Symmetric and antisymmetric kernels for machine learning problems in quantum physics and chemistry

S. Klus, P. Gelß, F. Nüske, F. Noé, Machine Learning: Science and Technology (2021)

Tensor-based computation of metastable and coherent sets

F. Nüske, P. Gelß, S. Klus, C. Clementi, Physica D: Nonlinear Phenomena (2021)

Spectral Properties of Effective Dynamics from Conditional Expectations

F. Nüske, P. Koltai, L. Boninsegna, C. Clementi, Entropy (2021)

<jats:p>The reduction of high-dimensional systems to effective models on a smaller set of variables is an essential task in many areas of science. For stochastic dynamics governed by diffusion processes, a general procedure to find effective equations is the conditioning approach. In this paper, we are interested in the spectrum of the generator of the resulting effective dynamics, and how it compares to the spectrum of the full generator. We prove a new relative error bound in terms of the eigenfunction approximation error for reversible systems. We also present numerical examples indicating that, if Kramers–Moyal (KM) type approximations are used to compute the spectrum of the reduced generator, it seems largely insensitive to the time window used for the KM estimators. We analyze the implications of these observations for systems driven by underdamped Langevin dynamics, and show how meaningful effective dynamics can be defined in this setting.</jats:p>


Data-driven approximation of the Koopman generator: Model reduction, system identification, and control

S. Klus, F. Nüske, S. Peitz, J. Niemann, C. Clementi, C. Schütte, Physica D: Nonlinear Phenomena (2020), 406

We derive a data-driven method for the approximation of the Koopman generator called gEDMD, which can be regarded as a straightforward extension of EDMD (extended dynamic mode decomposition). This approach is applicable to deterministic and stochastic dynamical systems. It can be used for computing eigenvalues, eigenfunctions, and modes of the generator and for system identification. In addition to learning the governing equations of deterministic systems, which then reduces to SINDy (sparse identification of nonlinear dynamics), it is possible to identify the drift and diffusion terms of stochastic differential equations from data. Moreover, we apply gEDMD to derive coarse-grained models of high-dimensional systems, and also to determine efficient model predictive control strategies. We highlight relationships with other methods and demonstrate the efficacy of the proposed methods using several guiding examples and prototypical molecular dynamics problems.

    Kernel-Based Approximation of the Koopman Generator and Schrödinger Operator

    S. Klus, F. Nüske, B. Hamzi, Entropy (2020)

    <jats:p>Many dimensionality and model reduction techniques rely on estimating dominant eigenfunctions of associated dynamical operators from data. Important examples include the Koopman operator and its generator, but also the Schrödinger operator. We propose a kernel-based method for the approximation of differential operators in reproducing kernel Hilbert spaces and show how eigenfunctions can be estimated by solving auxiliary matrix eigenvalue problems. The resulting algorithms are applied to molecular dynamics and quantum chemistry examples. Furthermore, we exploit that, under certain conditions, the Schrödinger operator can be transformed into a Kolmogorov backward operator corresponding to a drift-diffusion process and vice versa. This allows us to apply methods developed for the analysis of high-dimensional stochastic differential equations to quantum mechanical systems.</jats:p>


    Coarse-graining molecular systems by spectral matching

    F. Nüske, L. Boninsegna, C. Clementi, The Journal of Chemical Physics (2019)


    Data-Driven Model Reduction and Transfer Operator Approximation

    S. Klus, F. Nüske, P. Koltai, H. Wu, I. Kevrekidis, C. Schütte, F. Noé, Journal of Nonlinear Science (2018), pp. 985-1010

    Sparse learning of stochastic dynamical equations

    L. Boninsegna, F. Nüske, C. Clementi, The Journal of Chemical Physics (2018)

    Quantitative comparison of adaptive sampling methods for protein dynamics

    E. Hruska, J.R. Abella, F. Nüske, L.E. Kavraki, C. Clementi, The Journal of Chemical Physics (2018)

    Rapid Calculation of Molecular Kinetics Using Compressed Sensing

    F. Litzinger, L. Boninsegna, H. Wu, F. Nüske, R. Patel, R. Baraniuk, F. Noé, C. Clementi, Journal of Chemical Theory and Computation (2018), pp. 2771-2783


    Variational Koopman models: Slow collective variables and molecular kinetics from short off-equilibrium simulations

    H. Wu, F. Nüske, F. Paul, S. Klus, P. Koltai, F. Noé, The Journal of Chemical Physics (2017)

    Markov state models from short non-equilibrium simulations—Analysis and correction of estimation bias

    F. Nüske, H. Wu, J. Prinz, C. Wehmeyer, C. Clementi, F. Noé, The Journal of Chemical Physics (2017)


    Variational tensor approach for approximating the rare-event kinetics of macromolecular systems

    F. Nüske, R. Schneider, F. Vitalini, F. Noé, The Journal of Chemical Physics (2016)


    Variational Approach to Molecular Kinetics

    F. Nüske, B.G. Keller, G. Pérez-Hernández, A.S.J.S. Mey, F. Noé, Journal of Chemical Theory and Computation (2014), pp. 1739-1752


    A Variational Approach to Modeling Slow Processes in Stochastic Dynamical Systems

    F. Noé, F. Nüske, Multiscale Modeling & Simulation (2013), pp. 635-655

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