@inproceedings{30125,
  abstract     = {{We present an approach for guaranteed constraint satisfaction by means of data-based optimal control, where the model is unknown and has to be obtained from measurement data. To this end, we utilize the Koopman framework and an eDMD-based bilinear surrogate modeling approach for control systems to show an error bound on predicted observables, i.e., functions of the state. This result is then applied to the constraints of the optimal control problem to show that satisfaction of tightened constraints in the purely data-based surrogate model implies constraint satisfaction for the original system.}},
  author       = {{Schaller, Manuel and Worthmann, Karl and Philipp, Friedrich and Peitz, Sebastian and Nüske, Feliks}},
  booktitle    = {{IFAC-PapersOnLine}},
  number       = {{1}},
  pages        = {{169--174}},
  title        = {{{Towards reliable data-based optimal and predictive control using extended DMD}}},
  doi          = {{10.1016/j.ifacol.2023.02.029}},
  volume       = {{56}},
  year         = {{2023}},
}

@article{23428,
  abstract     = {{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.}},
  author       = {{Nüske, Feliks and Peitz, Sebastian and Philipp, Friedrich and Schaller, Manuel and Worthmann, Karl}},
  journal      = {{Journal of Nonlinear Science}},
  title        = {{{Finite-data error bounds for Koopman-based prediction and control}}},
  doi          = {{10.1007/s00332-022-09862-1}},
  volume       = {{33}},
  year         = {{2023}},
}

@article{29673,
  abstract     = {{Koopman operator theory has been successfully applied to problems from various research areas such as fluid dynamics, molecular dynamics, climate science, engineering, and biology. Applications include detecting metastable or coherent sets, coarse-graining, system identification, and control. There is an intricate connection between dynamical systems driven by stochastic differential equations and quantum mechanics. In this 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. Moreover, we 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.}},
  author       = {{Klus, Stefan and Nüske, Feliks and Peitz, Sebastian}},
  journal      = {{Journal of Physics A: Mathematical and Theoretical}},
  number       = {{31}},
  pages        = {{314002}},
  publisher    = {{IOP Publishing Ltd.}},
  title        = {{{Koopman analysis of quantum systems}}},
  doi          = {{10.1088/1751-8121/ac7d22}},
  volume       = {{55}},
  year         = {{2022}},
}

@article{24169,
  author       = {{Nüske, Feliks and Gelß, Patrick and Klus, Stefan and Clementi, Cecilia}},
  issn         = {{0167-2789}},
  journal      = {{Physica D: Nonlinear Phenomena}},
  title        = {{{Tensor-based computation of metastable and coherent sets}}},
  doi          = {{10.1016/j.physd.2021.133018}},
  year         = {{2021}},
}

@article{24170,
  author       = {{Klus, Stefan and Gelß, Patrick and Nüske, Feliks and Noé, Frank}},
  issn         = {{2632-2153}},
  journal      = {{Machine Learning: Science and Technology}},
  title        = {{{Symmetric and antisymmetric kernels for machine learning problems in quantum physics and chemistry}}},
  doi          = {{10.1088/2632-2153/ac14ad}},
  year         = {{2021}},
}

@article{21820,
  abstract     = {{<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>}},
  author       = {{Nüske, Feliks and Koltai, Péter and Boninsegna, Lorenzo and Clementi, Cecilia}},
  issn         = {{1099-4300}},
  journal      = {{Entropy}},
  title        = {{{Spectral Properties of Effective Dynamics from Conditional Expectations}}},
  doi          = {{10.3390/e23020134}},
  year         = {{2021}},
}

@article{21819,
  abstract     = {{<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>}},
  author       = {{Klus, Stefan and Nüske, Feliks and Hamzi, Boumediene}},
  issn         = {{1099-4300}},
  journal      = {{Entropy}},
  title        = {{{Kernel-Based Approximation of the Koopman Generator and Schrödinger Operator}}},
  doi          = {{10.3390/e22070722}},
  year         = {{2020}},
}

@article{16288,
  abstract     = {{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.}},
  author       = {{Klus, Stefan and Nüske, Feliks and Peitz, Sebastian and Niemann, Jan-Hendrik and Clementi, Cecilia and Schütte, Christof}},
  issn         = {{0167-2789}},
  journal      = {{Physica D: Nonlinear Phenomena}},
  title        = {{{Data-driven approximation of the Koopman generator: Model reduction, system identification, and control}}},
  doi          = {{10.1016/j.physd.2020.132416}},
  volume       = {{406}},
  year         = {{2020}},
}

@article{21944,
  author       = {{Nüske, Feliks and Boninsegna, Lorenzo and Clementi, Cecilia}},
  issn         = {{0021-9606}},
  journal      = {{The Journal of Chemical Physics}},
  title        = {{{Coarse-graining molecular systems by spectral matching}}},
  doi          = {{10.1063/1.5100131}},
  year         = {{2019}},
}

@article{21940,
  author       = {{Litzinger, Florian and Boninsegna, Lorenzo and Wu, Hao and Nüske, Feliks and Patel, Raajen and Baraniuk, Richard and Noé, Frank and Clementi, Cecilia}},
  issn         = {{1549-9618}},
  journal      = {{Journal of Chemical Theory and Computation}},
  pages        = {{2771--2783}},
  title        = {{{Rapid Calculation of Molecular Kinetics Using Compressed Sensing}}},
  doi          = {{10.1021/acs.jctc.8b00089}},
  year         = {{2018}},
}

@article{21941,
  author       = {{Klus, Stefan and Nüske, Feliks and Koltai, Péter and Wu, Hao and Kevrekidis, Ioannis and Schütte, Christof and Noé, Frank}},
  issn         = {{0938-8974}},
  journal      = {{Journal of Nonlinear Science}},
  pages        = {{985--1010}},
  title        = {{{Data-Driven Model Reduction and Transfer Operator Approximation}}},
  doi          = {{10.1007/s00332-017-9437-7}},
  year         = {{2018}},
}

@article{21942,
  author       = {{Boninsegna, Lorenzo and Nüske, Feliks and Clementi, Cecilia}},
  issn         = {{0021-9606}},
  journal      = {{The Journal of Chemical Physics}},
  title        = {{{Sparse learning of stochastic dynamical equations}}},
  doi          = {{10.1063/1.5018409}},
  year         = {{2018}},
}

@article{21943,
  author       = {{Hruska, Eugen and Abella, Jayvee R. and Nüske, Feliks and Kavraki, Lydia E. and Clementi, Cecilia}},
  issn         = {{0021-9606}},
  journal      = {{The Journal of Chemical Physics}},
  title        = {{{Quantitative comparison of adaptive sampling methods for protein dynamics}}},
  doi          = {{10.1063/1.5053582}},
  year         = {{2018}},
}

@article{21938,
  author       = {{Nüske, Feliks and Wu, Hao and Prinz, Jan-Hendrik and Wehmeyer, Christoph and Clementi, Cecilia and Noé, Frank}},
  issn         = {{0021-9606}},
  journal      = {{The Journal of Chemical Physics}},
  title        = {{{Markov state models from short non-equilibrium simulations—Analysis and correction of estimation bias}}},
  doi          = {{10.1063/1.4976518}},
  year         = {{2017}},
}

@article{21939,
  author       = {{Wu, Hao and Nüske, Feliks and Paul, Fabian and Klus, Stefan and Koltai, Péter and Noé, Frank}},
  issn         = {{0021-9606}},
  journal      = {{The Journal of Chemical Physics}},
  title        = {{{Variational Koopman models: Slow collective variables and molecular kinetics from short off-equilibrium simulations}}},
  doi          = {{10.1063/1.4979344}},
  year         = {{2017}},
}

@article{21937,
  author       = {{Nüske, Feliks and Schneider, Reinhold and Vitalini, Francesca and Noé, Frank}},
  issn         = {{0021-9606}},
  journal      = {{The Journal of Chemical Physics}},
  title        = {{{Variational tensor approach for approximating the rare-event kinetics of macromolecular systems}}},
  doi          = {{10.1063/1.4940774}},
  year         = {{2016}},
}

@article{21936,
  author       = {{Nüske, Feliks and Keller, Bettina G. and Pérez-Hernández, Guillermo and Mey, Antonia S. J. S. and Noé, Frank}},
  issn         = {{1549-9618}},
  journal      = {{Journal of Chemical Theory and Computation}},
  pages        = {{1739--1752}},
  title        = {{{Variational Approach to Molecular Kinetics}}},
  doi          = {{10.1021/ct4009156}},
  year         = {{2014}},
}

@article{21935,
  author       = {{Noé, Frank and Nüske, Feliks}},
  issn         = {{1540-3459}},
  journal      = {{Multiscale Modeling & Simulation}},
  pages        = {{635--655}},
  title        = {{{A Variational Approach to Modeling Slow Processes in Stochastic Dynamical Systems}}},
  doi          = {{10.1137/110858616}},
  year         = {{2013}},
}