TY - CONF AB - 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. AU - Schaller, Manuel AU - Worthmann, Karl AU - Philipp, Friedrich AU - Peitz, Sebastian AU - Nüske, Feliks ID - 30125 IS - 1 T2 - IFAC-PapersOnLine TI - Towards reliable data-based optimal and predictive control using extended DMD VL - 56 ER - TY - JOUR AB - 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. AU - Nüske, Feliks AU - Peitz, Sebastian AU - Philipp, Friedrich AU - Schaller, Manuel AU - Worthmann, Karl ID - 23428 JF - Journal of Nonlinear Science TI - Finite-data error bounds for Koopman-based prediction and control VL - 33 ER - TY - JOUR AB - 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. AU - Klus, Stefan AU - Nüske, Feliks AU - Peitz, Sebastian ID - 29673 IS - 31 JF - Journal of Physics A: Mathematical and Theoretical TI - Koopman analysis of quantum systems VL - 55 ER - TY - JOUR AU - Nüske, Feliks AU - Gelß, Patrick AU - Klus, Stefan AU - Clementi, Cecilia ID - 24169 JF - Physica D: Nonlinear Phenomena SN - 0167-2789 TI - Tensor-based computation of metastable and coherent sets ER - TY - JOUR AU - Klus, Stefan AU - Gelß, Patrick AU - Nüske, Feliks AU - Noé, Frank ID - 24170 JF - Machine Learning: Science and Technology SN - 2632-2153 TI - Symmetric and antisymmetric kernels for machine learning problems in quantum physics and chemistry ER - TY - JOUR AB - 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. AU - Nüske, Feliks AU - Koltai, Péter AU - Boninsegna, Lorenzo AU - Clementi, Cecilia ID - 21820 JF - Entropy SN - 1099-4300 TI - Spectral Properties of Effective Dynamics from Conditional Expectations ER - TY - JOUR AB - 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. AU - Klus, Stefan AU - Nüske, Feliks AU - Hamzi, Boumediene ID - 21819 JF - Entropy SN - 1099-4300 TI - Kernel-Based Approximation of the Koopman Generator and Schrödinger Operator ER - TY - JOUR AB - 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. AU - Klus, Stefan AU - Nüske, Feliks AU - Peitz, Sebastian AU - Niemann, Jan-Hendrik AU - Clementi, Cecilia AU - Schütte, Christof ID - 16288 JF - Physica D: Nonlinear Phenomena SN - 0167-2789 TI - Data-driven approximation of the Koopman generator: Model reduction, system identification, and control VL - 406 ER - TY - JOUR AU - Nüske, Feliks AU - Boninsegna, Lorenzo AU - Clementi, Cecilia ID - 21944 JF - The Journal of Chemical Physics SN - 0021-9606 TI - Coarse-graining molecular systems by spectral matching ER - TY - JOUR AU - Litzinger, Florian AU - Boninsegna, Lorenzo AU - Wu, Hao AU - Nüske, Feliks AU - Patel, Raajen AU - Baraniuk, Richard AU - Noé, Frank AU - Clementi, Cecilia ID - 21940 JF - Journal of Chemical Theory and Computation SN - 1549-9618 TI - Rapid Calculation of Molecular Kinetics Using Compressed Sensing ER - TY - JOUR AU - Klus, Stefan AU - Nüske, Feliks AU - Koltai, Péter AU - Wu, Hao AU - Kevrekidis, Ioannis AU - Schütte, Christof AU - Noé, Frank ID - 21941 JF - Journal of Nonlinear Science SN - 0938-8974 TI - Data-Driven Model Reduction and Transfer Operator Approximation ER - TY - JOUR AU - Boninsegna, Lorenzo AU - Nüske, Feliks AU - Clementi, Cecilia ID - 21942 JF - The Journal of Chemical Physics SN - 0021-9606 TI - Sparse learning of stochastic dynamical equations ER - TY - JOUR AU - Hruska, Eugen AU - Abella, Jayvee R. AU - Nüske, Feliks AU - Kavraki, Lydia E. AU - Clementi, Cecilia ID - 21943 JF - The Journal of Chemical Physics SN - 0021-9606 TI - Quantitative comparison of adaptive sampling methods for protein dynamics ER - TY - JOUR AU - Nüske, Feliks AU - Wu, Hao AU - Prinz, Jan-Hendrik AU - Wehmeyer, Christoph AU - Clementi, Cecilia AU - Noé, Frank ID - 21938 JF - The Journal of Chemical Physics SN - 0021-9606 TI - Markov state models from short non-equilibrium simulations—Analysis and correction of estimation bias ER - TY - JOUR AU - Wu, Hao AU - Nüske, Feliks AU - Paul, Fabian AU - Klus, Stefan AU - Koltai, Péter AU - Noé, Frank ID - 21939 JF - The Journal of Chemical Physics SN - 0021-9606 TI - Variational Koopman models: Slow collective variables and molecular kinetics from short off-equilibrium simulations ER - TY - JOUR AU - Nüske, Feliks AU - Schneider, Reinhold AU - Vitalini, Francesca AU - Noé, Frank ID - 21937 JF - The Journal of Chemical Physics SN - 0021-9606 TI - Variational tensor approach for approximating the rare-event kinetics of macromolecular systems ER - TY - JOUR AU - Nüske, Feliks AU - Keller, Bettina G. AU - Pérez-Hernández, Guillermo AU - Mey, Antonia S. J. S. AU - Noé, Frank ID - 21936 JF - Journal of Chemical Theory and Computation SN - 1549-9618 TI - Variational Approach to Molecular Kinetics ER - TY - JOUR AU - Noé, Frank AU - Nüske, Feliks ID - 21935 JF - Multiscale Modeling & Simulation SN - 1540-3459 TI - A Variational Approach to Modeling Slow Processes in Stochastic Dynamical Systems ER -