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 -