Feliks Nüske

Sebastian Peitz, Hans Harder, Feliks Nüske et al.

The Koopman operator has become an essential tool for data-driven analysis, prediction and control of complex systems. The main reason is the enormous potential of identifying linear function space representations of nonlinear dynamics from measurements. This equally applies to ordinary, stochastic, and partial differential equations (PDEs). Until now, with a few exceptions only, the PDE case is mostly treated rather superficially, and the specific structure of the underlying dynamics is largely ignored. In this paper, we show that symmetries in the system dynamics can be carried over to the Koopman operator, which allows us to significantly increase the model efficacy. Moreover, the situation where we only have access to partial observations (i.e., measurements, as is very common for experimental data) has not been treated to its full extent, either. Moreover, we address the highly-relevant case where we cannot measure the full state, where alternative approaches (e.g., delay coordinates) have to be considered. We derive rigorous statements on the required number of observables in this situation, based on embedding theory. We present numerical evidence using various numerical examples including the wave equation and the Kuramoto-Sivashinsky equation.

Vahid Nateghi, Lara Neureither, Selma Moqvist et al.

Coarse graining (CG) is an important task for efficient modeling and simulation of complex multi-scale systems, such as the conformational dynamics of biomolecules. This work presents a projection-based coarse-graining formalism for general underdamped Langevin dynamics. Following the Zwanzig projection approach, we derive a closed-form expression for the coarse grained dynamics. In addition, we show how the generator Extended Dynamic Mode Decomposition (gEDMD) method, which was developed in the context of Koopman operator methods, can be used to model the CG dynamics and evaluate its kinetic properties, such as transition timescales. Finally, we combine our approach with thermodynamic interpolation (TI), a generative approach to transform samples between thermodynamic conditions, to extend the scope of the approach across thermodynamic states without repeated numerical simulations. Using a two-dimensional model system, we demonstrate that the proposed method allows to accurately capture the thermodynamic and kinetic properties of the full-space model.

Stefan Klus, Patrick Gelß, Feliks Nüske et al.

We derive symmetric and antisymmetric kernels by symmetrizing and antisymmetrizing conventional kernels and analyze their properties. In particular, we compute the feature space dimensions of the resulting polynomial kernels, prove that the reproducing kernel Hilbert spaces induced by symmetric and antisymmetric Gaussian kernels are dense in the space of symmetric and antisymmetric functions, and propose a Slater determinant representation of the antisymmetric Gaussian kernel, which allows for an efficient evaluation even if the state space is high-dimensional. Furthermore, we show that by exploiting symmetries or antisymmetries the size of the training data set can be significantly reduced. The results are illustrated with guiding examples and simple quantum physics and chemistry applications.

Stefan Klus, Feliks Nüske, Boumediene Hamzi

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.

Stefan Klus, Feliks Nüske, Sebastian Peitz et al.

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.

Feliks Nüske, Patrick Gelß, Stefan Klus et al.

Recent years have seen rapid advances in the data-driven analysis of dynamical systems based on Koopman operator theory and related approaches. On the other hand, low-rank tensor product approximations -- in particular the tensor train (TT) format -- have become a valuable tool for the solution of large-scale problems in a number of fields. In this work, we combine Koopman-based models and the TT format, enabling their application to high-dimensional problems in conjunction with a rich set of basis functions or features. We derive efficient algorithms to obtain a reduced matrix representation of the system's evolution operator starting from an appropriate low-rank representation of the data. These algorithms can be applied to both stationary and non-stationary systems. We establish the infinite-data limit of these matrix representations, and demonstrate our methods' capabilities using several benchmark data sets.

Lorenzo Boninsegna, Feliks Nüske, Cecilia Clementi

With the rapid increase of available data for complex systems, there is great interest in the extraction of physically relevant information from massive datasets. Recently, a framework called Sparse Identification of Nonlinear Dynamics (SINDy) has been introduced to identify the governing equations of dynamical systems from simulation data. In this study, we extend SINDy to stochastic dynamical systems, which are frequently used to model biophysical processes. We prove the asymptotic correctness of stochastics SINDy in the infinite data limit, both in the original and projected variables. We discuss algorithms to solve the sparse regression problem arising from the practical implementation of SINDy, and show that cross validation is an essential tool to determine the right level of sparsity. We demonstrate the proposed methodology on two test systems, namely, the diffusion in a one-dimensional potential, and the projected dynamics of a two-dimensional diffusion process.

Hao Wu, Feliks Nüske, Fabian Paul et al.

Markov state models (MSMs) and Master equation models are popular approaches to approximate molecular kinetics, equilibria, metastable states, and reaction coordinates in terms of a state space discretization usually obtained by clustering. Recently, a powerful generalization of MSMs has been introduced, the variational approach (VA) of molecular kinetics and its special case the time-lagged independent component analysis (TICA), which allow us to approximate slow collective variables and molecular kinetics by linear combinations of smooth basis functions or order parameters. While it is known how to estimate MSMs from trajectories whose starting points are not sampled from an equilibrium ensemble, this has not yet been the case for TICA and the VA. Previous estimates from short trajectories, have been strongly biased and thus not variationally optimal. Here, we employ Koopman operator theory and ideas from dynamic mode decomposition (DMD) to extend the VA and TICA to non-equilibrium data. The main insight is that the VA and TICA provide a coefficient matrix that we call Koopman model, as it approximates the underlying dynamical (Koopman) operator in conjunction with the basis set used. This Koopman model can be used to compute a stationary vector to reweight the data to equilibrium. From such a Koopman-reweighted sample, equilibrium expectation values and variationally optimal reversible Koopman models can be constructed even with short simulations. The Koopman model can be used to propagate densities, and its eigenvalue decomposition provide estimates of relaxation timescales and slow collective variables for dimension reduction. Koopman models are generalizations of Markov state models, TICA and the linear VA and allow molecular kinetics to be described without a cluster discretization.