Péter Koltai

Gary Froyland, Oliver Junge, Péter Koltai

The long-term distributions of trajectories of a flow are described by invariant densities, i.e. fixed points of an associated transfer operator. In addition, global slowly mixing structures, such as almost-invariant sets, which partition phase space into regions that are almost dynamically disconnected, can also be identified by certain eigenfunctions of this operator. Indeed, these structures are often hard to obtain by brute-force trajectory-based analyses. In a wide variety of applications, transfer operators have proven to be very efficient tools for an analysis of the global behavior of a dynamical system. The computationally most expensive step in the construction of an approximate transfer operator is the numerical integration of many short term trajectories. In this paper, we propose to directly work with the infinitesimal generator instead of the operator, completely avoiding trajectory integration. We propose two different discretization schemes; a cell based discretization and a spectral collocation approach. Convergence can be shown in certain circumstances. We demonstrate numerically that our approach is much more efficient than the operator approach, sometimes by several orders of magnitude.

Stefan Klus, Péter Koltai, Christof Schütte

Information about the behavior of dynamical systems can often be obtained by analyzing the eigenvalues and corresponding eigenfunctions of linear operators associated with a dynamical system. Examples of such operators are the Perron-Frobenius and the Koopman operator. In this paper, we will review different methods that have been developed over the last decades to compute finite-dimensional approximations of these infinite-dimensional operators - e.g. Ulam's method and Extended Dynamic Mode Decomposition (EDMD) - and highlight the similarities and differences between these approaches. The results will be illustrated using simple stochastic differential equations and molecular dynamics examples.

Henry Schoeller, Robin Chemnitz, Péter Koltai et al.

We investigate the lifetime of dynamical regimes under the impact of noise motivated by low-dimensional models of the atmosphere. One may expect that the inclusion of noise tends to make the system leave prescribed regions of the state space faster. However, for relevant systems with complexities ranging from phenomenological toy models to reduced models of atmospheric dynamics, this intuition has proven misleading. As long as the noise is sufficiently small, the noisy system stays in regimes of interest on average longer than its deterministic counterpart, an effect we call ``stochastic inertia''. This phenomenon has been observed through extensive numerical simulations for different noise levels. We propose a numerical technique for testing the occurrence of stochastic inertia, constructing, for any fixed noise level, a Markov chain on the set of points given by a sufficiently long trajectory of the system without noise. The method is shown to correctly predict the presence of stochastic inertia in simple systems, and its utility is demonstrated on a paradigm model of atmospheric dynamics.

April Herwig, Matthew J. Colbrook, Oliver Junge et al.

Koopman operator theory enables linear analysis of nonlinear dynamical systems by lifting their evolution to infinite-dimensional function spaces. However, finite-dimensional approximations of Koopman and transfer (Frobenius--Perron) operators are prone to spectral pollution, introducing spurious eigenvalues that can compromise spectral computations. While recent advances have yielded provably convergent methods for Koopman operators, analogous tools for general transfer operators remain limited. In this paper, we present algorithms for computing spectral properties of transfer operators without spectral pollution, including extensions to the Hardy-Hilbert space. Case studies--ranging from families of Blaschke maps with known spectrum to a molecular dynamics model of protein folding--demonstrate the accuracy and flexibility of our approach. Notably, we demonstrate that spectral features can arise even when the corresponding eigenfunctions lie outside the chosen space, highlighting the functional-analytic subtleties in defining the "true" Koopman spectrum. Our methods offer robust tools for spectral estimation across a broad range of applications.

Niklas Wulkow, Péter Koltai, Vikram Sunkara et al.

We present a numerical method to model dynamical systems from data. We use the recently introduced method Scalable Probabilistic Approximation (SPA) to project points from a Euclidean space to convex polytopes and represent these projected states of a system in new, lower-dimensional coordinates denoting their position in the polytope. We then introduce a specific nonlinear transformation to construct a model of the dynamics in the polytope and to transform back into the original state space. To overcome the potential loss of information from the projection to a lower-dimensional polytope, we use memory in the sense of the delay-embedding theorem of Takens. By construction, our method produces stable models. We illustrate the capacity of the method to reproduce even chaotic dynamics and attractors with multiple connected components on various examples.

Mattes Mollenhauer, Péter Koltai

Given the joint distribution of two random variables $X,Y$ on some second countable locally compact Hausdorff space, we investigate the statistical approximation of the $L^2$-operator defined by $[Pf](x) := \mathbb{E}[ f(Y) \mid X = x ]$ under minimal assumptions. By modifying its domain, we prove that $P$ can be arbitrarily well approximated in operator norm by Hilbert-Schmidt operators acting on a reproducing kernel Hilbert space. This fact allows to estimate $P$ uniformly by finite-rank operators over a dense subspace even when $P$ is not compact. In terms of modes of convergence, we thereby obtain the superiority of kernel-based techniques over classically used parametric projection approaches such as Galerkin methods. This also provides a novel perspective on which limiting object the nonparametric estimate of $P$ converges to. As an application, we show that these results are particularly important for a large family of spectral analysis techniques for Markov transition operators. Our investigation also gives a new asymptotic perspective on the so-called kernel conditional mean embedding, which is the theoretical foundation of a wide variety of techniques in kernel-based nonparametric inference.

Mattes Mollenhauer, Stefan Klus, Christof Schütte et al.

We consider autocovariance operators of a stationary stochastic process on a Polish space that is embedded into a reproducing kernel Hilbert space. We investigate how empirical estimates of these operators converge along realizations of the process under various conditions. In particular, we examine ergodic and strongly mixing processes and obtain several asymptotic results as well as finite sample error bounds. We provide applications of our theory in terms of consistency results for kernel PCA with dependent data and the conditional mean embedding of transition probabilities. Finally, we use our approach to examine the nonparametric estimation of Markov transition operators and highlight how our theory can give a consistency analysis for a large family of spectral analysis methods including kernel-based dynamic mode decomposition.

Andreas Bittracher, Stefan Klus, Boumediene Hamzi et al.

We present a novel kernel-based machine learning algorithm for identifying the low-dimensional geometry of the effective dynamics of high-dimensional multiscale stochastic systems. Recently, the authors developed a mathematical framework for the computation of optimal reaction coordinates of such systems that is based on learning a parametrization of a low-dimensional transition manifold in a certain function space. In this article, we enhance this approach by embedding and learning this transition manifold in a reproducing kernel Hilbert space, exploiting the favorable properties of kernel embeddings. Under mild assumptions on the kernel, the manifold structure is shown to be preserved under the embedding, and distortion bounds can be derived. This leads to a more robust and more efficient algorithm compared to previous parametrization approaches.

Andreas Bittracher, Péter Koltai, Oliver Junge

Metastable behavior in dynamical systems may be a significant challenge for a simulation based analysis. In recent years, transfer operator based approaches to problems exhibiting metastability have matured. In order to make these approaches computationally feasible for larger systems, various reduction techniques have been proposed: For example, Schütte introduced a spatial transfer operator which acts on densities on configuration space, while Weber proposed to avoid trajectory simulation (like Froyland et al.) by considering a discrete generator. In this manuscript, we show that even though the family of spatial transfer operators is not a semigroup, it possesses a well defined generating structure. What is more, the pseudo generators up to order 4 in the Taylor expansion of this family have particularly simple, explicit expressions involving no momentum averaging. This makes collocation methods particularly easy to implement and computationally efficient, which in turn may open the door for further efficiency improvements in, e.g., the computational treatment of conformation dynamics. We experimentally verify the predicted properties of these pseudo generators by means of two academic examples.

Gero Friesecke, Oliver Junge, Péter Koltai

We propose a new approach to the transfer operator based analysis of the conformation dynamics of molecules. It is based on a statistical independence ansatz for the eigenfunctions of the operator related to a partitioning into subsystems. Numerical tests performed on small systems show excellent qualitative agreement between mean field and exact model, at greatly reduced computational cost.