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Entropic transfer operators for stochastic systems

Hancheng Bi, Clément Sarrazin, Bernhard Schmitzer, Thilo D. Stier

TL;DR

This work extends entropic transfer operators to stochastic and non-stationary dynamical systems by introducing a double-blurred regularization $T^{\varepsilon}=G^{\varepsilon}_{\nu\nu} T G^{\varepsilon}_{\mu\mu}$ and its empirical extension $T^{A,\varepsilon}_N$, providing quantitative Hilbert--Schmidt convergence guarantees. It establishes a rigorous pathway from finite data to a compact operator that preserves dynamics above length scales $\sqrt{\varepsilon}$ and analyzes spectral convergence, out-of-sample embedding, and the relation to prior work. The paper also demonstrates practical applicability with numerical experiments, including stochastic torus shifts and Rayleigh–Bénard convection data, highlighting scalability via GPU-accelerated Sinkhorn computations and providing a workflow that avoids discretization artefacts typical of Ulam-type approaches. The results show robust, mesh-free spectral analysis of stochastic dynamics, with interpretable embeddings and efficient extensions to unseen samples, offering a data-driven, scalable tool for exploring high-dimensional, noisy dynamical systems. Overall, the entropic transfer operator framework delivers a principled, scalable, and interpretable approach for spectral analysis of stochastic dynamical systems and non-stationary processes, with concrete convergence guarantees and practical demonstrations in fluid dynamics.

Abstract

Dynamical systems can be analyzed via their Frobenius-Perron transfer operator and its estimation from data is an active field of research. Recently entropic transfer operators have been introduced to estimate the operator of deterministic systems. The approach is based on the regularizing properties of entropic optimal transport plans. In this article we generalize the method to stochastic and non-stationary systems and give a quantitative convergence analysis of the empirical operator as the available samples increase. We introduce a way to extend the operator's eigenfunctions to previously unseen samples, such that they can be efficiently included into a spectral embedding. The practicality and numerical scalability of the method are demonstrated on a real-world fluid dynamics experiment.

Entropic transfer operators for stochastic systems

TL;DR

This work extends entropic transfer operators to stochastic and non-stationary dynamical systems by introducing a double-blurred regularization and its empirical extension , providing quantitative Hilbert--Schmidt convergence guarantees. It establishes a rigorous pathway from finite data to a compact operator that preserves dynamics above length scales and analyzes spectral convergence, out-of-sample embedding, and the relation to prior work. The paper also demonstrates practical applicability with numerical experiments, including stochastic torus shifts and Rayleigh–Bénard convection data, highlighting scalability via GPU-accelerated Sinkhorn computations and providing a workflow that avoids discretization artefacts typical of Ulam-type approaches. The results show robust, mesh-free spectral analysis of stochastic dynamics, with interpretable embeddings and efficient extensions to unseen samples, offering a data-driven, scalable tool for exploring high-dimensional, noisy dynamical systems. Overall, the entropic transfer operator framework delivers a principled, scalable, and interpretable approach for spectral analysis of stochastic dynamical systems and non-stationary processes, with concrete convergence guarantees and practical demonstrations in fluid dynamics.

Abstract

Dynamical systems can be analyzed via their Frobenius-Perron transfer operator and its estimation from data is an active field of research. Recently entropic transfer operators have been introduced to estimate the operator of deterministic systems. The approach is based on the regularizing properties of entropic optimal transport plans. In this article we generalize the method to stochastic and non-stationary systems and give a quantitative convergence analysis of the empirical operator as the available samples increase. We introduce a way to extend the operator's eigenfunctions to previously unseen samples, such that they can be efficiently included into a spectral embedding. The practicality and numerical scalability of the method are demonstrated on a real-world fluid dynamics experiment.

Paper Structure

This paper contains 47 sections, 26 theorems, 120 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Motivation and related work
  3. Dynamical systems and transfer operators.
  4. Approximation by compact operators and from discrete data.
  5. Entropic transfer operators.
  6. Contribution and outline
  7. Introduction of double-blurred entropic transfer operator.
  8. Quantitative probabilistic convergence analysis of $T^{A,\varepsilon}_N$ to $T^\varepsilon$.
  9. Further convergence results.
  10. Out-of-sample embedding.
  11. Examples and numerical experiments.
  12. Relation to beier2024transfer.
  13. Setting and notation
  14. Optimal transport and entropic regularization
  15. Transfer operators
  16. ...and 32 more sections

Key Result

Proposition 1.1

For $\mu, \nu \in \mathcal{P}(\mathcal{X})$, some cost function $c \in \mathcal{C}(\mathcal{X} \times \mathcal{X})$, and a regularization parameter $\varepsilon > 0$, the corresponding primal entropic optimal transport problem is given by where is the set of transport plans between $\mu$ and $\nu$ (recall that $P_{\mathcal{X}}^i$, $i=1,2$, are the projections from $\mathcal{X} \times \mathcal{X}

Figures (13)

  • Figure 1: Overview of operators defined in this paper. The table summarizes the different operators defined in this paper and the graph below lists the results on their relations.
  • Figure 2: Counterexample for convergence with single blurring. The leftmost panel shows the kernel of the true transfer operator $\widetilde{T}^{\varepsilon}$ w.r.t. $\mu\otimes\nu$ (it is uniform). The second panel shows the kernel of the discrete observed operator $T^N$ w.r.t. $\mu^N \otimes \nu^N$ (here $\nu^N = \nu$). The third panel shows how the kernel changes when the single blur operator is applied. A small amount of the mass that was previously mapped to the top row, is now mapped to the bottom row and vice versa. The rightmost panel shows the kernel of the fully assembled operator estimate w.r.t. $\mu\otimes\nu$. The kernel oscillates between $2(1-\varsigma)$ and $2\varsigma$ and therefore does not converge towards the kernel of $\widetilde{T}^{\varepsilon}$ in the $L^2$-norm as $N \to \infty$.
  • Figure 3: Integral kernels $t$, $t^\varepsilon$ and $t_N^\varepsilon$ for the system \ref{['eq:torus_pi']} for $\sigma = 0.05$, $\varepsilon = 0.01$, and various $N$. Yellow indicates high values, dark blue indicates zero; color scales are adjusted to each panel separately for better visibility.
  • Figure 4: $L^2$ distances between different integral kernels and parameters for the system \ref{['eq:torus_pi']}. Colors for encoding $\sigma$ and $\varepsilon$ are consistent in all panels. Vertical dashed lines indicate values of $\varepsilon$ used in other panels. Plots that involve empirical data show the estimated mean with $95\%$ confidence interval (based on $100$ simulations), all y-axes are in log scale.
  • Figure 5: The second and fourth dominant eigenfunctions (real part)for the system \ref{['eq:torus_pi_simple']} for $\sigma = 0.01$, $\varepsilon = 0.01$ on $(x_i)_i$ (red points), out-of-sample extension (blue line), and true eigenfunctions (grey line, aligned over the ambiguous phase shift).
  • ...and 8 more figures

Theorems & Definitions (60)

  • Proposition 1.1: Entropic optimal transport
  • Remark 1.2
  • Proposition 1.3
  • proof
  • Definition 1.5: Entropic transport kernel
  • Proposition 1.6: $p$-Wasserstein metric
  • Definition 1.7
  • Proposition 1.8
  • proof
  • Definition 1.9
  • ...and 50 more