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Data-driven Reduction of Transfer Operators for Particle Clustering Dynamics

Nathalie Wehlitz, Grigorios A. Pavliotis, Christof Schütte, Stefanie Winkelmann

TL;DR

The paper develops an operator-based framework to reduce clustering dynamics of interacting particles by projecting the particle-level transfer operator onto concentration space and then onto a coarse, data-driven partition of a low-dimensional manifold. Concentration data generated by a Dean–Kawasaki SPDE are embedded with Diffusion Maps to reveal intrinsic geometry, from which a Markov chain on coarse states is estimated and analyzed for metastability and timescales. The approach is demonstrated on multichromatic and Morse potentials, revealing dominant transition pathways, implied timescales, and early-warning structures prior to cluster collapse. The results provide an interpretable, computationally efficient reduced model that preserves key clustering features and can be extended to more complex potentials and higher dimensions, with potential applications in swarming, opinion dynamics, and biomolecular clustering.

Abstract

We develop an operator-based framework to coarse-grain interacting particle systems that exhibit clustering dynamics. Starting from the particle-based transfer operator, we first construct a sequence of reduced representations: the operator is projected onto concentrations and then further reduced by representing the concentration dynamics on a geometric low-dimensional manifold and an adapted finite-state discretization. The resulting coarse-grained transfer operator is finally estimated from dynamical simulation data by inferring the transition probabilities between the Markov states. Applied to systems with multichromatic and Morse interaction potentials, the reduced model reproduces key features of the clustering process, including transitions between cluster configurations and the emergence of metastable states. Spectral analysis and transition-path analysis of the estimated operator reveal implied time scales and dominant transition pathways, providing an interpretable and efficient description of particle-clustering dynamics.

Data-driven Reduction of Transfer Operators for Particle Clustering Dynamics

TL;DR

The paper develops an operator-based framework to reduce clustering dynamics of interacting particles by projecting the particle-level transfer operator onto concentration space and then onto a coarse, data-driven partition of a low-dimensional manifold. Concentration data generated by a Dean–Kawasaki SPDE are embedded with Diffusion Maps to reveal intrinsic geometry, from which a Markov chain on coarse states is estimated and analyzed for metastability and timescales. The approach is demonstrated on multichromatic and Morse potentials, revealing dominant transition pathways, implied timescales, and early-warning structures prior to cluster collapse. The results provide an interpretable, computationally efficient reduced model that preserves key clustering features and can be extended to more complex potentials and higher dimensions, with potential applications in swarming, opinion dynamics, and biomolecular clustering.

Abstract

We develop an operator-based framework to coarse-grain interacting particle systems that exhibit clustering dynamics. Starting from the particle-based transfer operator, we first construct a sequence of reduced representations: the operator is projected onto concentrations and then further reduced by representing the concentration dynamics on a geometric low-dimensional manifold and an adapted finite-state discretization. The resulting coarse-grained transfer operator is finally estimated from dynamical simulation data by inferring the transition probabilities between the Markov states. Applied to systems with multichromatic and Morse interaction potentials, the reduced model reproduces key features of the clustering process, including transitions between cluster configurations and the emergence of metastable states. Spectral analysis and transition-path analysis of the estimated operator reveal implied time scales and dominant transition pathways, providing an interpretable and efficient description of particle-clustering dynamics.
Paper Structure (48 sections, 54 equations, 10 figures, 2 tables)

This paper contains 48 sections, 54 equations, 10 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Model formulation
  3. Particle-based dynamics and SPDE approximation
  4. Approximation by the Dean--Kawasaki equation.
  5. Exemplary interaction potentials
  6. Parameter values.
  7. Analytical reduction of the transfer operator
  8. Transfer operator of the particle-based system
  9. Projection onto discrtized concentrations
  10. Transfer operator induced by the SPDE.
  11. Projection onto a coarse-gained subspace
  12. Abstract coarse partition.
  13. Coarse basis functions and subspace.
  14. Galerkin projection.
  15. Role in the full framework.
  16. ...and 33 more sections

Figures (10)

  • Figure 1: Hierarchy of operators. Starting from the Perron--Frobenius operator $\mathcal{P}_N^\tau$ of the particle-based dynamics \ref{['eq:Perron-Frobenius']}, we obtain the operator $\textbf{P}^\tau_{\!N}$ between particle concentrations \ref{['matrix_T']}. The reduction $P^\tau$\ref{['MLE_P']} of $\textbf{P}^\tau_{\!N}$ can be obtained by the following procedure: (I) Take concentrations of particle-based simulations or fluctuating densities obtained by solving the Dean--Kawasaki SPDE numerically. (II) Apply Diffusion Maps to get a geometric embedding of the high-dimensional manifold of concentrations (\ref{['sec:galerkin_diffmap']}) and discretize the resulting low-dimensional projection space. Finally, estimate the transition matrix $P^{\tau}$ of the reduced Markov chain using dynamical data (\ref{['sec:galerkin_msm']}).
  • Figure 2: Trajectory and Diffusion Map embedding for multichromatic potential \ref{['eq:F']} of \ref{['ex1']}. (a) Single SPDE simulation of clustering dynamics until $T=120$, after starting at $t=0$ in a uniform distribution. (b) Projection of particle densities from five independent simulations onto the first two Diffusion Map coordinates (Section \ref{['sec:galerkin_diffmap']}). Parameters are given on page \ref{['parameter_values']}, and the sampling interval length is $dt_{\text{diff}}=1$.
  • Figure 3: Trajectory and Diffusion Map embedding for Morse potential \ref{['eq:F_Morse']} of \ref{['ex2']}. (a) Single SPDE simulation of clustering dynamics until $T=2000$, after starting at $t=0$ in a uniform distribution. (b) Projection of particle densities from ten independent simulations onto the first two Diffusion Map coordinates (Section \ref{['sec:galerkin_diffmap']}). Parameters are given on page \ref{['parameter_values']}, and the sampling interval length is $dt_{\text{diff}}=20$.
  • Figure 4: Multichromatic interaction potential: Dynamics on the reduced space (uniform grid). (a) The relevant boxes are highlighted in orange, with higher transparency indicating lower stay probability. Green arrows illustrate the average transition direction, conditional on the process making a jump. (b) Metastable sets of Markov chain states identified using PCCA+ (set $M_1$ in blue, set $M_2$ in red). (c) Sets of Markov chain states representing the four-cluster concentrations ($A$ in blue) and the one-cluster concentrations ($B$ in red).
  • Figure 5: Multichromatic interaction potential: Dynamics on the reduced space (Voronoi cells). (a) The cells are highlighted in orange, with higher transparency indicating lower stay probability. Green arrows illustrate the average transition direction, conditional on the process making a jump. (b) Metastable sets of Markov chain states identified using PCCA+ (set $M_1$ in blue, set $M_2$ in red). (c) Sets of Markov chain states representing the four-cluster concentrations ($A$ in blue) and the one-cluster concentrations ($B$ in red).
  • ...and 5 more figures

Theorems & Definitions (6)

  • Example 1
  • Example 2
  • Remark 1
  • Remark 2: Other initial configurations
  • Remark 3: Relation between different timescales
  • Remark 4: Limitation of TPT