The Distributional Koopman Operator for Random Dynamical Systems

TL;DR

This work develops the Distributional Koopman Operator (DKO) to perform Koopman analysis on random dynamical systems when only distributional data is available, avoiding particle tracking. The DKO acts on observables of probability distributions via D_t h(pi) = h(T_t(pi)), with T_t the transfer operator, making the operator linear and semigroup-valued. A Hilbert-space framework is built by embedding distributional observables into random-measure spaces, and a data-driven DMD scheme is proven to converge to the best finite-rank DKO approximation as data grows. Numerical experiments on random circle rotations, stochastic differential equations, and real dust-plume data demonstrate accurate eigenstructure recovery, robustness to data quantity and noise, and the ability to predict not only means but variances of observables. Collectively, the results show that DKO extends Koopman analysis to distributional data, enabling trajectory-free analysis with nonlinear observables and practical applicability to uncertainty-rich systems.

Abstract

The Distributional Koopman Operator (DKO) is introduced as a way to perform Koopman analysis on random dynamical systems where only aggregate distribution data is available, thereby eliminating the need for particle tracking or detailed trajectory data. Our DKO generalizes the stochastic Koopman operator (SKO) to allow for observables of probability distributions, using the transfer operator to propagate these probability distributions forward in time. Like the SKO, the DKO is linear with semigroup properties, and we show that the dynamical mode decomposition (DMD) approximation can converge to the DKO in the large data limit. The DKO is particularly useful for random dynamical systems where trajectory information is unavailable.

Figures (6)

• Figure 1: Left: Ten eigenvalues of $S_m$ (blue triangles) and the $D_m$ (red rectangle) closest to $\lambda_1,\ldots,\lambda_{10}$, and compared against the values $\lambda_k = (\mathrm{i} - \mathrm{i}e^{\mathrm{i}k})/k$ (yellow star). Right: The eigenfunction of $S_m$ and $D_m$ associated with $\lambda_1$ (top row) and the eigenfunction associated with $\lambda_3$ (bottom row), compared against the analytically known functions.
• Figure 2: The MSE error between the true and the computed eigenvalues of $S_m$ and $D_m$ as the number of data points increases (left) and as the noise level in the trajectory data increases (right).
• Figure 3: DKO prediction for state-independent noise (left) and state-dependent noise (right) when the SDE drives the randomized dynamical system.
• Figure 4: Variance prediction for the SDE with state-independent noise (left) and the SDE with state-dependent noise (right). The test SDEs are shown in \ref{['subsec:noise_independent', 'subsec:noise_dependent']}) and experimental details are discussed in \ref{['subsec:variance']}.
• Figure 5: Left: One snapshot of the DustSCAN2022 data, showing dust clouds. Right: DKO eigenvalues (blue dots) along with the unit circle (black line).

Theorems & Definitions (21)

• Example 1: Random Rotations on the Circle
• Example 2: Stochastic Differential Equations
• Definition 1: Pushforward Measure
• Definition 2: Transfer Operator for an RDS
• Definition 3: SKO
• Definition 1: DKO
• Lemma 2
• Proof 1
• Lemma 3
• Proof 2