Semigroup-Based Koopman Analysis

• Semigroup-Based Framework is a data-driven method that directly approximates the stochastic Koopman semigroup for analyzing SDEs.
• It employs Stochastic Dynamic Mode Decomposition to avoid costly matrix exponentials, leveraging a linear-in-Δt approximation and rigorous convergence theory.
• Optional neural-network dictionary learning refines basis functions for improved spectral estimates and reliable identification of dominant dynamical features.

Searching arXiv for the target paper and closely related Koopman semigroup work. A semigroup-based framework, in the sense developed for stochastic Koopman analysis, is a data-driven methodology that treats the stochastic Koopman family $(\mathcal{K}^t)_{t\ge 0}$ itself as the primary object of approximation for stochastic dynamical systems, rather than first approximating the infinitesimal generator and then exponentiating it. In the formulation of Stochastic Dynamic Mode Decomposition (SDMD), the target system is an SDE

$$d\mathbf{X}_t=\mathbf{b}(\mathbf{X}_t)\,dt+\mathbf{\sigma}(\mathbf{X}_t)\,d\mathbf{W}_t,\qquad \mathbf{X}_0=\mathbf{x}\in M\subseteq\mathbb{R}^d,$$

and the associated stochastic Koopman operators are defined by conditional expectation on $\mathcal{F}=L^2_\rho(M)$. The framework is characterized by direct approximation of the one-step semigroup operator $\mathcal{K}^{\Delta t}$, explicit dependence on the sampling time $\Delta t$, avoidance of matrix exponential computations, optional neural-network-based dictionary learning, and a three-tier convergence theory in the limits $m\to\infty$, $\Delta t\to 0$, and $N\to\infty$ (Xu et al., 23 Jan 2025).

1. Semigroup formulation and operator-theoretic setting

The underlying stochastic dynamics are modeled by the Itô system

$$d\mathbf{X}_t = \mathbf{b}(\mathbf{X}_t)\,dt + \mathbf{\sigma}(\mathbf{X}_t)\,d\mathbf{W}_t,$$

with drift $\mathbf{b}$, diffusion $$d\mathbf{X}_t=\mathbf{b}(\mathbf{X}_t)\,dt+\mathbf{\sigma}(\mathbf{X}_t)\,d\mathbf{W}_t,\qquad \mathbf{X}_0=\mathbf{x}\in M\subseteq\mathbb{R}^d,$$0, and $$d\mathbf{X}_t=\mathbf{b}(\mathbf{X}_t)\,dt+\mathbf{\sigma}(\mathbf{X}_t)\,d\mathbf{W}_t,\qquad \mathbf{X}_0=\mathbf{x}\in M\subseteq\mathbb{R}^d,$$1-dimensional Wiener process $$d\mathbf{X}_t=\mathbf{b}(\mathbf{X}_t)\,dt+\mathbf{\sigma}(\mathbf{X}_t)\,d\mathbf{W}_t,\qquad \mathbf{X}_0=\mathbf{x}\in M\subseteq\mathbb{R}^d,$$2. On a probability space $$d\mathbf{X}_t=\mathbf{b}(\mathbf{X}_t)\,dt+\mathbf{\sigma}(\mathbf{X}_t)\,d\mathbf{W}_t,\qquad \mathbf{X}_0=\mathbf{x}\in M\subseteq\mathbb{R}^d,$$3 and a state-space measure $$d\mathbf{X}_t=\mathbf{b}(\mathbf{X}_t)\,dt+\mathbf{\sigma}(\mathbf{X}_t)\,d\mathbf{W}_t,\qquad \mathbf{X}_0=\mathbf{x}\in M\subseteq\mathbb{R}^d,$$4 on $$d\mathbf{X}_t=\mathbf{b}(\mathbf{X}_t)\,dt+\mathbf{\sigma}(\mathbf{X}_t)\,d\mathbf{W}_t,\qquad \mathbf{X}_0=\mathbf{x}\in M\subseteq\mathbb{R}^d,$$5, the framework works in

$$d\mathbf{X}_t=\mathbf{b}(\mathbf{X}_t)\,dt+\mathbf{\sigma}(\mathbf{X}_t)\,d\mathbf{W}_t,\qquad \mathbf{X}_0=\mathbf{x}\in M\subseteq\mathbb{R}^d,$$6

The stochastic Koopman operator family $$d\mathbf{X}_t=\mathbf{b}(\mathbf{X}_t)\,dt+\mathbf{\sigma}(\mathbf{X}_t)\,d\mathbf{W}_t,\qquad \mathbf{X}_0=\mathbf{x}\in M\subseteq\mathbb{R}^d,$$7 is defined for $$d\mathbf{X}_t=\mathbf{b}(\mathbf{X}_t)\,dt+\mathbf{\sigma}(\mathbf{X}_t)\,d\mathbf{W}_t,\qquad \mathbf{X}_0=\mathbf{x}\in M\subseteq\mathbb{R}^d,$$8 by

$$d\mathbf{X}_t=\mathbf{b}(\mathbf{X}_t)\,dt+\mathbf{\sigma}(\mathbf{X}_t)\,d\mathbf{W}_t,\qquad \mathbf{X}_0=\mathbf{x}\in M\subseteq\mathbb{R}^d,$$9

The basic assumption is that $\mathcal{F}=L^2_\rho(M)$0 is a strongly continuous semigroup of bounded operators on $\mathcal{F}=L^2_\rho(M)$1. Thus $\mathcal{F}=L^2_\rho(M)$2, the semigroup property

$\mathcal{F}=L^2_\rho(M)$3

holds, and strong continuity is expressed as

$\mathcal{F}=L^2_\rho(M)$4

In this setting, $\mathcal{F}=L^2_\rho(M)$5 is a Markov (Koopman) semigroup.

Its infinitesimal generator is

$\mathcal{F}=L^2_\rho(M)$6

with domain

$\mathcal{F}=L^2_\rho(M)$7

For $\mathcal{F}=L^2_\rho(M)$8, Itô’s formula gives

$\mathcal{F}=L^2_\rho(M)$9

Formally, the semigroup–generator relation is

$\mathcal{K}^{\Delta t}$0

and the spectral correspondence is

$\mathcal{K}^{\Delta t}$1

or equivalently

$\mathcal{K}^{\Delta t}$2

This operator-theoretic relation is conceptually central because the numerical framework is built around approximating the one-step operator $\mathcal{K}^{\Delta t}$3, then using semigroup composition for long-time propagation and finite-difference recovery of the generator: $\mathcal{K}^{\Delta t}$4 This suggests a shift in emphasis from generator approximation to direct semigroup approximation (Xu et al., 23 Jan 2025).

2. SDMD and direct approximation of the one-step semigroup

The core computational construction is Stochastic Dynamic Mode Decomposition, described as a semigroup-based extension of EDMD tailored to stochastic systems. One chooses dictionary functions $\mathcal{K}^{\Delta t}$5 and the finite-dimensional subspace

$\mathcal{K}^{\Delta t}$6

The associated Gram-type matrices are

$\mathcal{K}^{\Delta t}$7

Given i.i.d. samples $\mathcal{K}^{\Delta t}$8, one forms the data matrix $\mathcal{K}^{\Delta t}$9 and generator-evaluated matrix $\Delta t$0, and then the empirical matrices

$\Delta t$1

For small sampling time $\Delta t$2, the generator definition yields the stochastic Taylor expansion

$\Delta t$3

Substituting this into an EDMD-type least-squares representation and minimizing the residual leads to

$\Delta t$4

with minimizer

$\Delta t$5

Neglecting the higher-order term for sufficiently small $\Delta t$6 gives the SDMD approximation

$\Delta t$7

The defining methodological distinction is that SDMD approximates $\Delta t$8 directly. If one instead approximates the generator matrix $\Delta t$9, then time-$m\to\infty$0 propagation requires a matrix exponential $m\to\infty$1, which the framework describes as expensive and delicate when $m\to\infty$2 is large or stiff. SDMD replaces that route by the linear-in-$m\to\infty$3 approximation

$m\to\infty$4

so that long-time evolution is represented through repeated multiplication,

$m\to\infty$5

The explicit incorporation of $m\to\infty$6 into the approximation is therefore both structural and computational (Xu et al., 23 Jan 2025).

3. Dictionary design and neural-network-based dictionary learning

In its basic form, SDMD uses a fixed dictionary $m\to\infty$7, such as polynomials or Fourier modes. In that setting, $m\to\infty$8 is computed analytically from the generator formula, which requires gradients and Hessians of the basis functions together with the SDE coefficients $m\to\infty$9.

To reduce manual basis engineering, the framework introduces SDMD with Dictionary Learning (SDMD-DL). Here the dictionary is parameterized by a neural network

$\Delta t\to 0$0

The learning variant uses paired data $\Delta t\to 0$1, where $\Delta t\to 0$2 is the state evolved from $\Delta t\to 0$3 after time $\Delta t\to 0$4, and optimizes the semigroup regression objective

$\Delta t\to 0$5

At the same time, the Koopman approximation is parameterized as

$\Delta t\to 0$6

with

$\Delta t\to 0$7

The training loss is

$\Delta t\to 0$8

where $\Delta t\to 0$9 is a Tikhonov regularization parameter. Training proceeds by alternating updates of $N\to\infty$0 and the operator, and automatic differentiation is used to compute $N\to\infty$1, including Jacobians and Hessians.

Within the framework, this neural component does not replace the semigroup formulation; rather, it supplies a learned coordinate system in which the one-step operator $N\to\infty$2 can be estimated. A plausible implication is that basis adaptivity is treated as an approximation-space problem, while the semigroup structure remains the organizing principle (Xu et al., 23 Jan 2025).

4. Convergence theory and approximation limits

The convergence analysis is organized around the nested limit

$N\to\infty$3

At fixed $N\to\infty$4 and $N\to\infty$5, the population semigroup approximation is defined by

$N\to\infty$6

The empirical estimator is

$N\to\infty$7

Under boundedness assumptions $N\to\infty$8 and $N\to\infty$9, the paper gives concentration bounds

$$d\mathbf{X}_t = \mathbf{b}(\mathbf{X}_t)\,dt + \mathbf{\sigma}(\mathbf{X}_t)\,d\mathbf{W}_t,$$0

$$d\mathbf{X}_t = \mathbf{b}(\mathbf{X}_t)\,dt + \mathbf{\sigma}(\mathbf{X}_t)\,d\mathbf{W}_t,$$1

and, through a perturbation lemma for products $$d\mathbf{X}_t = \mathbf{b}(\mathbf{X}_t)\,dt + \mathbf{\sigma}(\mathbf{X}_t)\,d\mathbf{W}_t,$$2, a corresponding exponentially decaying tail bound for

$$d\mathbf{X}_t = \mathbf{b}(\mathbf{X}_t)\,dt + \mathbf{\sigma}(\mathbf{X}_t)\,d\mathbf{W}_t,$$3

As a consequence, $$d\mathbf{X}_t = \mathbf{b}(\mathbf{X}_t)\,dt + \mathbf{\sigma}(\mathbf{X}_t)\,d\mathbf{W}_t,$$4 almost surely as $$d\mathbf{X}_t = \mathbf{b}(\mathbf{X}_t)\,dt + \mathbf{\sigma}(\mathbf{X}_t)\,d\mathbf{W}_t,$$5.

For the sampling-time limit, define

$$d\mathbf{X}_t = \mathbf{b}(\mathbf{X}_t)\,dt + \mathbf{\sigma}(\mathbf{X}_t)\,d\mathbf{W}_t,$$6

If $$d\mathbf{X}_t = \mathbf{b}(\mathbf{X}_t)\,dt + \mathbf{\sigma}(\mathbf{X}_t)\,d\mathbf{W}_t,$$7 is the projected generator, then Theorem 4.6 states that, for fixed $$d\mathbf{X}_t = \mathbf{b}(\mathbf{X}_t)\,dt + \mathbf{\sigma}(\mathbf{X}_t)\,d\mathbf{W}_t,$$8,

$$d\mathbf{X}_t = \mathbf{b}(\mathbf{X}_t)\,dt + \mathbf{\sigma}(\mathbf{X}_t)\,d\mathbf{W}_t,$$9

Thus the semigroup-based approximation recovers the projected generator in the zero-sampling-time limit.

For the dictionary-size limit, the assumptions include $\mathbf{b}$0-orthogonal projections $\mathbf{b}$1 onto $\mathbf{b}$2 and graph-norm projections $\mathbf{b}$3 on $\mathbf{b}$4, with approximation properties

$\mathbf{b}$5

$\mathbf{b}$6

Under these conditions, Theorem 4.4 yields strong convergence of projected generators,

$\mathbf{b}$7

Semigroup convergence is then obtained through the First Trotter–Kato Approximation Theorem: if $\mathbf{b}$8 and $\mathbf{b}$9 are strongly continuous semigroups with exponential bounds and there is a core $$d\mathbf{X}_t=\mathbf{b}(\mathbf{X}_t)\,dt+\mathbf{\sigma}(\mathbf{X}_t)\,d\mathbf{W}_t,\qquad \mathbf{X}_0=\mathbf{x}\in M\subseteq\mathbb{R}^d,$$00 on which $$d\mathbf{X}_t=\mathbf{b}(\mathbf{X}_t)\,dt+\mathbf{\sigma}(\mathbf{X}_t)\,d\mathbf{W}_t,\qquad \mathbf{X}_0=\mathbf{x}\in M\subseteq\mathbb{R}^d,$$01, then

$$d\mathbf{X}_t=\mathbf{b}(\mathbf{X}_t)\,dt+\mathbf{\sigma}(\mathbf{X}_t)\,d\mathbf{W}_t,\qquad \mathbf{X}_0=\mathbf{x}\in M\subseteq\mathbb{R}^d,$$02

uniformly for $$d\mathbf{X}_t=\mathbf{b}(\mathbf{X}_t)\,dt+\mathbf{\sigma}(\mathbf{X}_t)\,d\mathbf{W}_t,\qquad \mathbf{X}_0=\mathbf{x}\in M\subseteq\mathbb{R}^d,$$03 in compact intervals.

The framework also states a spectral implication: strong convergence of $$d\mathbf{X}_t=\mathbf{b}(\mathbf{X}_t)\,dt+\mathbf{\sigma}(\mathbf{X}_t)\,d\mathbf{W}_t,\qquad \mathbf{X}_0=\mathbf{x}\in M\subseteq\mathbb{R}^d,$$04 and $$d\mathbf{X}_t=\mathbf{b}(\mathbf{X}_t)\,dt+\mathbf{\sigma}(\mathbf{X}_t)\,d\mathbf{W}_t,\qquad \mathbf{X}_0=\mathbf{x}\in M\subseteq\mathbb{R}^d,$$05, together with the relation

$$d\mathbf{X}_t=\mathbf{b}(\mathbf{X}_t)\,dt+\mathbf{\sigma}(\mathbf{X}_t)\,d\mathbf{W}_t,\qquad \mathbf{X}_0=\mathbf{x}\in M\subseteq\mathbb{R}^d,$$06

supports convergence of discrete-time eigenvalues and eigenfunctions to those of the true Koopman semigroup under typical spectral assumptions such as isolated eigenvalues. The paper notes, however, that the examples emphasize empirical spectral convergence rather than a fully explicit spectral perturbation theorem (Xu et al., 23 Jan 2025).

5. Spectral analysis, implementation, and canonical examples

Given a matrix approximation $$d\mathbf{X}_t=\mathbf{b}(\mathbf{X}_t)\,dt+\mathbf{\sigma}(\mathbf{X}_t)\,d\mathbf{W}_t,\qquad \mathbf{X}_0=\mathbf{x}\in M\subseteq\mathbb{R}^d,$$07 on $$d\mathbf{X}_t=\mathbf{b}(\mathbf{X}_t)\,dt+\mathbf{\sigma}(\mathbf{X}_t)\,d\mathbf{W}_t,\qquad \mathbf{X}_0=\mathbf{x}\in M\subseteq\mathbb{R}^d,$$08, SDMD computes eigenpairs

$$d\mathbf{X}_t=\mathbf{b}(\mathbf{X}_t)\,dt+\mathbf{\sigma}(\mathbf{X}_t)\,d\mathbf{W}_t,\qquad \mathbf{X}_0=\mathbf{x}\in M\subseteq\mathbb{R}^d,$$09

and reconstructs Koopman eigenfunctions as

$$d\mathbf{X}_t=\mathbf{b}(\mathbf{X}_t)\,dt+\mathbf{\sigma}(\mathbf{X}_t)\,d\mathbf{W}_t,\qquad \mathbf{X}_0=\mathbf{x}\in M\subseteq\mathbb{R}^d,$$10

The semigroup/generator spectral relation becomes

$$d\mathbf{X}_t=\mathbf{b}(\mathbf{X}_t)\,dt+\mathbf{\sigma}(\mathbf{X}_t)\,d\mathbf{W}_t,\qquad \mathbf{X}_0=\mathbf{x}\in M\subseteq\mathbb{R}^d,$$11

This makes the method semigroup-based but generator-aware: it estimates $$d\mathbf{X}_t=\mathbf{b}(\mathbf{X}_t)\,dt+\mathbf{\sigma}(\mathbf{X}_t)\,d\mathbf{W}_t,\qquad \mathbf{X}_0=\mathbf{x}\in M\subseteq\mathbb{R}^d,$$12 and then interprets its spectrum in generator coordinates. The resulting eigenvalues and eigenfunctions are used to identify dominant decay rates, oscillation frequencies, metastable structures, radial and angular modes, Hermite-like spectral patterns, and metastable wells with transitions.

Algorithm 1 in the paper summarizes the basic SDMD procedure. Given i.i.d. data, a dictionary, SDE coefficients $$d\mathbf{X}_t=\mathbf{b}(\mathbf{X}_t)\,dt+\mathbf{\sigma}(\mathbf{X}_t)\,d\mathbf{W}_t,\qquad \mathbf{X}_0=\mathbf{x}\in M\subseteq\mathbb{R}^d,$$13, sampling time $$d\mathbf{X}_t=\mathbf{b}(\mathbf{X}_t)\,dt+\mathbf{\sigma}(\mathbf{X}_t)\,d\mathbf{W}_t,\qquad \mathbf{X}_0=\mathbf{x}\in M\subseteq\mathbb{R}^d,$$14, and regularization $$d\mathbf{X}_t=\mathbf{b}(\mathbf{X}_t)\,dt+\mathbf{\sigma}(\mathbf{X}_t)\,d\mathbf{W}_t,\qquad \mathbf{X}_0=\mathbf{x}\in M\subseteq\mathbb{R}^d,$$15, one builds $$d\mathbf{X}_t=\mathbf{b}(\mathbf{X}_t)\,dt+\mathbf{\sigma}(\mathbf{X}_t)\,d\mathbf{W}_t,\qquad \mathbf{X}_0=\mathbf{x}\in M\subseteq\mathbb{R}^d,$$16 and $$d\mathbf{X}_t=\mathbf{b}(\mathbf{X}_t)\,dt+\mathbf{\sigma}(\mathbf{X}_t)\,d\mathbf{W}_t,\qquad \mathbf{X}_0=\mathbf{x}\in M\subseteq\mathbb{R}^d,$$17, computes $$d\mathbf{X}_t=\mathbf{b}(\mathbf{X}_t)\,dt+\mathbf{\sigma}(\mathbf{X}_t)\,d\mathbf{W}_t,\qquad \mathbf{X}_0=\mathbf{x}\in M\subseteq\mathbb{R}^d,$$18, and then forms the regularized approximation

$$d\mathbf{X}_t=\mathbf{b}(\mathbf{X}_t)\,dt+\mathbf{\sigma}(\mathbf{X}_t)\,d\mathbf{W}_t,\qquad \mathbf{X}_0=\mathbf{x}\in M\subseteq\mathbb{R}^d,$$19

followed by eigendecomposition. Algorithm 2 extends this to dictionary learning by iteratively updating $$d\mathbf{X}_t=\mathbf{b}(\mathbf{X}_t)\,dt+\mathbf{\sigma}(\mathbf{X}_t)\,d\mathbf{W}_t,\qquad \mathbf{X}_0=\mathbf{x}\in M\subseteq\mathbb{R}^d,$$20 and $$d\mathbf{X}_t=\mathbf{b}(\mathbf{X}_t)\,dt+\mathbf{\sigma}(\mathbf{X}_t)\,d\mathbf{W}_t,\qquad \mathbf{X}_0=\mathbf{x}\in M\subseteq\mathbb{R}^d,$$21.

The framework is validated on three canonical stochastic systems. For the 2D Stuart–Landau equation, a noisy limit-cycle oscillator, SDMD with a Fourier basis accurately approximates analytical generator eigenvalues $$d\mathbf{X}_t=\mathbf{b}(\mathbf{X}_t)\,dt+\mathbf{\sigma}(\mathbf{X}_t)\,d\mathbf{W}_t,\qquad \mathbf{X}_0=\mathbf{x}\in M\subseteq\mathbb{R}^d,$$22, particularly angular modes with $$d\mathbf{X}_t=\mathbf{b}(\mathbf{X}_t)\,dt+\mathbf{\sigma}(\mathbf{X}_t)\,d\mathbf{W}_t,\qquad \mathbf{X}_0=\mathbf{x}\in M\subseteq\mathbb{R}^d,$$23 and $$d\mathbf{X}_t=\mathbf{b}(\mathbf{X}_t)\,dt+\mathbf{\sigma}(\mathbf{X}_t)\,d\mathbf{W}_t,\qquad \mathbf{X}_0=\mathbf{x}\in M\subseteq\mathbb{R}^d,$$24, and captures eigenfunctions with the correct phase-rotational structure. For the 1D Ornstein–Uhlenbeck process

$$d\mathbf{X}_t=\mathbf{b}(\mathbf{X}_t)\,dt+\mathbf{\sigma}(\mathbf{X}_t)\,d\mathbf{W}_t,\qquad \mathbf{X}_0=\mathbf{x}\in M\subseteq\mathbb{R}^d,$$25

with generator

$$d\mathbf{X}_t=\mathbf{b}(\mathbf{X}_t)\,dt+\mathbf{\sigma}(\mathbf{X}_t)\,d\mathbf{W}_t,\qquad \mathbf{X}_0=\mathbf{x}\in M\subseteq\mathbb{R}^d,$$26

and analytical eigenpairs

$$d\mathbf{X}_t=\mathbf{b}(\mathbf{X}_t)\,dt+\mathbf{\sigma}(\mathbf{X}_t)\,d\mathbf{W}_t,\qquad \mathbf{X}_0=\mathbf{x}\in M\subseteq\mathbb{R}^d,$$27

SDMD-DL estimates Koopman eigenvalues and recovers generator eigenvalues close to $$d\mathbf{X}_t=\mathbf{b}(\mathbf{X}_t)\,dt+\mathbf{\sigma}(\mathbf{X}_t)\,d\mathbf{W}_t,\qquad \mathbf{X}_0=\mathbf{x}\in M\subseteq\mathbb{R}^d,$$28, while the eigenfunctions match Hermite-polynomial structure. For the 2D triple-well potential system

$$d\mathbf{X}_t=\mathbf{b}(\mathbf{X}_t)\,dt+\mathbf{\sigma}(\mathbf{X}_t)\,d\mathbf{W}_t,\qquad \mathbf{X}_0=\mathbf{x}\in M\subseteq\mathbb{R}^d,$$29

SDMD with a neural-network dictionary identifies the stationary mode $$d\mathbf{X}_t=\mathbf{b}(\mathbf{X}_t)\,dt+\mathbf{\sigma}(\mathbf{X}_t)\,d\mathbf{W}_t,\qquad \mathbf{X}_0=\mathbf{x}\in M\subseteq\mathbb{R}^d,$$30, a slow mode corresponding to transitions between deep wells, and a faster mode involving the shallower well, with eigenfunctions that partition the state space into metastable regions.

Across these examples, comparison with EDMD or gEDMD shows the semigroup-based method to be more numerically stable with respect to $$d\mathbf{X}_t=\mathbf{b}(\mathbf{X}_t)\,dt+\mathbf{\sigma}(\mathbf{X}_t)\,d\mathbf{W}_t,\qquad \mathbf{X}_0=\mathbf{x}\in M\subseteq\mathbb{R}^d,$$31, more accurate for stochastic dynamics, especially when coupled with dictionary learning, and computationally cheaper due to avoiding matrix exponentials (Xu et al., 23 Jan 2025).

6. Broader significance, limitations, and related developments

Within Koopman operator theory, the framework occupies an intermediate position. Classical DMD and EDMD approximate a single time-$$d\mathbf{X}_t=\mathbf{b}(\mathbf{X}_t)\,dt+\mathbf{\sigma}(\mathbf{X}_t)\,d\mathbf{W}_t,\qquad \mathbf{X}_0=\mathbf{x}\in M\subseteq\mathbb{R}^d,$$32 Koopman operator, typically in deterministic settings and often without explicit analysis of the $$d\mathbf{X}_t=\mathbf{b}(\mathbf{X}_t)\,dt+\mathbf{\sigma}(\mathbf{X}_t)\,d\mathbf{W}_t,\qquad \mathbf{X}_0=\mathbf{x}\in M\subseteq\mathbb{R}^d,$$33-dependence. Generator-based EDMD for stochastic systems approximates $$d\mathbf{X}_t=\mathbf{b}(\mathbf{X}_t)\,dt+\mathbf{\sigma}(\mathbf{X}_t)\,d\mathbf{W}_t,\qquad \mathbf{X}_0=\mathbf{x}\in M\subseteq\mathbb{R}^d,$$34 and then uses matrix exponentials for semigroup evolution. SDMD instead uses generator information to construct a semigroup approximation directly at the observed sampling interval. This suggests a computational alignment between continuous-time stochastic dynamics and discrete-time measurements that is built into the method rather than added afterward (Xu et al., 23 Jan 2025).

Its stated advantages are direct semigroup approximation, better numerical conditioning because $$d\mathbf{X}_t=\mathbf{b}(\mathbf{X}_t)\,dt+\mathbf{\sigma}(\mathbf{X}_t)\,d\mathbf{W}_t,\qquad \mathbf{X}_0=\mathbf{x}\in M\subseteq\mathbb{R}^d,$$35 is bounded whereas stochastic generators are often unbounded and stiff, explicit dependence on $$d\mathbf{X}_t=\mathbf{b}(\mathbf{X}_t)\,dt+\mathbf{\sigma}(\mathbf{X}_t)\,d\mathbf{W}_t,\qquad \mathbf{X}_0=\mathbf{x}\in M\subseteq\mathbb{R}^d,$$36, compatibility with neural-network dictionaries without explicit matrix exponentials in training, and rigorous convergence guarantees in the large-data, zero-sampling-time, and large-dictionary regimes. The stated limitations are also specific: high-dimensional state spaces remain practically challenging when large dictionaries are required; the convergence theory is focused essentially on point spectrum and strong operator convergence rather than continuous spectrum; the framework assumes a Markov semigroup and does not directly address non-Markovian or nonstationary settings; and neural architecture, regularization, and training design remain open variables.

The broader directions proposed include controlled and parameter-dependent stochastic systems, high-dimensional molecular dynamics and climate models, and reinforcement learning or Markov decision processes, where Koopman semigroups can be viewed as generalizations of transition operators. A related Koopman-semigroup line develops wavelet-based observables on $$d\mathbf{X}_t=\mathbf{b}(\mathbf{X}_t)\,dt+\mathbf{\sigma}(\mathbf{X}_t)\,d\mathbf{W}_t,\qquad \mathbf{X}_0=\mathbf{x}\in M\subseteq\mathbb{R}^d,$$37, proves that certain wavelet observables are eigenfunctions of the Koopman generator, and combines EDMD with a continuous wavelet transform in the cWDMD algorithm (Tilki et al., 14 May 2026). Taken together, these developments indicate that semigroup-based Koopman analysis is being treated not merely as notation for $$d\mathbf{X}_t=\mathbf{b}(\mathbf{X}_t)\,dt+\mathbf{\sigma}(\mathbf{X}_t)\,d\mathbf{W}_t,\qquad \mathbf{X}_0=\mathbf{x}\in M\subseteq\mathbb{R}^d,$$38, but as a primary approximation target whose algebraic structure, spectral interpretation, and numerical implementation can be exploited directly.