Generative Diffusion Surrogates with Analytical Variance Schedule

# Generative Diffusion Surrogates with Analytical Variance Schedule
Source: [https://arxiv.org/html/2609.01705](https://arxiv.org/html/2609.01705)
Patrick ReichherzerEmail:[p\.reichherzer@princeton\.edu](mailto:[email protected])Affiliation:Department of Physics, University of Oxford, Oxford OX1 3PU, UKAffiliation:Department of Astrophysical Sciences, Princeton University, Princeton NJ 08544, USAAffiliation:Max Planck Institute for Security and Privacy, Bochum 44799, GermanyDavid N\. HoskingAffiliation:Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, UKAffiliation:Gonville & Caius College, Cambridge CB2 1TA, UKSubir SarkarAffiliation:Department of Physics, University of Oxford, Oxford OX1 3PU, UK

###### Abstract

Stochastic transport describes physical systems in which an initially structured distribution spreads under unresolved forcing, scattering, or heterogeneous media\. Useful surrogates for such systems should be probabilistic, time\-resolved, and able to represent non\-Gaussian distributional structure\. Generative diffusion models, which corrupt data with Gaussian noise and learn a reverse flow back to structured states, have these properties\. Their noise schedules, however, are usually chosen heuristically: image and audio generation—the canonical use cases—provide no physical clock\. In transport, by contrast, the variance, or mean\-square displacement, is often known from macroscopic theory or empirical scaling even when the full distribution is not\. Here we prescribe the forward noising rate as the time derivative of this variance, turning generative time into a calibrated transport clock\. The variance path is enforced by construction, while the learned score field represents how non\-Gaussian structure inherited from entrance data is smoothed along that path, requiring no intermediate\-time physical transport data\. For ballistic\-to\-diffusive transport in turbulent plasmas, the surrogate matches test\-particle distributions, reproduces the laboratory\-measured variance scale, and tracks the simulated kurtosis evolution without schedule tuning, enabling calibrated emulation and likelihood\-based inference\.

## 1Introduction

Generative diffusion models—as widely used in AI image, audio and video generation—work by learning to undo noise\. Their forward diffusion process corrupts training data by successive addition of random noise\. A neural network is trained to reverse this corruption, so that a highly corrupted sample—even pure noise—can be transformed into a sample resembling the training data \(reverse diffusion process\)\. In this paper, we use the same machinery to model physical systems that evolve by the physical accumulation of random increments, i\.e\., stochastic transport, in which an initially organized distribution is acted on by unresolved fluctuations such as turbulent magnetic fields, scattering centers, porous heterogeneities, molecular collisions, or random forcing\.

A central design choice in diffusion models is the forward noise scheduleg⁡\(τ\)g\(\\tau\), which specifies the amount of noise added in the forward diffusion process as a function of a time\-like parameterτ\\tau\([Ho et al\., 2020](https://arxiv.org/html/2609.01705#bib.bib8);[Song et al\., 2021](https://arxiv.org/html/2609.01705#bib.bib9);[Nichol and Dhariwal, 2021](https://arxiv.org/html/2609.01705#bib.bib19);[Karras et al\., 2022](https://arxiv.org/html/2609.01705#bib.bib20)\)\. For the variance\-exploding \(VE\) formulation used in this work, the forward corruption is

d​𝐱τ=g~​\(τ\)​d​𝐖τ,d\\mathbf\{x\}\_\{\\tau\}\\;=\\;\\tilde\{g\}\(\\tau\)\\,d\\mathbf\{W\}\_\{\\tau\},\(1\)where𝐖τ\\mathbf\{W\}\_\{\\tau\}is a standard Wiener process with independent increments\. Ifσ~2​\(τ\)\\tilde\{\\sigma\}^\{2\}\(\\tau\)denotes the accumulated post\-entrance variance increment of the surrogate, then

σ~2​\(τ\)=∫0τg~2​\(u\)​𝑑u,g~2​\(τ\)=dd​τ​σ~2​\(τ\)\.\\tilde\{\\sigma\}^\{2\}\(\\tau\)=\\int\_\{0\}^\{\\tau\}\\tilde\{g\}^\{2\}\(u\)\\,du,\\qquad\\tilde\{g\}^\{2\}\(\\tau\)=\\frac\{d\}\{d\\tau\}\\tilde\{\\sigma\}^\{2\}\(\\tau\)\.\(2\)Thus, a nondecreasing macroscopic variance law directly determines the instantaneous noising rate of the diffusion model\.

In image, audio and video generation,g⁡\(τ\)g\(\\tau\)is usually chosen heuristically \(e\.g\., linear or cosine inτ\\tau\([Ho et al\., 2020](https://arxiv.org/html/2609.01705#bib.bib8);[Nichol and Dhariwal, 2021](https://arxiv.org/html/2609.01705#bib.bib19)\)\), and tuned to maximize output quality or likelihood\. For stochastic transport, by contrast, the variance, interpreted as the mean\-square displacement of transported particles, is often known more directly than the full distribution, either from macroscopic transport theory or from measured scaling laws\. Examples include non\-Fickian transport in porous media, modeled via fractional kinetics\([Berkowitz et al\., 2000](https://arxiv.org/html/2609.01705#bib.bib28);[Edery et al\., 2015](https://arxiv.org/html/2609.01705#bib.bib29)\); intracellular subdiffusion in crowded cytoplasm\([Weiss et al\., 2004](https://arxiv.org/html/2609.01705#bib.bib30);[Golding and Cox, 2006](https://arxiv.org/html/2609.01705#bib.bib31)\); superdiffusive dispersion in turbulence \(Richardson’s law\)\([Richardson, 1926](https://arxiv.org/html/2609.01705#bib.bib32)\); and charged\-particle transport in stochastic magnetic fields\. Analogous scaling laws also arise in broader stochastic\-spreading problems such as human mobility and rough volatility\([Brockmann et al\., 2006](https://arxiv.org/html/2609.01705#bib.bib33);[Gatheral et al\., 2018](https://arxiv.org/html/2609.01705#bib.bib43)\)\. Treatingτ\\tauas a normalized physical time, such a law can be written asσ~2​\(τ\)\\tilde\{\\sigma\}^\{2\}\(\\tau\)and used to calibrate the diffusion\-model clock at the level of the variance\.

In this work, we therefore consider generative diffusion models with the noise scheduleg~​\(τ\)\\tilde\{g\}\(\\tau\)anchored to a known physical variance lawσ~2​\(τ\)\\tilde\{\\sigma\}^\{2\}\(\\tau\)as surrogates for stochastic transport\. Anchoring the variance separates the scale dynamics from the remaining distributional shape: the variance is prescribed analytically, while the learned reverse model represents higher\-order structure, including non\-Gaussian structure inherited from the entrance data\. This calibration fixes the ensemble spreading at the variance level, but not the underlying microscopic dynamics: the synthetic Gaussian noising process does not, by itself, impose finite propagation speed, memory effects, or trajectory correlations\. Figure[1](https://arxiv.org/html/2609.01705#S1.F1)summarizes the mapping between diffusion\-model and stochastic\-transport terminology used throughout this work\.

![Refer to caption](https://arxiv.org/html/2609.01705v1/fig1.png)Figure 1:Diffusion–transport terminology map\.Individual curves show sample trajectoriesxτx\_\{\\tau\}; the blue density field shows the evolving marginalpτp\_\{\\tau\}\. A fixed\-τ\\tauslice is interpreted as a distribution over noised images in diffusion modeling and as the beam one\-particle probability density function in stochastic transport, connecting the entrance/data distributionP0P\_\{0\}to the Gaussianized limitP1P\_\{1\}\.This variance\-clock construction differs from existing uses of generative models in scientific surrogate modeling\. Diffusion models are increasingly used as scientific surrogates\([Thuerey et al\., 2021](https://arxiv.org/html/2609.01705#bib.bib13);[Greif et al\., 2023](https://arxiv.org/html/2609.01705#bib.bib12)\)including to synthesize particle trajectories from test\-particle simulations\([Martin et al\., 2025](https://arxiv.org/html/2609.01705#bib.bib37)\), but their generative timescale is typically algorithmic, where the forward schedule is chosen or tuned for objectives such as sample quality, likelihood, or sampling efficiency\([Ho et al\., 2020](https://arxiv.org/html/2609.01705#bib.bib8);[Song et al\., 2021](https://arxiv.org/html/2609.01705#bib.bib9);[Kingma et al\., 2021](https://arxiv.org/html/2609.01705#bib.bib15);[Dockhorn et al\., 2022](https://arxiv.org/html/2609.01705#bib.bib16);[Bao et al\., 2022](https://arxiv.org/html/2609.01705#bib.bib17);[Karras et al\., 2022](https://arxiv.org/html/2609.01705#bib.bib20)\)\. A separate line of scientific machine learning incorporates physical information by penalizing residuals of governing equations or imposing other physics\-based constraints\([Raissi et al\., 2019](https://arxiv.org/html/2609.01705#bib.bib21);[Thuerey et al\., 2021](https://arxiv.org/html/2609.01705#bib.bib13);[Karniadakis et al\., 2021](https://arxiv.org/html/2609.01705#bib.bib22);[Bastek et al\., 2024](https://arxiv.org/html/2609.01705#bib.bib14)\)\. Path\-based generative methods, including flow matching, stochastic interpolants, and Schrödinger bridges, learn vector fields or stochastic processes along prescribed probability paths between distributions\([Lipman et al\., 2023](https://arxiv.org/html/2609.01705#bib.bib23);[Albergo et al\., 2025](https://arxiv.org/html/2609.01705#bib.bib24);[De Bortoli et al\., 2021](https://arxiv.org/html/2609.01705#bib.bib2);[Duong et al\., 2025](https://arxiv.org/html/2609.01705#bib.bib18);[Baldan et al\., 2025](https://arxiv.org/html/2609.01705#bib.bib10)\)\. These paths need not coincide with the physical transport clock, although trajectory data or physical interpolants can be incorporated in scientific applications\([Wald and Steidl, 2025](https://arxiv.org/html/2609.01705#bib.bib4)\)\. Other approaches solve inverse problems by approximately reversing a simulator and correcting the result with a learned score\([Holzschuh et al\., 2023](https://arxiv.org/html/2609.01705#bib.bib11)\)\.

Operationally, we modify only the scalar schedule in the standard VE\-SDE framework\([Song et al\., 2021](https://arxiv.org/html/2609.01705#bib.bib9)\)\. The scheduleg~2​\(τ\)\\tilde\{g\}^\{2\}\(\\tau\)is derived directly from the macroscopic variance lawσ~2​\(τ\)\\tilde\{\\sigma\}^\{2\}\(\\tau\), calibrating the generative clock to physical time at the level of the second moment\. This preserves existing diffusion\-model infrastructure while enabling analytic moment tracking\. It also permits entrance\-only training: the network is trained using entrance samples and their synthetic corruptions along the prescribed variance path without requiring physical transport data at intermediate times\. Sampling at a prescribed readout timeτ⋆\\tau^\{\\star\}therefore gives the surrogate marginal associated withσ~2​\(τ⋆\)\\tilde\{\\sigma\}^\{2\}\(\\tau^\{\\star\}\)\.

In the following, we develop a framework for integrating macroscopic transport laws into generative diffusion models\. We provide the theoretical basis in §[2\.1](https://arxiv.org/html/2609.01705#S2.SS1), apply the method to telegraph\-type ballistic\-to\-diffusive transport in §[2\.2](https://arxiv.org/html/2609.01705#S2.SS2), demonstrate entrance\-only training in §[2\.3](https://arxiv.org/html/2609.01705#S2.SS3), and use the learned model as a differentiable surrogate for laboratory plasma transport in §[2\.4](https://arxiv.org/html/2609.01705#S2.SS4)\. In §[2\.5](https://arxiv.org/html/2609.01705#S2.SS5), we apply the same construction to heavy\-tailed and intermittent transport, where early\-time transport statistics are strongly non\-Gaussian\. The resulting surrogate is indexed by physical time and is suitable for calibrated scientific emulation and likelihood\-based inference\.

## 2Results

### 2\.1Central Limit Theorem motivates physics\-anchored Gaussian surrogates

In many physical systems, macroscopic transport arises from the accumulation of microscopic perturbations, such as scattering events or stochastic collisions, that are correlated over a finite length scaleℓc\\ell\_\{c\}or, equivalently, a correlation timeτc\\tau\_\{c\}\. Below this scale, the microphysics may be complex and strongly non\-Gaussian\. If coarse\-grained perturbations separated byℓc\\ell\_\{c\}are weakly dependent and have finite second moment, the central limit theorem \(CLT\) motivates a Gaussian approximation to the accumulated transport kernel at scales large compared withℓc\\ell\_\{c\}\. AfterN∼τ/τcN\\\!\\sim\\\!\\tau/\\tau\_\{c\}approximately decorrelated contributions on the normalized clock with finite fourth moment, standardized higher cumulants relax; in particular, the excess kurtosis of the accumulated kernel falls as1/N1/N\(Methods, §[4\.3](https://arxiv.org/html/2609.01705#S4.SS3)\)\.

We use this Gaussianization as a coarse\-grained modeling assumption\. Sub\-ℓc\\ell\_\{c\}non\-Gaussian structure is represented through the entrance distributionP0P\_\{0\}, while subsequent post\-entrance spreading is approximated by a Gaussian kernel whose variance is prescribed by the macroscopic transport law\. Thus non\-Gaussian structure in the model marginals is inherited fromP0P\_\{0\}and then diluted by the prescribed variance growth, unless the physical transport kernel itself remains strongly non\-Gaussian\. We exploit this construction in §[2\.5](https://arxiv.org/html/2609.01705#S2.SS5)to propagate intermittency from sub\-ℓc\\ell\_\{c\}field statistics, while the anchored variance law supplies the scale evolution\.

The Gaussianized spreading aboveℓc\\ell\_\{c\}is encoded by the accumulated post\-entrance varianceσ~2​\(τ\)\\tilde\{\\sigma\}^\{2\}\(\\tau\)\. If a macroscopic law is written for the total variance, thenσ~2​\(τ\)\\tilde\{\\sigma\}^\{2\}\(\\tau\)should be read as the variance increment relative to the entrance variance\. We implement this variance increment using the VE process introduced in Eqs\. \([1](https://arxiv.org/html/2609.01705#S1.E1)\)–\([2](https://arxiv.org/html/2609.01705#S1.E2)\), now initialized with

𝐱0∼P0\.\\mathbf\{x\}\_\{0\}\\sim P\_\{0\}\.WithCov⁡\[d​𝐖τ\]=I​d​τ\\mathrm\{Cov\}\[d\\mathbf\{W\}\_\{\\tau\}\]=I\\,d\\tau, the resulting marginal law at each fixedτ\\taucan equivalently be written as

𝐱τ=𝐱0\+σ~​\(τ\)​ϵ,ϵ∼𝒩⁡\(0,I\)\.\\mathbf\{x\}\_\{\\tau\}=\\mathbf\{x\}\_\{0\}\+\\tilde\{\\sigma\}\(\\tau\)\\boldsymbol\{\\epsilon\},\\qquad\\boldsymbol\{\\epsilon\}\\sim\\mathcal\{N\}\(0,I\)\.Each synthetic Gaussian increment represents the coarse\-grained, decorrelated part of the transport only at the level of the surrogate\. The Markov approximation amounts to neglecting memory beyondτc\\tau\_\{c\}in these coarse\-grained increments\. The marginal distributionpτ​\(𝐱\)p\_\{\\tau\}\(\\mathbf\{x\}\)generated by Eq\. \([1](https://arxiv.org/html/2609.01705#S1.E1)\) is therefore the convolution ofP0P\_\{0\}with a Gaussian kernel \(Methods, §[4\.1](https://arxiv.org/html/2609.01705#S4.SS1)\)\. Starting fromP0P\_\{0\}, each state accumulates independent Gaussian displacement with per\-coordinate varianceσ~2​\(τ\)\\tilde\{\\sigma\}^\{2\}\(\\tau\):

pτ​\(𝐱\)=\(P0∗𝒩⁡\(0,σ~2​\(τ\)​I\)\)​\(𝐱\)\.p\_\{\\tau\}\(\\mathbf\{x\}\)=\(P\_\{0\}\*\\mathcal\{N\}\(0,\\tilde\{\\sigma\}^\{2\}\(\\tau\)I\)\)\(\\mathbf\{x\}\)\.\(3\)Thus the total per\-coordinate variance of the model marginal isσ02\+σ~2​\(τ\)\\sigma\_\{0\}^\{2\}\+\\tilde\{\\sigma\}^\{2\}\(\\tau\), whereσ02=Var⁡\(P0\)\\sigma\_\{0\}^\{2\}=\\mathrm\{Var\}\(P\_\{0\}\)denotes the entrance variance\.

The VE process therefore acts as a Gaussian surrogate for the post\-entrance transport kernel\. When the physical transport kernel is centered and isotropic, this Gaussian is the moment\-matched member of the centered isotropic Gaussian family and minimizes cross\-entropy within that family \(equivalently, forward KL when the physical kernel has a density and finite differential entropy\) \(Methods, §[4\.2](https://arxiv.org/html/2609.01705#S4.SS2)\)\. The associated score is𝐬⁡\(𝐱,τ\)=∇𝐱​log​pτ​\(𝐱\)\\mathbf\{s\}\(\\mathbf\{x\},\\tau\)=\\nabla\_\{\\mathbf\{x\}\}\\log p\_\{\\tau\}\(\\mathbf\{x\}\), which we approximate by a neural network𝐬θ\\mathbf\{s\}\_\{\\theta\}\. Because the variance is enforced analytically by the schedule, this network is not required to learn the second\-moment evolution\. Instead, it represents the modeled evolution of the remaining distributional shape, including non\-Gaussian structure inherited fromP0P\_\{0\}\. The trained score field makes the surrogate useful for inversion and likelihood\-based inference \(see §[2\.3](https://arxiv.org/html/2609.01705#S2.SS3)\)\.

The Gaussian convolution \([3](https://arxiv.org/html/2609.01705#S2.E3)\) yields a closed\-form relaxation of entrance kurtosis\. We quantify tail heaviness using the excess kurtosisK:=κ4/σ4K:=\\kappa\_\{4\}/\\sigma^\{4\}, whereκ4\\kappa\_\{4\}is the fourth cumulant; a Gaussian hasK=0K=0, while heavy\-tailed distributions haveK\>0K\>0\. LetK0K\_\{0\}be the entrance excess kurtosis\. Because the Gaussian kernel contributes no fourth cumulant, the fourth cumulant of the model marginal is inherited fromP0P\_\{0\}, while the variance grows asσ02\+σ~2​\(τ\)\\sigma\_\{0\}^\{2\}\+\\tilde\{\\sigma\}^\{2\}\(\\tau\)\. The model excess kurtosis therefore relaxes deterministically as

KM​\(τ\)=K0​\(σ02σ02\+σ~2​\(τ\)\)2,K\_\{M\}\(\\tau\)\\;=\\;K\_\{0\}\\left\(\\frac\{\\sigma\_\{0\}^\{2\}\}\{\\sigma\_\{0\}^\{2\}\+\\tilde\{\\sigma\}^\{2\}\(\\tau\)\}\\right\)^\{\\\!2\},\(4\)entirely governed by the anchored variance pathσ~2​\(τ\)\\tilde\{\\sigma\}^\{2\}\(\\tau\)\.

For centered transport, the mean and variance are fixed by construction\. The residual mismatch between the physical kernel and the Gaussian surrogate therefore lies in unresolved distributional shape, including higher cumulants when they exist\. We quantify the leading mismatch relevant here using the kurtosis gap, defined in Methods, §[4\.3](https://arxiv.org/html/2609.01705#S4.SS3), Eq\. \([11](https://arxiv.org/html/2609.01705#S4.E11)\)\.

### 2\.2Variance laws as noise schedules

In the coarse\-grained picture of §[2\.1](https://arxiv.org/html/2609.01705#S2.SS1),τc\\tau\_\{c\}separates correlated from approximately decorrelated perturbations\. For telegraph\-type transport, the same relaxation scale separates two transport regimes\. Here and below,τc\\tau\_\{c\}is expressed on the same normalized clock asτ\\tau\. Unlike ordinary Fickian diffusion, where the accumulated variance grows linearly \(σ~2∝τ\\tilde\{\\sigma\}^\{2\}\\\!\\propto\\\!\\tau\), many real systems show a ballistic\-to\-diffusive transition\.

The two regimes have distinct signatures\. At short times \(τ≪τc\\tau\\ll\\tau\_\{c\}\), correlations persist: the direction or velocity has not yet decorrelated, so displacements grow coherently\. This is the ballistic regime, where variance grows quadratically \(σ~2∝τ2\\tilde\{\\sigma\}^\{2\}\\propto\\tau^\{2\}\)\. Onceτ≫τc\\tau\\gg\\tau\_\{c\}, many approximately independent perturbations have accumulated and the variance grows linearly \(σ~2∝τ\\tilde\{\\sigma\}^\{2\}\\propto\\tau\)\. This is the ordinary diffusive regime\.

A minimal model for this crossover is the Goldstein–Kac, or telegraph, process\([Goldstein, 1951](https://arxiv.org/html/2609.01705#bib.bib34)\)\. In one dimension, a particle moves at finite speedccwhile its direction decorrelates on a correlation timetct\_\{c\}\. In dimensional variables, the corresponding densityq⁡\(x,t\)q\(x,t\)satisfies, up to convention\-dependent normalizations oftct\_\{c\},

∂2q∂t2\+1tc​∂q∂t=c2​∂2q∂x2\.\\frac\{\\partial^\{2\}q\}\{\\partial t^\{2\}\}\+\\frac\{1\}\{t\_\{c\}\}\\frac\{\\partial q\}\{\\partial t\}=c^\{2\}\\,\\frac\{\\partial^\{2\}q\}\{\\partial x^\{2\}\}\.\(5\)The second\-order time derivative yields finite\-speed, persistent transport with ordinary diffusion as its long\-time, large\-scale limit\.

In this work we do not use the full telegraph kernel as the VE surrogate kernel\. We use only its mean\-square displacement to define the variance clock\. For a point source with symmetric initial velocities, the post\-entrance variance increment has the form

⟨x2\(t\)⟩=2D\[t−tc\(1−e−t/tc\)\]\.\\langle x^\{2\}\(t\)\\rangle=2D\\left\[t\-t\_\{c\}\\left\(1\-e^\{\-t/t\_\{c\}\}\\right\)\\right\]\.After normalizing physical time bytobst\_\{\\rm obs\}, so thatτ=t/tobs\\tau=t/t\_\{\\rm obs\}andτc=tc/tobs\\tau\_\{c\}=t\_\{c\}/t\_\{\\rm obs\}, and rescaling the final accumulated variance to unity, this becomes

σ~2\(τ\)=C−1\[τ−τc\(1−e−τ/τc\)\],\\tilde\{\\sigma\}^\{2\}\(\\tau\)\\;=\\;C^\{\-1\}\\left\[\\tau\-\\tau\_\{c\}\\\!\\left\(1\-e^\{\-\\tau/\\tau\_\{c\}\}\\right\)\\right\],\(6\)whereC=1−τc\(1−e−1/τc\)C=1\-\\tau\_\{c\}\(1\-e^\{\-1/\\tau\_\{c\}\}\)enforcesσ~2​\(1\)=1\\tilde\{\\sigma\}^\{2\}\(1\)=1\.

Applying the anchoring principle \([2](https://arxiv.org/html/2609.01705#S1.E2)\), the VE noise schedule is

g~2\(τ\)=dd​τσ~2\(τ\)=C−1\[1−e−τ/τc\]\.\\tilde\{g\}^\{2\}\(\\tau\)\\;=\\;\\frac\{d\}\{d\\tau\}\\tilde\{\\sigma\}^\{2\}\(\\tau\)\\;=\\;C^\{\-1\}\\left\[1\-e^\{\-\\tau/\\tau\_\{c\}\}\\right\]\.\(7\)
A VE model equipped with this schedule has the prescribed ballistic\-to\-diffusive variance crossover by construction\. Figure[2](https://arxiv.org/html/2609.01705#S2.F2)tests whether this prescribed marginal path is realized after score training and reverse sampling, and how it differs from common heuristic schedules and from a finite\-speed telegraph kernel\.

Panel[2](https://arxiv.org/html/2609.01705#S2.F2)a compares the accumulated variance of sampled models with the analytical telegraph prediction, plotted as a running diffusion coefficient\. The black dashed curve is the telegraph variance law of Eq\. \([6](https://arxiv.org/html/2609.01705#S2.E6)\); the colored curves are empirical variances measured from generated samples\. VE\-Telegraph ODE and VE\-Telegraph SDE use the same anchored schedule, Eq\. \([7](https://arxiv.org/html/2609.01705#S2.E7)\), but different reverse samplers: the deterministic probability\-flow ODE and the stochastic reverse SDE\. VE\-Linear, VE\-Cosine and VE\-Karras/EDM use standard heuristic VE schedules under the same training budget\. These schedules are valid generative choices, but their parameterτ\\tauis not the telegraph physical clock, so they do not align with the timing and shape of the ballistic\-to\-diffusive crossover\. The telegraph\-anchored schedule is therefore the variance law used below for the ballistic\-to\-diffusive plasma application \(Methods, §[4\.7](https://arxiv.org/html/2609.01705#S4.SS7)and §[4\.9](https://arxiv.org/html/2609.01705#S4.SS9)\)\.

Panels[2](https://arxiv.org/html/2609.01705#S2.F2)b–d test the higher\-order statistics of the same sampled marginals\. The solid black curve is the closed\-form prediction of the Gaussian VE surrogate, Eq\. \([4](https://arxiv.org/html/2609.01705#S2.E4)\); the colored curves are the excess kurtosis measured from trained sampled models for three entrance Student\-ttdistributions\. Agreement of the VE\-Telegraph ODE and VE\-Telegraph SDE curves with the solid line shows that the learned score and reverse samplers reproduce the intended Gaussian\-convolution marginal path\. This is the non\-trivial part of the ablation: the variance law is imposed by the scalar schedule, but the kurtosis relaxation is not used as a training target\.

The dotted black curve gives a different reference: the variance\-matched Kac/telegraph kernel derived in Methods, §[4\.4](https://arxiv.org/html/2609.01705#S4.SS4)\. This kernel corresponds to a simple particle\-level model for the telegraph equation: a particle moves at finite speed and changes direction at Poisson\-distributed times\. Its parameters are chosen so that it has the same variance path as Eq\. \([6](https://arxiv.org/html/2609.01705#S2.E6)\)\. The comparison therefore shows that distinct transport mechanisms can share the same variance law while differing in their propagators, approach to the diffusive limit, and memory\([Liang and Oh, 2026](https://arxiv.org/html/2609.01705#bib.bib3)\)\. The Gaussian VE surrogate and the Kac kernel have the same mean\-square displacement, but different higher moments\. The Kac/telegraph kernel has bounded support\([Goldstein, 1951](https://arxiv.org/html/2609.01705#bib.bib34)\)and can transiently exhibit negative excess kurtosis\. This is a controlled example of higher\-cumulant mismatch between the physical kernel and the Gaussian VE surrogate\. The departure is visible in Fig\.[2](https://arxiv.org/html/2609.01705#S2.F2)b–d, most clearly in panel c, as the gap between the dotted \(Kac\) and solid \(VE\) theory curves\.

Figure 2:Schedule ablation under controlled training budget \(physics\-anchored vs\. heuristic schedules\)\.a\)Running diffusion coefficient\[σ~2​\(τ\)−σ~2​\(0\)\]/τ\[\\tilde\{\\sigma\}^\{2\}\(\\tau\)\-\\tilde\{\\sigma\}^\{2\}\(0\)\]/\\taumeasured from model samples as a function of normalized timeτ\\tau\. Six different model configurations are shown: three heuristic schedules \(Linear, Cosine, Karras/EDM\), the Telegraph schedule \(PF\-ODE and reverse\-SDE\), and the Kac flow \(a non\-VE baseline\)\. The analytical prediction of the Telegraph equation \(Eq\. \([6](https://arxiv.org/html/2609.01705#S2.E6)\)\) is marked as a black dashed line\.
b–d\)Excess kurtosisK⁡\(τ\)K\(\\tau\)as functions ofτ\\taufor three different entrance Student\-ttdistributions withK0∈\{0\.5,1\.0,1\.5\}K\_\{0\}\\in\\\{0\.5,1\.0,1\.5\\\}\(black stars on the left edge\)\. The analytical VE prediction is shown as a black line \(Eq\. \([4](https://arxiv.org/html/2609.01705#S2.E4)\)\); the dotted black line shows the Kac/telegraph reference computed from the analytical kernel moments \(Eq\. \([15](https://arxiv.org/html/2609.01705#S4.E15)\), Methods, §[4\.4](https://arxiv.org/html/2609.01705#S4.SS4)\)\. Implementation details are given in Algorithm[1](https://arxiv.org/html/2609.01705#alg1)\. The standard deviations are shown as colored bands around the mean values\.Table[1](https://arxiv.org/html/2609.01705#S2.T1)quantifies these comparisons across readout times and entrance kurtoses using a seed\-normalized squared\-deviation statisticQQ\. The telegraph PF\-ODE and reverse\-SDE samplers give the smallest deviations from the anchored VE reference\. For the Kac flow, the small variance deviation confirms the imposed variance match, while the larger kurtosis deviation reflects the different higher\-cumulant structure of the finite\-speed kernel\.

Table 1:Physical\-time alignment diagnostic for the schedule ablation \(Fig\.[2](https://arxiv.org/html/2609.01705#S2.F2)\)\.For each moment we reportQ=Nτ−1​∑i\(y¯i−yiref\)2/max⁡\[si2,0\.01​⟨s2⟩\]Q=N\_\{\\tau\}^\{\-1\}\\sum\_\{i\}\(\\bar\{y\}\_\{i\}\-y\_\{i\}^\{\\rm ref\}\)^\{2\}/\\max\[s\_\{i\}^\{2\},0\.01\\langle s^\{2\}\\rangle\], wherey¯i\\bar\{y\}\_\{i\}andsi2s\_\{i\}^\{2\}are the across\-seed mean and variance\. The statistic is evaluated at the available logarithmically spaced readout points inτ∈\[3×10−3,1\]\\tau\\in\[3\\times 10^\{\-3\},1\]and averaged over 10 training runs andK0∈\{0\.5,1\.0,1\.5\}K\_\{0\}\\in\\\{0\.5,1\.0,1\.5\\\}\. Smaller values indicate closer alignment\.The anchoring principle \([2](https://arxiv.org/html/2609.01705#S1.E2)\) applies to any known, nondecreasing variance path, whether available in closed form or as a differentiable numerical representation\. Anomalous diffusion is a natural extension: for power\-law scalingσ~2​\(τ\)∝τα\\tilde\{\\sigma\}^\{2\}\(\\tau\)\\propto\\tau^\{\\alpha\}, the schedule isg~2​\(τ\)∝α​τα−1\\tilde\{g\}^\{2\}\(\\tau\)\\propto\\alpha\\tau^\{\\alpha\-1\}\([Metzler and Klafter, 2000](https://arxiv.org/html/2609.01705#bib.bib27);[Thiel and Sokolov, 2014](https://arxiv.org/html/2609.01705#bib.bib44)\)\. This includes subdiffusive scaling \(α<1\\alpha<1\), as in some CTRW models, and superdiffusive scaling \(α\>1\\alpha\>1\); fractional Brownian motion provides Gaussian examples withα=2​H\\alpha=2H\. Forα<1\\alpha<1, the schedule is integrable but singular atτ=0\\tau=0, so practical implementations start from a finiteτmin\\tau\_\{\\min\}or regularize the schedule near the origin\. Fractional telegraph models also admit closed\-form variance expressions\([Orsingher and Beghin, 2009](https://arxiv.org/html/2609.01705#bib.bib45)\)\. The practical constraint is a finite second moment: ideal Lévy flights are incompatible, so one anchors to a tempered or truncated law instead\. In these cases, the VE construction should be interpreted as a variance\-calibrated marginal surrogate, not as a pathwise representation of the underlying non\-Markovian dynamics\.

### 2\.3Entrance\-only training and differentiable inference

Reverse sampling in a diffusion model must start from a terminal state in which the data structure has been washed out\. In image generation, the forward corruption is simply run until this is the case\. In our setting, the physical window closes earlier, at the detector readoutτ⋆\\tau^\{\\star\}, before the entrance structure is gone\. We therefore extend the corruption beyond the detector along the same anchored variance law, until the accumulated variance dominates the entrance variance\. This continuation,τ∈\(τ⋆,1\]\\tau\\in\(\\tau^\{\\star\},1\], is the virtual extension of the experiment, as illustrated in Fig\.[3](https://arxiv.org/html/2609.01705#S2.F3)\. It requires no data, because corrupting entrance samples to anyτ\\tauneeds only Gaussian noise of known variance\. Because training and sampling share the sameτ\\tau\-anchored marginalspτp\_\{\\tau\}, we integrate the PF\-ODE backward from the fully decorrelated stateP1P\_\{1\}\(Fig\.[3](https://arxiv.org/html/2609.01705#S2.F3)c\) and halt atτ⋆\\tau^\{\\star\}, obtainingpτ⋆p\_\{\\tau^\{\\star\}\}without intermediate data\.

![Refer to caption](https://arxiv.org/html/2609.01705v1/fig3.png)Figure 3:Physics\-anchored diffusion maps experimental timescales to generative processes\.a, Initial probability distributionP0P\_\{0\}at experimental source \(τ=0\\tau=0\)\.b, Target detector distributionPτ⋆P\_\{\\tau^\{\\star\}\}at physical readoutτ⋆\\tau^\{\\star\}\(e\.g\., one correlation time\)\.c, Fully decorrelated terminal stateP1P\_\{1\}at normalized endpoint \(τ=1\\tau=1\)\. The forward physical process evolvesP0→Pτ⋆P\_\{0\}\\to P\_\{\\tau^\{\\star\}\}in real time\. The generative model trains on the full spanP0→P1P\_\{0\}\\to P\_\{1\}using the physics\-anchored scheduleg~2​\(τ\)=d​σ~2/d​τ\\tilde\{g\}^\{2\}\(\\tau\)=d\\tilde\{\\sigma\}^\{2\}/d\\tau\. The segment beyond the detector is a virtual extension of the experiment\. At inference, integrating the reverse probability\-flow ODE fromP1P\_\{1\}and halting atτ⋆\\tau^\{\\star\}recovers the detector\-time marginal under the anchored surrogate clock without intermediate training data\. In regimes where the transport kernel is near\-Gaussian, this coincides with the physical detector distribution up to higher\-cumulant mismatch\. This decoupling enables entrance\-only training: the network learns shape evolution while the schedule analytically satisfies scale\. Algorithm[1](https://arxiv.org/html/2609.01705#alg1)formalizes the training and readout\-time sampling steps shown schematically in \(a–c\)\.Given entrance samples, samples frompτ⋆p\_\{\\tau^\{\\star\}\}could also be produced by direct forward corruption\. The trained score field supplies what direct corruption alone does not: a differentiable transport map, PF\-ODE likelihood estimates under the terminal\-density assumptions specified in Methods, gradient\-based parameter inference, and model\-based backward propagation of observed detector ensembles to earlier times\. The model is therefore best viewed as a stochastic propagator of distributions\.

The physics\-anchored schedule ensures that the model marginalspτp\_\{\\tau\}are indexed by physical time and have the prescribed accumulated transport varianceσ~2​\(τ\)\\tilde\{\\sigma\}^\{2\}\(\\tau\)at eachτ\\tau\. When the physical transport kernel is close to Gaussian, as expected after decorrelation overℓc\\ell\_\{c\}, these anchored marginals provide a variance\-exact approximation to the physical marginal\. Remaining discrepancies lie in unresolved distributional shape, including higher cumulants when they exist, and are assessed here using the kurtosis gap \(Methods, Eq\. \([11](https://arxiv.org/html/2609.01705#S4.E11)\)\)\.

The training scheme does not require a known analytical expression for the forward convolution\. It requires only samples from a well\-defined entrance distributionP0P\_\{0\}and the ability to add Gaussian corruptions with the prescribed variance\. For any positive corruption variance, Gaussian smoothing gives a regular model marginal, and denoising score matching targets the score of this full, generally non\-Gaussian marginal distribution\([Vincent, 2011](https://arxiv.org/html/2609.01705#bib.bib42);[Song et al\., 2021](https://arxiv.org/html/2609.01705#bib.bib9)\)\.

We verified entrance\-only learning across entrance kurtoses using Student\-ttfamilies with fixed entrance variance\. As implied by Eq\. \([4](https://arxiv.org/html/2609.01705#S2.E4)\),K⁡\(τ\)K\(\\tau\)preserves the entrance ordering and relaxes deterministically along the telegraph\-anchored path \(Fig\.[2](https://arxiv.org/html/2609.01705#S2.F2)b–d\)\. When the physical kernel is approximately Gaussian, the model evolution matches the physical one at the level of the resolved marginal statistics\. Residual kernel non\-Gaussianity is quantified by the kurtosis gap \(Methods, Eq\. \([11](https://arxiv.org/html/2609.01705#S4.E11)\)\)\.

The learned score field can also support parameter inference from detector data through the differentiable likelihood estimate obtained with the PF\-ODE \(Eq\. \([18](https://arxiv.org/html/2609.01705#S4.E18)\)\)\. IfK0K\_\{0\}is not known, it can be treated as a parameter of the entrance familyP0​\(⋅,K0\)P\_\{0\}\(\\,\\cdot\\,;K\_\{0\}\), equivalently by conditioning the score model onK0K\_\{0\}as described in Methods, §[4\.5](https://arxiv.org/html/2609.01705#S4.SS5), and estimated by maximizing Eq\. \([18](https://arxiv.org/html/2609.01705#S4.E18)\) from an observation snapshot\.

For the zero\-drift VE process in Eq\. \([1](https://arxiv.org/html/2609.01705#S1.E1)\), the reverse\-time SDE is

d​𝐱τ=−g~2​\(τ\)​∇𝐱​log⁡pτ​\(𝐱τ\)​d​τ\+g~​\(τ\)​d​𝐖¯τ,d​τ<0\.d\\mathbf\{x\}\_\{\\tau\}=\-\\tilde\{g\}^\{2\}\(\\tau\)\\nabla\_\{\\mathbf\{x\}\}\\log p\_\{\\tau\}\(\\mathbf\{x\}\_\{\\tau\}\)\\,d\\tau\+\\tilde\{g\}\(\\tau\)d\\bar\{\\mathbf\{W\}\}\_\{\\tau\},\\qquad d\\tau<0\.The corresponding probability\-flow ODE is

d​𝐱τ=−12​g~2​\(τ\)​∇𝐱​log⁡pτ​\(𝐱τ\)​d​τ,d​τ<0,d\\mathbf\{x\}\_\{\\tau\}=\-\\frac\{1\}\{2\}\\tilde\{g\}^\{2\}\(\\tau\)\\nabla\_\{\\mathbf\{x\}\}\\log p\_\{\\tau\}\(\\mathbf\{x\}\_\{\\tau\}\)\\,d\\tau,\\qquad d\\tau<0,with the score replaced in practice by𝐬θ​\(𝐱τ,τ\)\\mathbf\{s\}\_\{\\theta\}\(\\mathbf\{x\}\_\{\\tau\},\\tau\)\. In both forms,g~2​\(τ\)\\tilde\{g\}^\{2\}\(\\tau\)ties the transport map to the prescribed variance law, while the network supplies the unresolved distributional structure\. The learned score remains useful in settings such as inhomogeneous transport or nonlinear detector models, where analytic convolution is unavailable\.

### 2\.4Application to turbulent plasma transport

We now apply the physics\-anchored VE model to the problem of charged\-particle transport in turbulent magnetic fields, a scenario relevant for ultra\-high\-energy cosmic rays, solar energetic particles, and fusion plasmas\. Since these systems are challenging to probe directly, scaled laboratory astrophysics experiments are used to investigate transport under controlled conditions\([Bott et al\., 2017](https://arxiv.org/html/2609.01705#bib.bib6);[Tzeferacos et al\., 2018](https://arxiv.org/html/2609.01705#bib.bib7);[Chen et al\., 2020](https://arxiv.org/html/2609.01705#bib.bib25)\)\. The experimental technique, proton radiography \(Fig\.[4](https://arxiv.org/html/2609.01705#S2.F4)a, c\), uses a proton beam to probe the turbulence, measuring the resulting flux patternΨ⁡\(𝐫\)\\Psi\(\\mathbf\{r\}\)\(in contrast, we useΨ¯​\(r\)\\overline\{\\Psi\}\(r\)for the azimuthally averaged profile\) at a detector\. Here𝐫=\(x,y\)\\mathbf\{r\}=\(x,y\)are detector‑plane coordinates andr=‖𝐫‖r=\\\|\\mathbf\{r\}\\\|denotes radial distance\.

![Refer to caption](https://arxiv.org/html/2609.01705v1/fig4.png)Figure 4:Physics‑anchored surrogate validation in turbulent plasmas\.a\)A narrow particle beam probes a magnetized turbulence snapshot, inducing stochastic transverse accelerationaa\.b\)Evolution of the PDFs of acceleration, velocityv⟂v\_\{\\perp\}, and displacementx⟂x\_\{\\perp\}\.c\)Experimental diagnostic \(proton radiography\) \(see[Chen et al\. \(2020\)](https://arxiv.org/html/2609.01705#bib.bib25)\)\.d\)Reverse PF‑ODE sampling visualization\.e\)RMS transverse\-speed magnitudev⟂,rmsv\_\{\\perp,\\mathrm\{rms\}\}versus physical path lengthℓ\\ell\. The simulated variance increases super\-diffusively at largeℓ\\ellbecause deflections then add up correlatedly due to the periodic continuation of the magnetic field\.f\)Excess kurtosis of the signed transverse componentvxv\_\{x\}versusℓ\\ell\. Panel f contains test\-particle and surrogate results only\. Black points in panel e: experiment of[Chen et al\. \(2020\)](https://arxiv.org/html/2609.01705#bib.bib25)\. Colored bands/lines: test\-particle simulation in the MHD field\. The purple curve in panel f shows the sampled kurtosis evolution of the scalar anchored VE surrogate using the nearest trained single\-component entrance\-kurtosis condition\. The surrogate captures the laboratory\-measured variance scale and the simulated component\-kurtosis evolution without schedule tuning\. In panel e the model variance is not plotted separately because it is the imposed anchored variance path\. Bands ineandfare bootstrap 68% intervals resampled over the 20 independent simulation runs\.Turbulent magnetic fields have a typical correlation lengthℓc\\ell\_\{c\}\(the average size of a turbulent cell\)\. We concentrate on the high\-rigidity limit, in which the particle gyroradiusrg=p/\(\|q\|​B\)r\_\{g\}=p/\(\|q\|B\)is much larger than the correlation length \(rg≫ℓcr\_\{g\}\\gg\\ell\_\{c\}\)\. In this limit, the physical picture is intuitive \(Fig\.[4](https://arxiv.org/html/2609.01705#S2.F4)b\): a high\-energy particle traverses a turbulent cell \(sizeℓc\\ell\_\{c\}\) quickly, giving rise to only a small angular deflection\. The overall transport results from the accumulation of many such small\-angle kicks across numerous independent cells\. This scenario motivates the telegraph\-type finite\-speed transport framework \(§[2\.2](https://arxiv.org/html/2609.01705#S2.SS2)\)\. The physical time required to cross one cell definestc=ℓc/vt\_\{c\}=\\ell\_\{c\}/v, corresponding to the normalized correlation timeτc=tc/tobs\\tau\_\{c\}=t\_\{c\}/t\_\{\\rm obs\}\. The correlation lengthℓc\\ell\_\{c\}plays the role of the coarse\-graining scale of §[2\.1](https://arxiv.org/html/2609.01705#S2.SS1)that determines the timeτc\\tau\_\{c\}needed to cross the coarse\-graining scale\. The ballistic\-to\-diffusive crossover of §[2\.2](https://arxiv.org/html/2609.01705#S2.SS2)therefore applies directly, with the telegraph variance law \([6](https://arxiv.org/html/2609.01705#S2.E6)\) governing the transition\. By anchoring the VE schedule to this law, the generative model is aligned with physical time at the level of the transport variance, without heuristic schedule tuning\.

The nature of magnetic turbulence adds an additional complication \(see Methods, §[4\.9](https://arxiv.org/html/2609.01705#S4.SS9)for details\)\. It is often intermittent, meaning the magnetic field exhibits rare, strong fluctuations rather than smooth variations\. This results in heavy tails in the distribution of magnetic fieldsP⁡\(B⟂\)P\(B\_\{\\perp\}\), characterized by positive excess kurtosisK\>0K\>0\. The particle transport also inherits these heavy tails due to rare large\-angle scattering events\. However, the choice of diagnostic determines whether these non\-Gaussian features are observable \(Fig\.[4](https://arxiv.org/html/2609.01705#S2.F4)e, f\)\. A reliable experimental diagnostic is the20%20\\%contour of the flux,C20C\_\{20\}, which measures the broadening of the beam core while suppressing the influence of the tails \(see\([Chen et al\., 2020](https://arxiv.org/html/2609.01705#bib.bib25), Appendix C\.2\)for details\)\. This observable is Gaussian withK≈0K\\approx 0because the beam radius is much larger than the transverse correlation length\. The contour effectively averages over many independent turbulent patches\. By the CLT \(here applied to spatial averaging across the beam, distinct from the temporal accumulation in §[2\.1](https://arxiv.org/html/2609.01705#S2.SS1)\), this spatial averaging Gaussianizes the core statistics\. For this specific observable, the moment\-matched Gaussian surrogate is optimal within the class of Gaussian approximations \(Methods, §[4\.2](https://arxiv.org/html/2609.01705#S4.SS2)\)\. In contrast, the azimuthally averaged profileΨ¯​\(r\)\\overline\{\\Psi\}\(r\)probes the full displacement PDF, including the tails, and remains sensitive to the intermittency inherited from the magnetic field statistics \(Methods, §[4\.8](https://arxiv.org/html/2609.01705#S4.SS8)\), as shown next\.

### 2\.5Modeling intermittency with the anchored VE surrogate

Section[2\.1](https://arxiv.org/html/2609.01705#S2.SS1)introduced the modeling assumption used throughout the paper: sub\-ℓc\\ell\_\{c\}non\-Gaussian structure is represented through the entrance distributionP0P\_\{0\}, while subsequent coarse\-grained spreading is approximated by a Gaussian transport kernel with physics\-anchored variance\. We now give this construction a concrete physical realization\.

In the high\-rigidity regime of §[2\.4](https://arxiv.org/html/2609.01705#S2.SS4), a particle receives a small angular kick when crossing one turbulent cell\. This kick is proportional to the path\-averaged perpendicular magnetic fieldB¯⟂\\overline\{B\}\_\{\\perp\}over a distance of orderℓc\\ell\_\{c\}\. For the scalar surrogate, we use one signed transverse component and representP0P\_\{0\}by the trained Student\-ttentrance condition whose kurtosis is closest to the value measured at the coarse\-graining scale\. Because the anchored surrogate is defined only forτ≥τc\\tau\\geq\\tau\_\{c\}, the entrance statistic must be read at the corresponding path lengthℓ=ℓc\\ell=\\ell\_\{c\}: we therefore takeK0K\_\{0\}as the excess kurtosis of the signed transverse velocity component of the test\-particle ensemble atℓ=ℓc\\ell=\\ell\_\{c\}, givingK0=1\.4±0\.2K\_\{0\}=1\.4\\pm 0\.2\(mean±\\pmstandard error over the 20 independent runs\)\. This is the particle\-level realization of the field statistic of Methods §[4\.8](https://arxiv.org/html/2609.01705#S4.SS8), where the linear kick relation givesK⁡\(Δ​v⟂​\(τc\)\)=K⁡\(B¯⟂\)K\(\\Delta v\_\{\\perp\}\(\\tau\_\{c\}\)\)=K\(\\overline\{B\}\_\{\\perp\}\)\. This identification injects the non\-Gaussian shape, including the heavy tails inherited from intermittent turbulence withK0\>0K\_\{0\}\\\!\>\\\!0, at the coarse\-graining scaleℓc\\ell\_\{c\}, before subsequent multi\-cell spreading\. The physics\-anchored telegraph schedule then fixes the variance growth and governs the relaxation toward the diffusive regime at the level of the second moment\.

We validate this construction in Fig\.[4](https://arxiv.org/html/2609.01705#S2.F4)e and f\. The purple curve in Fig\.[4](https://arxiv.org/html/2609.01705#S2.F4)f shows the sampled kurtosis evolution predicted by the VE surrogate using the nearest trained entrance\-kurtosis condition\. The variance scale in Fig\.[4](https://arxiv.org/html/2609.01705#S2.F4)e is fixed by the telegraph\-anchored schedule\. Panel f shows that the entrance statistics, propagated with the anchored variance law, capture the overall trend of the observed kurtosis evolution\. In the diffusive regime, the empirical residual is\|Kphys−KM\|=0\.20±0\.05\|K\_\{\\mathrm\{phys\}\}\-K\_\{M\}\|=0\.20\\pm 0\.05\. The contribution from residual kernel non\-Gaussianity is quantified by Eq\. \([11](https://arxiv.org/html/2609.01705#S4.E11)\) \(Methods, §[4\.8](https://arxiv.org/html/2609.01705#S4.SS8)\)\.

## 3Discussion

We have introduced a framework that links macroscopic physical transport laws directly to the variance schedule of a diffusion model\. This physics\-anchoring aligns the generative and physical timescales by construction, at the level of the variance\. The variance path is enforced analytically\. The score network is then needed only for the remaining non\-Gaussian structure\. Because training and sampling share the sameτ\\tau\-anchored marginals, the reverse PF\-ODE acts as a physics\-timed denoiser: integrating backward from a detector ensemble atτ⋆\\tau^\{\\star\}recovers earlier surrogate marginals and—by stopping early—undoes only late\-time noise\. Along this reverse evolution the variance path and the kurtosis relaxation are preserved \(Fig\.[2](https://arxiv.org/html/2609.01705#S2.F2)\), so reconstruction is distributional \(more detailed than moment inversion only\)\.

The resulting Gaussian surrogate accurately describes diagnostics that average many independent contributions \(e\.g\., the laboratory 20% contour via the CLT\)\. For Student\-ttentrance distributions matched in variance and kurtosis, the VE model reproduces the predicted entrance\-kurtosis relaxation \(§[2\.5](https://arxiv.org/html/2609.01705#S2.SS5)\)\. The kurtosis gap quantifies the contribution from a non\-Gaussian physical transport kernel\.

This distinction is sharpest when comparing against the Kac/telegraph propagator\. Since the test\-particle marginals show positive excess kurtosis \(Fig\.[4](https://arxiv.org/html/2609.01705#S2.F4)f\), the data favor the Gaussian\-kernel surrogate in this regime\. The observed positive non\-Gaussianity can be represented through the entrance statistics and does not require a bounded\-support transport kernel\.

Because our contribution is a scalar pathσ~2​\(τ\)\\tilde\{\\sigma\}^\{2\}\(\\tau\), it can be used as a drop\-in schedule in any VE sampler \(e\.g\., Euler–Maruyama, predictor\-corrector\([Song et al\., 2021](https://arxiv.org/html/2609.01705#bib.bib9)\), Heun/second\-order, or the EDM framework\([Karras et al\., 2022](https://arxiv.org/html/2609.01705#bib.bib20)\)\)\. Figure[2](https://arxiv.org/html/2609.01705#S2.F2)already demonstrates both PF\-ODE and reverse\-SDE sampling with the same schedule\. In all physics results we train and sample on the sameτ\\tau‑anchored path to keep the generative clock calibrated to physical time at the level of the variance\. In a non\-physical Swiss\-roll benchmark, the telegraph path provides a competitive precision–recall balance across the tested neighborhood sizes \(Fig\.[5](https://arxiv.org/html/2609.01705#S3.F5); Methods, §[4\.10](https://arxiv.org/html/2609.01705#S4.SS10)\)\. A theoretical explanation remains an open question\.

![Refer to caption](https://arxiv.org/html/2609.01705v1/fig5.png)Figure 5:Non\-physics example\.Top:Precision–Recall curves on 2‑D swiss roll comparing telegraph \(τc=0\.2\\tau\_\{c\}=0\.2\) to karras EDM, linear, and cosine schedules\. Points along each curve are colored by neighborhood sizekkin the nearest\-center k\-NN P/R diagnostic defined in Methods\. Largerkkevaluates global structure\. Shaded bands show±1​σ\\pm 1\\sigmaover five independent sampling seeds\. Telegraph provides a competitive precision\-recall balance across the tested neighborhood sizes\.Bottom:Examples of generated samples at different times\. All curves share one model trained once withσ\\sigma‑conditioning; we vary only the sampling path \(solver, NFE, seeds, and noise bounds are identical\), isolating the effect of the schedule\.Returning to the scientific setting, the learned score field provides differentiable likelihood estimates via the PF\-ODE and can, in principle, support gradient\-based inference for physical parameters entering the schedule or conditioning the score model\. This capability extends to complex scenarios \(e\.g\., heteroscedastic transport or nonlinear detectors\) where analytic convolution does not exist, while naturally maintaining the correct variance scaling \(Fig\.[4](https://arxiv.org/html/2609.01705#S2.F4)\)\. Note that joint identifiability of multiple physical parameters requires either multiple temporal snapshots or additional constraints\.

The variance\-anchoring method we propose complements alternative techniques that learn explicit temporal variations in transport maps or trajectories, including flow matching\([Lipman et al\., 2023](https://arxiv.org/html/2609.01705#bib.bib23);[Baldan et al\., 2025](https://arxiv.org/html/2609.01705#bib.bib10)\), for which[Wald and Steidl \(2025\)](https://arxiv.org/html/2609.01705#bib.bib4)provides a unified overview, and reverse\-time simulators\([Holzschuh et al\., 2023](https://arxiv.org/html/2609.01705#bib.bib11)\)\. Those approaches typically use trajectory data or interpolated pairs at multiple intermediate times to learn the complete transport map\. Our focus on distributional snapshots allows for a highly data\-efficient paradigm: training requires entrance samples and the prescribed variance law\. This makes our method particularly suited to experimental conditions in which intermediate\-time transport data are unavailable\. The trade\-off is that we predict the distribution of particles, not their individual paths\. The Kac flow baseline in Fig\.[2](https://arxiv.org/html/2609.01705#S2.F2)is itself a flow\-matching model, providing a direct controlled comparison: it matches the telegraph variance but predicts sub\-Gaussian intermediate marginals inconsistent with the observed positive kurtosis \(Fig\.[4](https://arxiv.org/html/2609.01705#S2.F4)f\)\. The combination of physically anchored schedules and path\-learning networks is likely to lead to models which are compatible both with the macroscopic laws and the microscopic correlations\. Learned or non\-Brownian 1\-D noising processes\([Chemseddine et al\., 2025](https://arxiv.org/html/2609.01705#bib.bib5)\)offer another route to data\-adapted schedules that could be combined with physics\-anchored variance paths\.

The application domains listed in the Introduction \(anomalous diffusion, rough volatility, intracellular transport\) are natural candidates\. For processes with non\-Gaussian kernels \(e\.g\., CTRW propagators in porous media\([Berkowitz et al\., 2000](https://arxiv.org/html/2609.01705#bib.bib28)\)\), the leading fourth\-order mismatch is quantified by the kurtosis gap \([11](https://arxiv.org/html/2609.01705#S4.E11)\)\. The entrance\-statistics mechanism of §[2\.5](https://arxiv.org/html/2609.01705#S2.SS5)captures leading non\-Gaussian features\. Wherever a system’s variance growth is governed by a known law, that law can serve as the clock of a generative model\. Such a distributional propagator, calibrated to physical time at the variance level, could serve as a transport module within a broader scientific world model\. The network is then left to learn only what the law does not already know\.

## 4Methods

Table[2](https://arxiv.org/html/2609.01705#S4.T2)collects the principal symbols; each is also defined at first use\.

Table 2:Principal notation\.SymbolMeaningUnitsDefinedGenerative modelτ\\taunormalized time∈\[0,1\]\\in\[0,1\]—§ 1τ⋆\\tau^\{\\star\}Readout \(detector\) time—§ 1σ~2​\(τ\)\\tilde\{\\sigma\}^\{2\}\(\\tau\)Acc\. post\-entrance variance,σ~2​\(1\)=1\\tilde\{\\sigma\}^\{2\}\(1\)\\\!=\\\!1—Eq\. \([2](https://arxiv.org/html/2609.01705#S1.E2)\)g~​\(τ\)\\tilde\{g\}\(\\tau\)Noise schedule,g~2=d​σ~2/d​τ\\tilde\{g\}^\{2\}\\\!=\\\!d\\tilde\{\\sigma\}^\{2\}/d\\tau—Eq\. \([2](https://arxiv.org/html/2609.01705#S1.E2)\)P0P\_\{0\}Entrance distribution—Eq\. \([1](https://arxiv.org/html/2609.01705#S1.E1)\)pτp\_\{\\tau\}Marginal at timeτ\\tau—Eq\. \([3](https://arxiv.org/html/2609.01705#S2.E3)\)𝐬θ\\mathbf\{s\}\_\{\\theta\}Learned score≈∇𝐱​log​pτ\\approx\\nabla\_\{\\mathbf\{x\}\}\\\!\\log p\_\{\\tau\}\[𝐱\]−1\[\\mathbf\{x\}\]^\{\-1\}§[4\.1](https://arxiv.org/html/2609.01705#S4.SS1)KKExcess kurtosisκ4/σ4\\kappa\_\{4\}/\\sigma^\{4\}—Eq\. \([4](https://arxiv.org/html/2609.01705#S2.E4)\)𝒯⁡\(τ\)\\mathcal\{T\}\(\\tau\)Physical transport kernel—§[4\.2](https://arxiv.org/html/2609.01705#S4.SS2)𝐖τ\\mathbf\{W\}\_\{\\tau\}Standard Wiener process—Eq\. \([1](https://arxiv.org/html/2609.01705#S1.E1)\)Physical transportτc\\tau\_\{c\}Correlation / crossover time—§[2\.1](https://arxiv.org/html/2609.01705#S2.SS1)ℓc\\ell\_\{c\}Correlation lengthμ\\mum§[2\.1](https://arxiv.org/html/2609.01705#S2.SS1)rgr\_\{g\}Gyroradiusp/\(q​B\)p/\(qB\)μ\\mum§[2\.4](https://arxiv.org/html/2609.01705#S2.SS4)DvD\_\{v\}Velocity\-space diffusion coeff\.m2s\-3§[4\.7](https://arxiv.org/html/2609.01705#S4.SS7)κ\\kappaSpatial diffusion coefficientm2s\-1§[4\.7](https://arxiv.org/html/2609.01705#S4.SS7)BrmsB\_\{\\mathrm\{rms\}\}RMS magnetic fieldkGTable[4](https://arxiv.org/html/2609.01705#S4.T4)### 4\.1The VE framework and Gaussian convolution

We adopt the standard VE framework\([Song et al\., 2021](https://arxiv.org/html/2609.01705#bib.bib9)\)\. The forward SDE and resulting Gaussian convolution are given in Eqs\. \([1](https://arxiv.org/html/2609.01705#S1.E1)\) and \([3](https://arxiv.org/html/2609.01705#S2.E3)\); here we record the training objective\. We setg~2​\(τ\)=dd​τ​σ~2​\(τ\)\\tilde\{g\}^\{2\}\(\\tau\)=\\frac\{d\}\{d\\tau\}\\tilde\{\\sigma\}^\{2\}\(\\tau\)so that, by Itô isometry, the accumulated noise has varianceσ~2​\(τ\)\\tilde\{\\sigma\}^\{2\}\(\\tau\)\. Writing𝐱τ=𝐱0\+σ~​\(τ\)​ϵ\\mathbf\{x\}\_\{\\tau\}=\\mathbf\{x\}\_\{0\}\+\\tilde\{\\sigma\}\(\\tau\)\\boldsymbol\{\\epsilon\}withϵ∼𝒩⁡\(0,I\)\\boldsymbol\{\\epsilon\}\\sim\\mathcal\{N\}\(0,I\), denoising score matching\([Vincent, 2011](https://arxiv.org/html/2609.01705#bib.bib42);[Song et al\., 2021](https://arxiv.org/html/2609.01705#bib.bib9)\)trains a score network𝐬θ​\(𝐱,τ\)≈∇𝐱​log​pτ​\(𝐱\)\\mathbf\{s\}\_\{\\theta\}\(\\mathbf\{x\},\\tau\)\\approx\\nabla\_\{\\mathbf\{x\}\}\\log p\_\{\\tau\}\(\\mathbf\{x\}\)by minimizing

𝔼​‖𝐬θ​\(𝐱τ,τ\)−\(𝐱0−𝐱τ\)/σ~2​\(τ\)‖2\.\\mathbb\{E\}\\,\\big\\\|\\mathbf\{s\}\_\{\\theta\}\(\\mathbf\{x\}\_\{\\tau\},\\tau\)\-\(\\mathbf\{x\}\_\{0\}\-\\mathbf\{x\}\_\{\\tau\}\)/\\tilde\{\\sigma\}^\{2\}\(\\tau\)\\big\\\|^\{2\}\.\(8\)At each fixedτ\\tau, the network learns the global score of the marginalpτp\_\{\\tau\}, with adjacent times coupled through shared parameters\. Equation \([16](https://arxiv.org/html/2609.01705#S4.E16)\) in §[4\.5](https://arxiv.org/html/2609.01705#S4.SS5)gives theg~2\\tilde\{g\}^\{2\}\-weighted form used in practice\.

### 4\.2Optimality within Gaussian surrogates

Among centered, isotropic Gaussian approximations to the physical transport kernel𝒯⁡\(τ\)\\mathcal\{T\}\(\\tau\), the moment\-matched Gaussian uniquely minimizes cross\-entropy\. When𝒯⁡\(τ\)\\mathcal\{T\}\(\\tau\)has a density and finite differential entropy, this is equivalently the unique minimizer of forward KL\. After convolution withP0P\_\{0\}, the resulting marginal error is controlled by theW2W\_\{2\}contraction bound below\.

From optimal transport theory\([Villani, 2009](https://arxiv.org/html/2609.01705#bib.bib35)\), convolution by an independent variable contracts theW2W\_\{2\}distance: for any kernelsν1,ν2\\nu\_\{1\},\\nu\_\{2\},

W2​\(P0∗ν1,P0∗ν2\)≤W2​\(ν1,ν2\)\.W\_\{2\}\(P\_\{0\}\*\\nu\_\{1\},\\,P\_\{0\}\*\\nu\_\{2\}\)\\leq W\_\{2\}\(\\nu\_\{1\},\\nu\_\{2\}\)\.\(9\)
Let\(𝐘1,𝐘2\)\(\\mathbf\{Y\}\_\{1\},\\mathbf\{Y\}\_\{2\}\)be an optimalW2W\_\{2\}\-coupling ofν1\\nu\_\{1\}andν2\\nu\_\{2\}, and let𝐗0∼P0\\mathbf\{X\}\_\{0\}\\\!\\sim\\\!P\_\{0\}independently\. Setting𝐙i=𝐗0\+𝐘i\\mathbf\{Z\}\_\{i\}=\\mathbf\{X\}\_\{0\}\+\\mathbf\{Y\}\_\{i\}produces marginalsP0∗νiP\_\{0\}\*\\nu\_\{i\}with‖𝐙1−𝐙2‖=‖𝐘1−𝐘2‖\\\|\\mathbf\{Z\}\_\{1\}\-\\mathbf\{Z\}\_\{2\}\\\|=\\\|\\mathbf\{Y\}\_\{1\}\-\\mathbf\{Y\}\_\{2\}\\\|\(the shared𝐗0\\mathbf\{X\}\_\{0\}cancels\), soW22​\(P0∗ν1,P0∗ν2\)≤𝔼​‖𝐙1−𝐙2‖2=W22​\(ν1,ν2\)W\_\{2\}^\{2\}\(P\_\{0\}\*\\nu\_\{1\},P\_\{0\}\*\\nu\_\{2\}\)\\leq\\mathbb\{E\}\\\|\\mathbf\{Z\}\_\{1\}\-\\mathbf\{Z\}\_\{2\}\\\|^\{2\}=W\_\{2\}^\{2\}\(\\nu\_\{1\},\\nu\_\{2\}\)\. Settingν1=𝒯⁡\(τ\)\\nu\_\{1\}=\\mathcal\{T\}\(\\tau\)\(physical kernel\) andν2=𝒩⁡\(0,σ~2​\(τ\)​I\)\\nu\_\{2\}=\\mathcal\{N\}\(0,\\tilde\{\\sigma\}^\{2\}\(\\tau\)I\)\(Gaussian surrogate kernel\) yields

W2​\(PτPhys,pτ\)≤W2​\(𝒯⁡\(τ\),𝒩⁡\(0,σ~2​\(τ\)​I\)\),W\_\{2\}\(P^\{\\mathrm\{Phys\}\}\_\{\\tau\},\\,p\_\{\\tau\}\)\\leq W\_\{2\}\\\!\\big\(\\mathcal\{T\}\(\\tau\),\\,\\mathcal\{N\}\(0,\\tilde\{\\sigma\}^\{2\}\(\\tau\)I\)\\big\),\(10\)so the approximation error is controlled by how far the physical transport kernel𝒯⁡\(τ\)\\mathcal\{T\}\(\\tau\)deviates from Gaussian\.

### 4\.3Kurtosis evolution and error quantification

We quantify higher‑order statistics using excess kurtosisK=κ4/σ4K=\\kappa\_\{4\}/\\sigma^\{4\}\. From the Gaussian convolution structure, the model’s kurtosis obeys the closed‑form evolution given in Eq\. \([4](https://arxiv.org/html/2609.01705#S2.E4)\) of the main text\.

For the physical processPτPhys=P0∗𝒯⁡\(τ\)P^\{\\mathrm\{Phys\}\}\_\{\\tau\}=P\_\{0\}\*\\mathcal\{T\}\(\\tau\), cumulant additivity givesκ4Phys=κ4​\(P0\)\+κ4​\(𝒯\)\\kappa\_\{4\}^\{\\mathrm\{Phys\}\}=\\kappa\_\{4\}\(P\_\{0\}\)\+\\kappa\_\{4\}\(\\mathcal\{T\}\), while the VE surrogate hasκ4M=κ4​\(P0\)\\kappa\_\{4\}^\{M\}=\\kappa\_\{4\}\(P\_\{0\}\)\. We define the kernel’s excess kurtosis asK𝒯​\(τ\):=κ4​\(𝒯⁡\(τ\)\)/σ~4​\(τ\)K\_\{\\mathcal\{T\}\}\(\\tau\):=\\kappa\_\{4\}\(\\mathcal\{T\}\(\\tau\)\)/\\tilde\{\\sigma\}^\{4\}\(\\tau\)\(subscript𝒯\\mathcal\{T\}distinguishes this from the marginal kurtosisKK\)\. The resulting excess\-kurtosis difference is

Kphys​\(τ\)−KM​\(τ\)=K𝒯​\(τ\)​\(σ~2​\(τ\)σ02\+σ~2​\(τ\)\)2\.K\_\{\\mathrm\{phys\}\}\(\\tau\)\-K\_\{M\}\(\\tau\)=K\_\{\\mathcal\{T\}\}\(\\tau\)\\left\(\\frac\{\\tilde\{\\sigma\}^\{2\}\(\\tau\)\}\{\\sigma\_\{0\}^\{2\}\+\\tilde\{\\sigma\}^\{2\}\(\\tau\)\}\\right\)^\{\\\!2\}\.\(11\)The gap vanishes if the kernel Gaussianizes \(K𝒯→0K\_\{\\mathcal\{T\}\}\\\!\\to\\\!0\) and is suppressed early whenσ~2≪σ02\\tilde\{\\sigma\}^\{2\}\\ll\\sigma\_\{0\}^\{2\}\.

### 4\.4Kac/telegraph kernel: density, moments, and parameter selection

The Kac flow baseline is built on the Goldstein–Kac telegraph process\([Goldstein, 1951](https://arxiv.org/html/2609.01705#bib.bib34);[Duong et al\., 2025](https://arxiv.org/html/2609.01705#bib.bib18)\)\. A particle moves in one dimension at constant speedccand reverses direction at the events of a Poisson process with rateaa\. The initial velocity is±c\\pm cwith equal probability\. The resulting displacement kernel at timettis

pK​\(x,t\)\\displaystyle p\_\{\\mathrm\{K\}\}\(x,t\)=e−a​t2​\[δ⁡\(x−c​t\)\+δ⁡\(x\+c​t\)\]\\displaystyle=\\frac\{e^\{\-at\}\}\{2\}\\big\[\\delta\(x\-ct\)\+\\delta\(x\+ct\)\\big\]\+a​e−a​t2​c\[I0\(acη\)\+c​tηI1\(acη\)\]𝟏\{\|x\|<ct\},\\displaystyle\\;\+\\frac\{a\\,e^\{\-at\}\}\{2c\}\\left\[I\_\{0\}\\\!\\left\(\\tfrac\{a\}\{c\}\\eta\\right\)\+\\frac\{ct\}\{\\eta\}\\,I\_\{1\}\\\!\\left\(\\tfrac\{a\}\{c\}\\eta\\right\)\\right\]\\mathbf\{1\}\_\{\\\{\|x\|<ct\\\}\},\(12\)whereη=c2​t2−x2\\eta=\\sqrt\{c^\{2\}t^\{2\}\-x^\{2\}\}andI0I\_\{0\},I1I\_\{1\}are modified Bessel functions of the first kind\. The kernel has bounded support\|x\|≤c​t\|x\|\\leq ct\.

The kernel moments follow from the velocity autocorrelation⟨v⁡\(t\)​v​\(t\+s\)⟩=c2​e−2​a​s\\langle v\(t\)\\,v\(t\+s\)\\rangle=c^\{2\}e^\{\-2as\}\. The fourth moment is obtained by integratingx4​pK​\(x,t\)x^\{4\}p\_\{\\mathrm\{K\}\}\(x,t\), including the two point masses in Eq\. \([12](https://arxiv.org/html/2609.01705#S4.E12)\)\. With the abbreviationu:=2​a​tu:=2at, the kernel variance is

σK2​\(t\)=c22​a2​\(u−1\+e−u\),\\sigma^\{2\}\_\{\\mathrm\{K\}\}\(t\)=\\frac\{c^\{2\}\}\{2a^\{2\}\}\\left\(u\-1\+e^\{\-u\}\\right\),\(13\)and the kernel excess kurtosis, which plays the role ofK𝒯K\_\{\\mathcal\{T\}\}in Eq\. \([11](https://arxiv.org/html/2609.01705#S4.E11)\), is

K𝒯Kac​\(t\)=3​u2−4​u\+6−\(6\+2​u\)​e−u\(u−1\+e−u\)2−3\.K\_\{\\mathcal\{T\}\}^\{\\mathrm\{Kac\}\}\(t\)=3\\,\\frac\{u^\{2\}\-4u\+6\-\(6\+2u\)\\,e^\{\-u\}\}\{\\left\(u\-1\+e^\{\-u\}\\right\)^\{2\}\}\-3\.\(14\)It equals−2\-2fort→0t\\to 0, where the kernel is a symmetric two\-point distribution, and relaxes to zero from below asK𝒯Kac≃−3/\(at\)K\_\{\\mathcal\{T\}\}^\{\\mathrm\{Kac\}\}\\simeq\-3/\(at\)fora​t≫1at\\gg 1\.

The dotted reference curves in Fig\.[2](https://arxiv.org/html/2609.01705#S2.F2)b–d show the marginal excess kurtosis of the Kac model\. It follows from cumulant additivity forP0∗pKP\_\{0\}\*p\_\{\\mathrm\{K\}\}and combines Eq\. \([4](https://arxiv.org/html/2609.01705#S2.E4)\) with the gap term of Eq\. \([11](https://arxiv.org/html/2609.01705#S4.E11)\):

KKac​\(τ\)=K0​σ04\+K𝒯Kac​\(τ\)​σ~4​\(τ\)\(σ02\+σ~2​\(τ\)\)2\.K\_\{\\mathrm\{Kac\}\}\(\\tau\)=\\frac\{K\_\{0\}\\,\\sigma\_\{0\}^\{4\}\+K\_\{\\mathcal\{T\}\}^\{\\mathrm\{Kac\}\}\(\\tau\)\\,\\tilde\{\\sigma\}^\{4\}\(\\tau\)\}\{\\left\(\\sigma\_\{0\}^\{2\}\+\\tilde\{\\sigma\}^\{2\}\(\\tau\)\\right\)^\{2\}\}\.\(15\)
The parameters\(a,c\)\(a,c\)are not fitted\. On the normalized clock, they are fixed by requiring the Kac process to share the anchored telegraph variance path\. This givesa=1/\(2​τc\)a=1/\(2\\tau\_\{c\}\)andc2=C−1/\(2​τc\)c^\{2\}=C^\{\-1\}/\(2\\tau\_\{c\}\), for which Eq\. \([13](https://arxiv.org/html/2609.01705#S4.E13)\) reduces to Eq\. \([6](https://arxiv.org/html/2609.01705#S2.E6)\) withu=τ/τcu=\\tau/\\tau\_\{c\}andσ~2​\(1\)=1\\tilde\{\\sigma\}^\{2\}\(1\)=1\. The Kac baseline is variance\-matched to the VE\-telegraph schedule by construction, and Table[1](https://arxiv.org/html/2609.01705#S2.T1)compares the resulting Gaussian and finite\-speed constructions\.

The learned Kac\-flow baseline follows the conditional\-flow\-matching construction of[Duong et al\. \(2025\)](https://arxiv.org/html/2609.01705#bib.bib18): a bounded velocity network is regressed against the analytical conditional Kac velocity, and sampling integrates the learned flow ODE backward from the terminal marginal\.

### 4\.5Diffusion model implementation and baseline comparisons

We use the VE framework instead of Variance Preserving \(VP/DDPM\) models because the transport problems considered here exhibit growing variance\. In scattering processes, the spatial variance grows naturally with time \(σ~2​\(τ\)\\tilde\{\\sigma\}^\{2\}\(\\tau\)increases\), matching the VE formulationd​𝐱τ=g~​\(τ\)​d​𝐖τd\\mathbf\{x\}\_\{\\tau\}=\\tilde\{g\}\(\\tau\)\\,d\\mathbf\{W\}\_\{\\tau\}, withg~2​\(τ\)=d​σ~2​\(τ\)/d​τ\\tilde\{g\}^\{2\}\(\\tau\)=d\\tilde\{\\sigma\}^\{2\}\(\\tau\)/d\\tau\. In contrast, VP models constrain variance to unity \(σ2=1\\sigma^\{2\}=1\) via a mean\-reverting drift, which would obscure the direct physical interpretation of the transport coordinates\. For finite\-speed transport,σ~2​\(τ\)\\tilde\{\\sigma\}^\{2\}\(\\tau\)follows the normalized telegraph law \(Eq\.[6](https://arxiv.org/html/2609.01705#S2.E6)in the main text\)\. Baseline schedules use linearσ~2​\(τ\)∝τ\\tilde\{\\sigma\}^\{2\}\(\\tau\)\\propto\\tau\(pure diffusion at allτ\\tau\), cosine schedules\([Nichol and Dhariwal, 2021](https://arxiv.org/html/2609.01705#bib.bib19)\), and a Karras/EDM\-inspired power\-law variance scheduleσ~2​\(τ\)∝τρ\\tilde\{\\sigma\}^\{2\}\(\\tau\)\\propto\\tau^\{\\rho\}withρ=7\\rho=7\([Karras et al\., 2022](https://arxiv.org/html/2609.01705#bib.bib20)\)\. All models share the same score network architecture \(a 10\-block FiLM\-MLP with GroupNorm, using sinusoidal time embeddings\([Song et al\., 2021](https://arxiv.org/html/2609.01705#bib.bib9)\)and FiLM conditioning; approximately1\.16×1061\.16\\times 10^\{6\}learnable parameters\)\. The physics\-anchored variant fixesσ~2​\(τ\)\\tilde\{\\sigma\}^\{2\}\(\\tau\)via the physical law; architecture and computational cost are unchanged—only the scalar scheduleg~2​\(τ\)\\tilde\{g\}^\{2\}\(\\tau\)differs\.

The score network optionally conditions on entrance kurtosisK0K\_\{0\}through a small MLP that mapslog⁡\(K0\)\\log\(K\_\{0\}\)\(normalized to zero mean and unit variance\) to a 32\-dimensional embedding, which modulates hidden features via FiLM layers alongside the 64\-dimensional sinusoidal time embedding\. This enables a single trained model to serve multiple entrance distributions without retraining\.

The complete training and sampling procedure is summarized in Algorithm[1](https://arxiv.org/html/2609.01705#alg1); all hyperparameters and architecture details are listed in Table[3](https://arxiv.org/html/2609.01705#S4.T3)\. Training minimizes theg~2\\tilde\{g\}^\{2\}\-weighted denoising score\-matching loss,

ℒ⁡\(θ\)=𝔼τ,𝐱0,ϵ​\[g~2​\(τ\)​∥𝐬θ​\(𝐱τ,τ\)−\(𝐱0−𝐱τ\)/σ~2​\(τ\)∥2\],\\mathcal\{L\}\(\\theta\)=\\mathbb\{E\}\_\{\\tau,\\mathbf\{x\}\_\{0\},\\boldsymbol\{\\epsilon\}\}\\\!\\Big\[\\,\\tilde\{g\}^\{2\}\(\\tau\)\\;\\big\\lVert\\,\\mathbf\{s\}\_\{\\theta\}\(\\mathbf\{x\}\_\{\\tau\},\\tau\)\-\(\\mathbf\{x\}\_\{0\}\-\\mathbf\{x\}\_\{\\tau\}\)/\\tilde\{\\sigma\}^\{2\}\(\\tau\)\\,\\big\\rVert^\{2\}\\Big\],\(16\)with𝐱τ=𝐱0\+σ~​\(τ\)​ϵ\\mathbf\{x\}\_\{\\tau\}=\\mathbf\{x\}\_\{0\}\+\\tilde\{\\sigma\}\(\\tau\)\\boldsymbol\{\\epsilon\}andϵ∼𝒩⁡\(𝟎,I\)\\boldsymbol\{\\epsilon\}\\sim\\mathcal\{N\}\(\\mathbf\{0\},I\)\. Theg~2\\tilde\{g\}^\{2\}prefactor is standard signal\-to\-noise weighting\([Song et al\., 2021](https://arxiv.org/html/2609.01705#bib.bib9);[Karras et al\., 2022](https://arxiv.org/html/2609.01705#bib.bib20)\); it upweights time steps where the schedule is steepest, ensuring the network allocates capacity where the marginal changes fastest\.

Table 3:Training and sampling configuration\.All VE variants share identical architecture and budget; only the scheduleg~2​\(τ\)\\tilde\{g\}^\{2\}\(\\tau\)differs\. The Kac flow baseline uses the same architecture with a bounded output layer \(c⋅tanhc\\\!\\cdot\\\!\\tanh\)\.At inference we draw terminal states from the exact forward marginalp1p\_\{1\}implied by the VE process \(i\.e\.𝐱1=𝐱0\+σ~​\(1\)​ϵ\\mathbf\{x\}\_\{1\}=\\mathbf\{x\}\_\{0\}\+\\tilde\{\\sigma\}\(1\)\\boldsymbol\{\\epsilon\}with𝐱0∼P0\\mathbf\{x\}\_\{0\}\\sim P\_\{0\}\) and integrate the PF\-ODE backward with an Euler solver \(T=500T=500steps\), halting at the desired readoutτ⋆\\tau^\{\\star\}\. Uncertainty is quantified over 10 independent seeds\.

Algorithm 1Physics\-anchored VE diffusion: training and sampling\.1:Input:Variance path

σ~2​\(τ\)\\tilde\{\\sigma\}^\{2\}\(\\tau\)from physical law, with

σ~2​\(1\)=1\\tilde\{\\sigma\}^\{2\}\(1\)=1; entrance samples

𝐱0∼P0\\mathbf\{x\}\_\{0\}\\\!\\sim\\\!P\_\{0\}
2:Define:

g~2​\(τ\)=d​σ~2/d​τ\\tilde\{g\}^\{2\}\(\\tau\)=d\\tilde\{\\sigma\}^\{2\}/d\\tau;

σ~​\(τ\)=σ~2​\(τ\)\\tilde\{\\sigma\}\(\\tau\)=\\sqrt\{\\smash\[b\]\{\\tilde\{\\sigma\}^\{2\}\(\\tau\)\}\}
3:

4:

5:Training\(entrance\-only denoising score matching\)

6:

7:repeat

8:Sample

𝐱0∼P0\\mathbf\{x\}\_\{0\}\\\!\\sim\\\!P\_\{0\}
9:Sample

τ∼𝒰\(0,1\]\\tau\\\!\\sim\\\!\\mathcal\{U\}\(0,1\]
10:Sample

ϵ∼𝒩⁡\(𝟎,I\)\\boldsymbol\{\\epsilon\}\\\!\\sim\\\!\\mathcal\{N\}\(\\mathbf\{0\},I\)
11:

𝐱τ←𝐱0\+σ~​\(τ\)​ϵ\\mathbf\{x\}\_\{\\tau\}\\leftarrow\\mathbf\{x\}\_\{0\}\+\\tilde\{\\sigma\}\(\\tau\)\\,\\boldsymbol\{\\epsilon\}⊳\\trianglerightNoisy state at physical timeτ\\tau

12:

ℒ←g~2​\(τ\)​∥𝐬θ​\(𝐱τ,τ\)−\(𝐱0−𝐱τ\)/σ~2​\(τ\)∥2\\mathcal\{L\}\\;\\leftarrow\\;\\tilde\{g\}^\{2\}\(\\tau\)\\;\\big\\lVert\\,\\mathbf\{s\}\_\{\\theta\}\(\\mathbf\{x\}\_\{\\tau\},\\tau\)\-\(\\mathbf\{x\}\_\{0\}\-\\mathbf\{x\}\_\{\\tau\}\)/\\tilde\{\\sigma\}^\{2\}\(\\tau\)\\,\\big\\rVert^\{2\}⊳\\trianglerightg~2\\tilde\{g\}^\{2\}\-weighted loss

13:Update

θ\\thetavia AdamW on

∇θℒ\\nabla\_\{\\theta\}\\mathcal\{L\}; update EMA shadow

14:untilconverged

15:

16:

17:Sampling\(physics\-timed PF\-ODE at readoutτ⋆\\tau^\{\\star\}\)

18:

19:Draw

𝐱0∼P0\\mathbf\{x\}\_\{0\}\\\!\\sim\\\!P\_\{0\},

ϵ∼𝒩⁡\(𝟎,I\)\\boldsymbol\{\\epsilon\}\\\!\\sim\\\!\\mathcal\{N\}\(\\mathbf\{0\},I\); set

𝐱1←𝐱0\+σ~​\(1\)​ϵ\\mathbf\{x\}\_\{1\}\\leftarrow\\mathbf\{x\}\_\{0\}\+\\tilde\{\\sigma\}\(1\)\\,\\boldsymbol\{\\epsilon\}⊳\\trianglerightTerminal state fromp1p\_\{1\}

20:Integrate backward from

τ=1\\tau\\\!=\\\!1to

τ=τ⋆\\tau\\\!=\\\!\\tau^\{\\star\}:

21:

d​𝐱d​τ=−12​g~2​\(τ\)​𝐬θ​\(𝐱,τ\)\\displaystyle\\frac\{d\\mathbf\{x\}\}\{d\\tau\}=\-\\tfrac\{1\}\{2\}\\,\\tilde\{g\}^\{2\}\(\\tau\)\\,\\mathbf\{s\}\_\{\\theta\}\(\\mathbf\{x\},\\tau\)
22:return

𝐱τ⋆∼pτ⋆\\mathbf\{x\}\_\{\\tau^\{\\star\}\}\\sim p\_\{\\tau^\{\\star\}\}⊳\\trianglerightAnchored surrogate marginal at detector time

Key point:only the scalar scheduleg~2​\(τ\)\\tilde\{g\}^\{2\}\(\\tau\)changes relative to a standard VE model; the network architecture, optimizer, and sampler are unchanged\. Optionally, the score network conditions on entrance kurtosisK0K\_\{0\}via a FiLM embedding \(see text\), enabling a single model for multiple entrance distributions\.

### 4\.6Likelihood computation and differentiable inference

The probability\-flow ODE corresponding to Eq\. \([1](https://arxiv.org/html/2609.01705#S1.E1)\) is

d​𝐱d​τ=−12​g~2​\(τ\)​𝐬θ​\(𝐱,τ\)\.\\frac\{d\\mathbf\{x\}\}\{d\\tau\}=\-\\tfrac\{1\}\{2\}\\tilde\{g\}^\{2\}\(\\tau\)\\,\\mathbf\{s\}\_\{\\theta\}\(\\mathbf\{x\},\\tau\)\.\(17\)The instantaneous change\-of\-variables formula\([Song et al\., 2021](https://arxiv.org/html/2609.01705#bib.bib9)\)yields the PF‑ODE model log‑likelihood:

log⁡pτ⋆​\(𝐱τ⋆\)=log⁡p1​\(𝐱1\)−∫τ⋆112​g~2​\(τ\)​div​𝐬θ​\(𝐱τ,τ\)​𝑑τ,\\log p\_\{\\tau^\{\\star\}\}\(\\mathbf\{x\}\_\{\\tau^\{\\star\}\}\)=\\log p\_\{1\}\(\\mathbf\{x\}\_\{1\}\)\-\\int\_\{\\tau^\{\\star\}\}^\{1\}\\\!\\tfrac\{1\}\{2\}\\tilde\{g\}^\{2\}\(\\tau\)\\,\\mathrm\{div\}\\,\\mathbf\{s\}\_\{\\theta\}\(\\mathbf\{x\}\_\{\\tau\},\\tau\)\\,d\\tau,\(18\)where the divergence is estimated using Hutchinson’s trace estimator\([Hutchinson, 1990](https://arxiv.org/html/2609.01705#bib.bib38)\)\. This expression yields the exact marginal log\-likelihood in the limit where𝐬θ\\mathbf\{s\}\_\{\\theta\}matches the true score and the terminal densityp1p\_\{1\}is specified; otherwise it provides a differentiable likelihood estimate\. Physical parameters that enter the schedule \(e\.g\.,τc\\tau\_\{c\}\) may be included inσ~2​\(τ,τc\)\\tilde\{\\sigma\}^\{2\}\(\\tau;\\tau\_\{c\}\), while entrance parameters such asK0K\_\{0\}may parameterizeP0​\(⋅,K0\)P\_\{0\}\(\\,\\cdot\\,;K\_\{0\}\)or condition𝐬θ\\mathbf\{s\}\_\{\\theta\}; such parameters can then be estimated by maximizing Eq\. \([18](https://arxiv.org/html/2609.01705#S4.E18)\)\.

### 4\.7Charged\-particle transport through magnetized turbulence

Throughout this paper,τ∈\[0,1\]\\tau\\in\[0,1\]denotes normalized time\. Physical timettrelates toτ\\tauvia the total observation window \(the normalized correlation time isτc=tc/tobs\\tau\_\{c\}=t\_\{c\}/t\_\{\\mathrm\{obs\}\}wheretc=ℓc/vt\_\{c\}=\\ell\_\{c\}/vis the cell\-crossing time\)\.

The transverse acceleration of a charged particle along its path through magnetized turbulence𝐁\\mathbf\{B\}is

a⁡\(t\)≡d​v⟂d​t≃q​vm​B⟂\.a\(t\)\\;\\equiv\\;\\frac\{dv\_\{\\perp\}\}\{dt\}\\;\\simeq\\;\\frac\{q\\,v\}\{m\}\\,B\_\{\\perp\}\.\(19\)Sincev⟂​\(t\)=∫0ta⟂​\(s\)​𝑑sv\_\{\\perp\}\(t\)=\\int\_\{0\}^\{t\}a\_\{\\perp\}\(s\)\\,ds, the velocity variance is a double integral of the acceleration autocorrelation:

σv⟂2​\(t\)=∫0t∫0tCa​\(s−s′\)​𝑑s​d​s′=∫0t\(t−Δ​t\)​Ca​\(Δ​t\)​𝑑Δ​t\.\\sigma^\{2\}\_\{v\_\{\\perp\}\}\(t\)\\;=\\;\\int\_\{0\}^\{t\}\\\!\\\!\\int\_\{0\}^\{t\}\\\!C\_\{a\}\(s\{\-\}s^\{\\prime\}\)\\,ds\\,ds^\{\\prime\}\\;=\\;2\\\!\\int\_\{0\}^\{t\}\\\!\(t\{\-\}\\Delta t\)\\,C\_\{a\}\(\\Delta t\)\\,d\\Delta t\.\(20\)Fort≪tct\\\!\\ll\\\!t\_\{c\}the autocorrelation is nearly constant,Ca≈⟨a2⟩C\_\{a\}\\\!\\approx\\\!\\langle a^\{2\}\\rangle, givingσv⟂2≈⟨a2⟩​t2\\sigma^\{2\}\_\{v\_\{\\perp\}\}\\\!\\approx\\\!\\langle a^\{2\}\\rangle\\,t^\{2\}\(ballistic\)\. Fort≫tct\\\!\\gg\\\!t\_\{c\}the factor\(t−Δ​t\)≈t\(t\{\-\}\\Delta t\)\\\!\\approx\\\!tover the correlation width and the integral saturates, givingσv⟂2≈2​Dv​t\\sigma^\{2\}\_\{v\_\{\\perp\}\}\\\!\\approx\\\!2\\,D\_\{v\}\\,t\(diffusive\), where

Dv=∫0∞Ca​\(Δ​t\)​𝑑Δ​t≈⟨a2⟩​ℓcv\.D\_\{v\}\\;=\\;\\int\_\{0\}^\{\\infty\}C\_\{a\}\(\\Delta t\)\\,d\\Delta t\\;\\approx\\;\\langle a^\{2\}\\rangle\\,\\frac\{\\ell\_\{c\}\}\{v\}\.\(21\)Equivalently, from a random‑walk estimate one obtainsDv≃q2​Brms2​ℓc​v/\(2​m2\)D\_\{v\}\\simeq q^\{2\}B\_\{\\mathrm\{rms\}\}^\{2\}\\,\\ell\_\{c\}\\,v/\(2m^\{2\}\)\. Here, we used the Taylor\-Green–Kubo formula \(see\([Kubo, 1957](https://arxiv.org/html/2609.01705#bib.bib1)\), and references therein\) andCa​\(Δ​t\):=⟨a⟂​\(t\)​a⟂​\(t\+Δ​t\)⟩C\_\{a\}\(\\Delta t\)\\;:=\\;\\big\\langle a\_\{\\perp\}\(t\)\\,a\_\{\\perp\}\(t\+\\Delta t\)\\big\\rangleis the stationary autocorrelation of the transverse accelerationa⟂​\(t\)a\_\{\\perp\}\(t\)\.

In physical timett\(with correlation timetc=ℓc/vt\_\{c\}=\\ell\_\{c\}/v\), the expected crossovers follow:

σv⟂2​\(t\)∝\{⟨a2⟩​t2,t≪tc\(ballistic\),Dv​t,t≫tc\(diffusive\)\.\\sigma^\{2\}\_\{v\_\{\\perp\}\}\(t\)\\ \\propto\\ \\begin\{cases\}\\langle a^\{2\}\\rangle\\,t^\{2\},&t\\ll t\_\{c\}\\quad\\text\{\(ballistic\)\},\\\\\[2\.0pt\] D\_\{v\}\\,t,&t\\gg t\_\{c\}\\quad\\text\{\(diffusive\)\}\.\\end\{cases\}\(22\)The corresponding displacement scaling in the telegraph \(finite\-speed\) transport picture is:

σx⟂2​\(t\)∝\{t2,t≪tc\(ballistic\),κ​t,t≫tc\(diffusive\),\\sigma^\{2\}\_\{x\_\{\\perp\}\}\(t\)\\ \\propto\\ \\begin\{cases\}t^\{2\},&t\\ll t\_\{c\}\\quad\\text\{\(ballistic\)\},\\\\\[2\.0pt\] \\kappa\\,t,&t\\gg t\_\{c\}\\quad\\text\{\(diffusive\)\},\\end\{cases\}\(23\)Upon normalizingτ=t/tobs\\tau=t/t\_\{\\mathrm\{obs\}\}, these give the short\- and long\-time limits of Eq\. \([6](https://arxiv.org/html/2609.01705#S2.E6)\)\. Here,κ\\kappais the spatial diffusion coefficient,⟨a2⟩\\langle a^\{2\}\\rangleis absorbed intoDvD\_\{v\}via the Green–Kubo integral \(Dv≈⟨a2⟩​ℓc/vD\_\{v\}\\approx\\langle a^\{2\}\\rangle\\,\\ell\_\{c\}/v\), andκ=v4/\(6​Dv\)\\kappa=v^\{4\}/\(6D\_\{v\}\)absorbs all remaining acceleration dependence into a single transport coefficient\.

Note thatκ\\kappacan be inferred in two equivalent ways: \(i\) from the long\-time slopeσx⟂2​\(t\)=2​κ​t\\sigma^\{2\}\_\{x\_\{\\perp\}\}\(t\)=2\\kappa\\,t; or \(ii\) from the measuredDvD\_\{v\}via small‑angle scattering,

κ=v23​ν=v46​Dv,\\kappa=\\frac\{v^\{2\}\}\{3\\,\\nu\}=\\frac\{v^\{4\}\}\{6\\,D\_\{v\}\},\(24\)whereν\\nuis the pitch‑angle scattering rate\.

### 4\.8Kurtosis inheritance and relaxation in turbulent transport

In the high\-rigidity limit \(rg≫ℓcr\_\{g\}\\gg\\ell\_\{c\}\), transport proceeds via independent scattering events as particles traverse turbulent cells\. Each signed transverse\-component kick is proportional to the path\-averaged transverse fieldB¯⟂\\overline\{B\}\_\{\\perp\}:

Δ​v⟂​\(τc\)∝B¯⟂\.\\Delta v\_\{\\perp\}\(\\tau\_\{c\}\)\\propto\\overline\{B\}\_\{\\perp\}\.\(25\)This linear relationship preserves kurtosis:K⁡\(Δ​v⟂​\(τc\)\)=K⁡\(B¯⟂\)K\(\\Delta v\_\{\\perp\}\(\\tau\_\{c\}\)\)=K\(\\overline\{B\}\_\{\\perp\}\)\.

After accumulatingN=ℓ/ℓcN=\\ell/\\ell\_\{c\}approximately independent kicks, cumulant additivity givesκ4​\(v⟂total\)=N​κ4​\(B¯⟂\)\\kappa\_\{4\}\(v\_\{\\perp\}^\{\\mathrm\{total\}\}\)=N\\,\\kappa\_\{4\}\(\\overline\{B\}\_\{\\perp\}\), while variance scales asσ2=N​σkick2\\sigma^\{2\}=N\\,\\sigma^\{2\}\_\{\\mathrm\{kick\}\}\. Since excess kurtosis is normalized by variance squared \(K=κ4/σ4K=\\kappa\_\{4\}/\\sigma^\{4\}\),

K⁡\(v⟂total\)=N​κ4​\(B¯⟂\)\(N​σkick2\)2=1N​K​\(B¯⟂\),K\\big\(v\_\{\\perp\}^\{\\mathrm\{total\}\}\\big\)=\\frac\{N\\,\\kappa\_\{4\}\(\\overline\{B\}\_\{\\perp\}\)\}\{\(N\\,\\sigma^\{2\}\_\{\\mathrm\{kick\}\}\)^\{2\}\}=\\frac\{1\}\{N\}\\,K\(\\overline\{B\}\_\{\\perp\}\),\(26\)describing the slow Gaussianization observed as the distribution propagates through the turbulent medium \(Fig\.[4](https://arxiv.org/html/2609.01705#S2.F4)f\)\. This1/N1/Ndecay holds once transport has entered the decorrelated regime \(ℓ≳ℓc\\ell\\gtrsim\\ell\_\{c\}\)\. Forℓ≲ℓc\\ell\\lesssim\\ell\_\{c\}\(the ballistic/crossover regime\), finite\-speed effects can make the true kernel non\-Gaussian; in that regime the VE construction serves as a variance\-correct Gaussian surrogate, with kernel mismatch quantified by Eq\. \([11](https://arxiv.org/html/2609.01705#S4.E11)\)\.

It is instructive to compare this physical1/N1/Nlaw with the surrogate\. Settingσ02=σkick2\\sigma\_\{0\}^\{2\}=\\sigma^\{2\}\_\{\\mathrm\{kick\}\}andσ~2=\(N−1\)​σkick2\\tilde\{\\sigma\}^\{2\}=\(N\{\-\}1\)\\,\\sigma^\{2\}\_\{\\mathrm\{kick\}\}in Eq\. \([4](https://arxiv.org/html/2609.01705#S2.E4)\) givesKM=K0/N2K\_\{M\}=K\_\{0\}/N^\{2\}: the surrogate carries only the entrance contribution toκ4\\kappa\_\{4\}and therefore relaxes one power ofNNfaster than Eq\. \([26](https://arxiv.org/html/2609.01705#S4.E26)\)\. The kurtosis gap restores the difference exactly: the kick\-accumulation kernel hasK𝒯=K0/\(N−1\)K\_\{\\mathcal\{T\}\}=K\_\{0\}/\(N\{\-\}1\), for which Eq\. \([11](https://arxiv.org/html/2609.01705#S4.E11)\) evaluates toK0​\(N−1\)/N2K\_\{0\}\(N\{\-\}1\)/N^\{2\}, so thatKMK\_\{M\}plus the gap reproduces the physicalK0/NK\_\{0\}/N\. This component\-wise comparison is evaluated directly in Fig\.[4](https://arxiv.org/html/2609.01705#S2.F4)f\.

### 4\.9MHD\-grid turbulence \(periodic cube\)

We use a pre\-computed realization of a three\-dimensional solenoidal magnetic field from the Johns Hopkins Turbulence Databases\([Eyink et al\., 2013](https://arxiv.org/html/2609.01705#bib.bib41);[Aluie and others, 2019](https://arxiv.org/html/2609.01705#bib.bib40)\)on a uniform periodic grid of sizeN3N^\{3\}\(N=1024N=1024\)\. The vector field is then loaded intoCRPropawithin aMagneticFieldGrid\([Alves Batista et al\., 2022](https://arxiv.org/html/2609.01705#bib.bib39)\)\. The box size is set from the integral scale quoted for this dataset,ℓint=0\.217​Lbox\\ell\_\{\\rm int\}=0\.217\\,L\_\{\\rm box\}, so thatLbox=ℓc/0\.217=461​μL\_\{\\rm box\}=\\ell\_\{c\}/0\.217=461\\,\\mum forℓc=100​μ\\ell\_\{c\}=100\\,\\mum\. The field is rescaled so that the mean field magnitude⟨\|𝐁\|⟩\\langle\|\\mathbf\{B\}\|\\rangleequals the target value \(for this realizationBrms/⟨\|𝐁\|⟩=1\.17B\_\{\\rm rms\}/\\langle\|\\mathbf\{B\}\|\\rangle=1\.17\)\. Reference parameters are listed in Table[4](https://arxiv.org/html/2609.01705#S4.T4)\.

Table 4:Reference parameters for the turbulence benchmark and detector geometry\.Experimental values are taken from[Chen et al\. \(2020\)](https://arxiv.org/html/2609.01705#bib.bib25)where indicated\.
### 4\.10Swiss roll ablation andkk‑NN P/R analysis

We evaluate schedule choice on a 2\-D swiss roll manifold\. The score network is trained once withσ\\sigma\-conditioning \(the network learns score functions at many noise levelsσ\\sigmasimultaneously, without committing to a specific time\-to\-noise mapping\) usingσ∼LogNormal⁡\(μ=−1,σ=1\)\\sigma\\sim\\mathrm\{LogNormal\}\(\\mu=\-1,\\sigma=1\)and sinusoidal embeddings\. At sampling, we vary only the pathσ⁡\(τ\)\\sigma\(\\tau\)—Telegraph, Karras EDM, Linear, or Cosine—using identical Karras\-Heun ODE solvers, NFE, seeds, and noise bounds\([Karras et al\., 2022](https://arxiv.org/html/2609.01705#bib.bib20);[Nichol and Dhariwal, 2021](https://arxiv.org/html/2609.01705#bib.bib19)\)\.

For evaluation, we drawMMground\-truth points from the swiss roll andMMgenerated points \(sameMMfor all schedules\)\. We repeat sampling with five independent seeds and report mean±1​σ\\pm 1\\sigma\. For each neighborhood sizek∈\{1,…,50\}k\\in\\\{1,\\dots,50\\\}, we compute the nearest\-center precision/recall diagnostic defined below, adapted from[Kynkäänniemi et al\. \(2019\)](https://arxiv.org/html/2609.01705#bib.bib26)\.

Letrdata​\(xi\)r\_\{\\mathrm\{data\}\}\(x\_\{i\}\)be the distance from data pointxix\_\{i\}to itskk\-th nearest neighbor in the real set, and letdg→d​\(x^j\)d\_\{\\mathrm\{g\\to d\}\}\(\\hat\{x\}\_\{j\}\)be the distance from generated pointx^j\\hat\{x\}\_\{j\}to its nearest real neighbor\. Precision measures what fraction of generated samples lie within the k\-NN bubble of their nearest real neighbor:

Preck=1M∑j=1M\[dg→d\(x^j\)≤rdata\(NNdata\(x^j\)\)\]\.\\mathrm\{Prec\}\_\{k\}=\\frac\{1\}\{M\}\\sum\_\{j=1\}^\{M\}\\mathbf\{1\}\\\!\\left\[d\_\{\\mathrm\{g\\to d\}\}\(\\hat\{x\}\_\{j\}\)\\leq r\_\{\\mathrm\{data\}\}\\\!\\big\(\\mathrm\{NN\}\_\{\\mathrm\{data\}\}\(\\hat\{x\}\_\{j\}\)\\big\)\\right\]\.\(27\)High precision means generated samples stay close to the real manifold \(no spurious modes\)\. Similarly, letrgen​\(x^j\)r\_\{\\mathrm\{gen\}\}\(\\hat\{x\}\_\{j\}\)be the distance from generated pointx^j\\hat\{x\}\_\{j\}to itskk\-th nearest generated neighbor, anddd→g​\(xi\)d\_\{\\mathrm\{d\\to g\}\}\(x\_\{i\}\)the distance from real pointxix\_\{i\}to its nearest generated neighbor\. Recall measures what fraction of real samples have a generated neighbor within their k\-NN bubble:

Reck=1M∑i=1M\[dd→g\(xi\)≤rgen\(NNgen\(xi\)\)\]\.\\mathrm\{Rec\}\_\{k\}=\\frac\{1\}\{M\}\\sum\_\{i=1\}^\{M\}\\mathbf\{1\}\\\!\\left\[d\_\{\\mathrm\{d\\to g\}\}\(x\_\{i\}\)\\leq r\_\{\\mathrm\{gen\}\}\\\!\\big\(\\mathrm\{NN\}\_\{\\mathrm\{gen\}\}\(x\_\{i\}\)\\big\)\\right\]\.\(28\)High recall means the generator covers all regions of the real distribution \(no missing modes\)\. Smallkktests strict local fidelity \(sharpness\), while largekktests global structure \(coverage\)\. At very local scales \(smallkk\), EDM attains high recall quickly \(broad coverage\) but at the cost of precision \(lower fidelity\), consistent with early aggressive denoising\. Telegraph strikes a better local balance\. Across five independent sampling seeds, the telegraph path attains higher recall than cosine at smallkkat comparable precision and maintains higher precision than EDM at every neighborhood size, while cosine retains a modest precision lead at intermediatekk; telegraph and cosine both approach high precision and near\-unit recall fork≳40k\\\!\\gtrsim\\\!40\. Quantitatively, atk=5k\{=\}5, telegraph attains recall0\.830\.83vs\.0\.720\.72for cosine at comparable precision \(0\.430\.43vs\.0\.440\.44\); atk=20k\{=\}20, cosine leads in precision \(0\.910\.91vs\.0\.860\.86\)\. The linear schedule has precision<0\.16<0\.16fork≤20k\\\!\\leq\\\!20\.

## Data availability

The source data underlying all figures and tables, together with the pre\-computed simulation and ablation outputs generated in this study, have been deposited in Zenodo\([Reichherzer, 2026](https://arxiv.org/html/2609.01705#bib.bib36)\)\. The forced\-MHD turbulence field used in this study is available from the Johns Hopkins Turbulence Database under DOI[https://doi\.org/10\.7281/T1930RBS](https://doi.org/10.7281/T1930RBS)\. The experimental comparison data were taken from Ref\.[Chen et al\. \(2020\)](https://arxiv.org/html/2609.01705#bib.bib25)\.

## Code availability

The code used to generate the results and figures has been deposited in Zenodo\([Reichherzer, 2026](https://arxiv.org/html/2609.01705#bib.bib36)\)and is available under the MIT License\. Test\-particle simulations use the open\-sourceCRPropaframework\([Alves Batista et al\., 2022](https://arxiv.org/html/2609.01705#bib.bib39)\)\.

## Acknowledgments

We thank Archie F\. A\. Bott, Robert J\. Ewart, and Jonathan Citrin for valuable comments\.

## Funding

The work of P\.R\. was funded through the Walter Benjamin Fellowship by the Deutsche Forschungsgemeinschaft \(DFG, German Research Foundation\)—grant numbers 518672034 and 593157299; P\.R\. was also supported by a postdoc fellowship of the German Academic Exchange Service \(DAAD\); The work of G\.G\. was partially supported by UKRI under grants no\. ST/W000903/1 and EP/Y035038/1\.

## Author contributions statement

P\.R\. conceived the method, developed the theoretical framework, wrote all code, performed the simulations and analysis, and wrote the original manuscript\. D\.N\.H\. refined the theoretical formulation and substantially revised the manuscript’s narrative, structure, and presentation\. G\.G\. and S\.S\. provided scientific guidance and contributed to the interpretation of the laboratory validation through their expertise in the experimental data and associated uncertainties\. All authors discussed the results and reviewed and edited the manuscript\.

## Competing interests statement

The authors declare no competing interests\.

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