Generative Diffusion Surrogates with Analytical Variance Schedule

Patrick Reichherzer; Gianluca Gregori; David N. Hosking; Subir Sarkar

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Generative Diffusion Surrogates with Analytical Variance Schedule

Patrick Reichherzer, Gianluca Gregori, David N. Hosking, Subir Sarkar

arXiv:2609.01705v1Sep 1, 2026cs.LGastro-ph.IMphysics.plasm-ph

TL;DR

Stochastic transport needs probabilistic, time-resolved surrogates that capture non-Gaussian structure, but diffusion-model schedules are usually heuristic and lack a physical clock. The paper anchors the forward schedule to a known transport variance law, leaving the score to model residual shape; the resulting surrogate matches transport distributions, variance scaling, and kurtosis evolution while supporting calibrated emulation and likelihood-based inference.

Problem
Diffusion-model noise schedules are usually heuristic, whereas stochastic transport often provides a known variance law without a known full distribution.
Method
The method derives the VE diffusion schedule from the macroscopic variance path, analytically fixing second-moment evolution while learning remaining non-Gaussian structure from entrance data.
Results
The surrogate preserves the variance path and kurtosis relaxation, matches positive-kurtosis test-particle marginals in the studied regime, and uses the anchored schedule across samplers.
Takeaways & Limitations
A known variance-growth law can calibrate a generative model’s physical clock for transport emulation and differentiable likelihood-based inference.
Takeaways & Limitations
Matching the variance law does not reproduce microscopic transport mechanisms such as finite propagation speed, bounded support, memory, or trajectory correlations.

Abstract

from arXiv · show

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.

1 Introduction

The paper adapts generative diffusion models to stochastic transport by anchoring their noise schedule to a known macroscopic variance law. This calibrates generative time to physical time while leaving the learned score to represent higher-order, non-Gaussian structure.

1 Introduction: Generative diffusion models reverse Gaussian corruption and can model stochastic transport driven by unresolved random increments.The paper applies this machinery to distributions affected by turbulent fields, scattering, heterogeneities, collisions, or random forcing.
1 Introduction: The proposed schedule derives the instantaneous noising rate from the derivative of the accumulated macroscopic variance.In the VE-SDE formulation, a nondecreasing variance law directly determines the forward noising rate.
1 Introduction: Transport variance, often available from theory or empirical scaling when the full distribution is unknown, provides a physical basis for calibrating diffusion time.Examples include fractional kinetics, intracellular subdiffusion, Richardson’s law, and charged-particle transport in stochastic magnetic fields.
1 Introduction: Anchoring the schedule prescribes variance evolution analytically while the reverse score models remaining distributional structure, including non-Gaussian features inherited from entrance data.The construction calibrates ensemble spreading without imposing microscopic properties such as finite propagation speed, memory, or trajectory correlations.
1 Introduction: Entrance-only training uses initial samples and synthetic corruptions along the prescribed variance path, avoiding physical transport data at intermediate times.The approach preserves standard diffusion-model infrastructure while enabling analytic moment tracking.
1 Introduction: The framework is designed for physical-time-indexed surrogates suitable for calibrated scientific emulation and likelihood-based inference.The paper applies the construction to ballistic-to-diffusive and heavy-tailed or intermittent transport.

2 Results

The framework uses a prescribed macroscopic variance path to calibrate diffusion-model time while learning higher-order distributional structure from entrance data. Experiments show that anchored schedules reproduce the intended marginal evolution, while Gaussian-kernel limitations emerge when physical transport has distinct higher cumulants.

2.1 Central Limit Theorem motivates physics-anchored Gaussian surrogates: Non-Gaussian entrance structure is retained in the model marginals and diluted deterministically as prescribed variance grows.The model’s residual physical mismatch is quantified through the kurtosis gap when higher cumulants differ.
2.1 Central Limit Theorem motivates physics-anchored Gaussian surrogates: The VE process approximates post-entrance transport with a Gaussian kernel whose variance increment follows the macroscopic transport law.For centered isotropic transport, this is the moment-matched Gaussian surrogate within the centered isotropic Gaussian family.
2.2 Variance laws as noise schedules: The anchored VE schedule realizes the ballistic-to-diffusive variance crossover by construction, providing the path tested against analytical telegraph predictions.Figure 2 compares sampled accumulated variance with the analytical telegraph law under controlled training budgets and reverse samplers.
2.2 Variance laws as noise schedules: Anchored PF-ODE and reverse-SDE models reproduce the intended Gaussian-convolution marginal path, although kurtosis relaxation is not a training target.This tests whether score learning and reverse sampling recover higher-order evolution implied by the anchored variance path.
2.2 Variance laws as noise schedules: Variance-matched Gaussian VE and Kac/telegraph kernels can share mean-square displacement while differing in propagators, memory, support, and higher moments.The Kac kernel has bounded support and can transiently exhibit negative excess kurtosis, producing a theory-curve gap from the Gaussian surrogate.
2.2 Variance laws as noise schedules: Table comparisons show the telegraph PF-ODE and reverse-SDE samplers have the smallest deviations from the anchored VE reference.For Kac flow, small variance deviation confirms variance matching, while larger kurtosis deviation reflects finite-speed higher-cumulant structure.

3 Discussion

The framework calibrates diffusion-model time to macroscopic variance, leaving the score network to represent residual distributional structure. In turbulent-plasma validation, it reproduces laboratory variance scaling and simulated kurtosis evolution, while supporting flexible sampling and inference.

3 Discussion: The variance path is enforced analytically, while the score network represents remaining non-Gaussian structure and enables physics-timed reverse evolution.Training and sampling share the same variance-anchored marginals, so stopping reverse integration can undo only late-time noise.
3 Discussion: The surrogate captures the laboratory-measured variance scale and simulated component-kurtosis evolution without schedule tuning.Figure 4 compares RMS transverse-speed variance against physical path length and excess kurtosis of the signed transverse component.
3 Discussion: The Gaussian surrogate accurately describes diagnostics that average many independent contributions, while the kurtosis gap quantifies non-Gaussian physical-kernel effects.For Student-t entrance distributions matched in variance and kurtosis, the VE model reproduces predicted entrance-kurtosis relaxation.
3 Discussion: The Gaussian-kernel surrogate is favored over the Kac/telegraph propagator because test-particle marginals show positive excess kurtosis.The observed non-Gaussianity can be represented through entrance statistics rather than a bounded-support transport kernel.
3 Discussion: The scalar variance path is a drop-in schedule for VE samplers and gives a competitive precision–recall balance on the Swiss-roll benchmark.The benchmark varies only the sampling path while holding the trained model, solver, NFE, seeds, and noise bounds fixed; a theoretical explanation remains open.
3 Discussion: The learned score field provides differentiable PF-ODE likelihood estimates that can in principle support gradient-based inference, although joint parameter identifiability needs additional data or constraints.The stated extension includes parameters entering the schedule or conditioning the score model.
3 Discussion: The approach targets data-efficient distributional snapshots rather than trajectory data or interpolated pairs at multiple intermediate times.Its application scope includes anomalous diffusion, rough volatility, intracellular transport, and other systems with known variance growth.

4 Methods

The methods derive a VE diffusion schedule from a prescribed variance law and analyze the resulting Gaussian-kernel approximation. They also formulate Kac-flow comparisons and track higher-order statistics through excess-kurtosis evolution.

4 Methods: The instantaneous noising rate is set to the derivative of the accumulated macroscopic variance, directly calibrating the diffusion-model clock.For the VE formulation, this construction ensures the accumulated noise has the prescribed variance.
4 Methods: The score network learns the global score of each noisy marginal, with adjacent diffusion times coupled through shared parameters.The practical training objective uses a variance-weighted form of the score-matching loss.
4 Methods: The moment-matched centered isotropic Gaussian uniquely minimizes cross-entropy against the physical kernel and, under stated regularity conditions, forward KL.After convolution with the entrance distribution, the approximation error is controlled by W2 contraction.
4 Methods: Model excess kurtosis follows a closed-form evolution, while the physical-kernel kurtosis gap vanishes as the kernel Gaussianizes and is suppressed when ˜σ2 ≪σ2.The kernel excess kurtosis is defined separately from marginal kurtosis to quantify transport-kernel non-Gaussianity.
4 Methods: The Kac baseline models constant-speed one-dimensional motion with Poisson direction reversals and is learned using conditional flow matching.Its displacement kernel has bounded support, |x| ≤ ct.
4 Methods: The Kac kernel excess kurtosis equals −2 as t →0 and relaxes toward zero from below, with KT ≃ −3/(at) for at ≫1.These kernel statistics enter the marginal-kurtosis comparison through cumulant additivity.
4 Methods: Kac parameters are fixed rather than fitted by matching the anchored telegraph variance path, yielding a = 1/(2τc) and c2 = C−1/(2τc).This makes the baseline variance-matched to the VE-telegraph schedule by construction.

4.5 Diffusion model implementation and baseline comparisons

The implementation changes only the VE scalar noise schedule, deriving it from the transport variance while retaining standard diffusion-model components and enabling entrance-only training and calibrated sampling.

Model choice: The VE formulation is selected because transport variance grows with time, whereas VP models constrain variance to unity through mean-reverting drift.The schedule satisfies dx_τ = ˜g(τ)dW_τ with ˜g²(τ) = d˜σ²(τ)/dτ.
Training: Training uses ˜g²-weighted denoising score matching on entrance samples and synthetic corruptions generated along the prescribed variance path.The weighting emphasizes time steps where the schedule is steepest.
Sampling: At inference, terminal states are drawn from the exact VE forward marginal, then the probability-flow ODE is integrated backward to the desired readout time.The implementation uses an Euler solver with T = 500 steps and quantifies uncertainty over 10 independent seeds.
Model choice: Only the scalar schedule ˜g²(τ) changes relative to a standard VE model; the architecture, optimizer, and sampler remain unchanged.The optional FiLM embedding of entrance kurtosis K0 allows one model to serve multiple entrance distributions.
Likelihood: The probability-flow ODE also provides a differentiable log-likelihood estimate, becoming exact when the score and terminal density are specified correctly.Divergence is estimated with Hutchinson’s trace estimator, and schedule or entrance parameters can be optimized through likelihood.

4.7 Charged-particle transport through magnetized turbulence

Charged-particle transport is modeled through acceleration correlations with finite correlation time, producing ballistic short-time spreading and diffusive long-time spreading linked by transport coefficients.

Transport scaling: Normalized time is τ = t/t_obs, with correlation time τ_c = t_c/t_obs and t_c = ℓ_c/v.The correlation scale sets the transition between short- and long-time transport regimes.
Transport scaling: The transverse velocity variance is a double integral of the acceleration autocorrelation, linking microscopic turbulence to particle spreading.The autocorrelation enters through the factor (t − Δt) over the integration interval.
Transport scaling: For t ≪ t_c, nearly constant acceleration correlation yields ballistic scaling σ²_v⊥ ≈ ⟨a²⟩t².This regime reflects finite-speed motion before decorrelation becomes important.
Transport scaling: For t ≫ t_c, the correlation integral saturates and produces diffusive growth proportional to t.The long-time coefficient is represented by D_v, with κ absorbing the remaining acceleration dependence.
Transport coefficients: The diffusion coefficient κ can be inferred either from the long-time displacement-variance slope or from measured D_v through small-angle scattering.These provide two equivalent routes to the same transport coefficient.

4.8 Kurtosis inheritance and relaxation in turbulent transport

Turbulent scattering preserves entrance kurtosis for an individual kick but relaxes accumulated transport toward Gaussianity, with a distinct correction required for the surrogate’s kurtosis evolution.

Kurtosis inheritance: In the high-rigidity limit, particles experience approximately independent scattering events as they traverse turbulent cells.Each signed transverse kick is proportional to the path-averaged transverse magnetic field.
Kurtosis inheritance: The linear kick–field relationship preserves kurtosis at one correlation time: K(Δv_⊥(τ_c)) = K(B_⊥).Thus non-Gaussian field statistics can enter the transport through the kick distribution.
Kurtosis relaxation: After N approximately independent kicks, the fourth cumulant and variance are additive, while excess kurtosis is normalized by variance squared.This scaling produces Gaussianization as the number of decorrelated kicks increases.
Scope: For ℓ ≲ ℓ_c, finite-speed effects can make the true kernel non-Gaussian, so the VE model acts as a variance-correct Gaussian surrogate with quantified kernel mismatch.The 1/N physical decay applies after transport enters the decorrelated regime.
Kurtosis relaxation: The physical kick-accumulation kernel follows K_T = K_0/(N−1), whereas the surrogate’s entrance contribution alone follows K_M = K_0/N².The surrogate therefore relaxes one power of N faster before the kurtosis gap is included.

4.9 MHD-grid turbulence (periodic cube)

The turbulence experiments use a periodic three-dimensional magnetic-field realization and compare schedule paths under controlled sampling conditions using nearest-neighbor precision and recall.

MHD-grid turbulence: The magnetic field is a 3-D solenoidal realization from the Johns Hopkins Turbulence Databases on a uniform periodic grid with N = 1024.The field is loaded into CRPropa and rescaled to the target mean magnitude.
Schedule comparison: Schedule choice is evaluated by varying only σ(τ) among Telegraph, Karras EDM, Linear, and Cosine paths with identical solver, NFE, seeds, and noise bounds.The score network is trained once with σ-conditioning rather than a fixed time-to-noise mapping.
Evaluation: Precision measures generated samples inside real-data k-NN neighborhoods, while recall measures real samples with generated neighbors inside generated-data k-NN neighborhoods.Small k emphasizes local fidelity and large k emphasizes global coverage.
Results: At k = 5, Telegraph reaches recall 0.83 versus 0.72 for Cosine at comparable precision, 0.43 versus 0.44.At k = 20, Cosine leads precision with 0.91 versus 0.86 for Telegraph; Linear has precision < 0.16 for k ≤ 20.

Data availability

The study makes its figure, table, simulation, and ablation data available through Zenodo, while also identifying external turbulence and experimental data sources.

Source data underlying all figures and tables, along with pre-computed simulation and ablation outputs, were deposited in Zenodo.
The forced-MHD turbulence field used in the study is available from the Johns Hopkins Turbulence Database.The cited DOI is https://doi.org/10.7281/T1930RBS.
Experimental comparison data were taken from Ref..

Code availability

The study’s result-generation code is deposited in Zenodo, distributed under the MIT License, and uses the open-source CRPropa framework for test-particle simulations.

The code used to generate the results and figures has been deposited in Zenodo.
The deposited code is available under the MIT License.
Test-particle simulations use the open-source CRPropa framework.

Funding

The authors report fellowship and institutional grant support from the DFG, DAAD, and UKRI.

P.R. received Walter Benjamin Fellowship funding from the DFG under grant numbers 518672034 and 593157299.
P.R. was also supported by a postdoc fellowship from the German Academic Exchange Service.
G.G. received partial support from UKRI under grants ST/W000903/1 and EP/Y035038/1.

Author contributions statement

P.R. led the method, theory, coding, simulations, analysis, and manuscript, while the other authors refined the formulation, guided the science, and supported interpretation of laboratory validation.

P.R. conceived the method, developed the theoretical framework, wrote the code, performed 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 interpreting laboratory validation data and associated uncertainties.