PaperlayerView exact revision on arXiv

Source-linked AI summary

Generative Modeling of Stochastic Dynamics for Long-Time Evolution

Yang-yang Tan, Jinyang Li, Lingxiao Wang

arXiv:2610.00546v1cond-mat.stat-mechcs.LGhep-lat

TL;DR

The paper addresses how to predict stochastic dynamics when the governing equation is unknown. It learns finite-time transition kernels from configuration pairs and iterates them, recovering long-time behavior in Model B and driven colloids. The results include agreement with direct simulations and experimental measurements beyond the training window.

  • Problem

    Time-resolved configurations are available for stochastic systems, but their effective equations of motion are often unknown.

    Source: p. 1

  • Method

    A conditional diffusion model learns a fixed-lag transition kernel from configuration pairs, which is iterated to predict later configurations.

    Source: p. 2

  • Results

    The learned kernels recover Model B critical scaling and t^1/3 coarsening, with effective z = 3.792(19) versus 3.812(20) for direct integration, and predict driven-colloid current and passage time within experimental uncertainty.

    Source: pp. 2, 4, 5

  • Takeaways & Limitations

    Short-time observations can support predictions of emergent stochastic dynamics far beyond the training lag, including on larger lattices and for initial ensembles absent from training.

    Source: pp. 2, 5

    The state must carry the variables needed for a Markov description; which slow variables are needed differs across dynamic universality classes.

    Source: pp. 2, 5

Abstract

from arXiv · show

Exact stochastic equations for non-equilibrium dynamics are rarely accessible. We show that the long-time evolution of stochastic dynamics can be predicted from configuration pairs at a fixed short time lag, without knowledge of the equation of motion. Generative diffusion models learn the finite-time transition kernel from these pairs, and iterating it propagates the dynamics far beyond the training lag. For two-dimensional Model B, the diffusive dynamics of a conserved order parameter, the learned kernels reproduce dynamic critical scaling and self-similar $t^{1/3}$ coarsening. Agreement with direct simulations persists on lattices twice the largest training size and for initial ensembles absent from training. For driven colloids in a periodic optical potential, ten minutes of measured trajectories suffice to predict the particle current and mean passage time over the next twenty minutes within experimental uncertainty. Short-time observations thus contain the information needed to predict emergent non-equilibrium dynamics at much longer times.

END MATTER

The learned kernels reproduce measured colloid motion beyond the training window, including driven current and passage-time behavior. The section also reports computational comparisons with stochastic solvers.

Computational cost

  • For the same ensembles, evolution time, and comparable accuracy, the learned kernel is faster than every public stochastic solver tested.The fused SOSRA implementation is faster in equilibrium and comparable in cost out of equilibrium; Table I covers 512 configurations on L = 64 from t = 0 to 100.Source: END MATTER · p. 9, Table I · p. 9

Colloid experiment

  • The figure compares DM and Langevin predictions with experimental data across mean position, displacement distributions, phase-dependent displacement rates, and passage times.Its upper and lower rows show undriven and driven motion, respectively; the dotted line marks the end of training.Source: Figure 5 · p. 9
  • Separate kernels trained on position pairs at δt = 10 s predict 50 min of undriven and 20 min of driven motion from positions measured at 10 min.The particles move in a periodic optical potential; translating the sample cell biases hopping and generates a non-equilibrium current.Source: Colloid experiment · p. 9
  • Under driving, predicted mean speed agrees with experiment within 1%, and the model closely reproduces measured displacement distributions and passage-time statistics.Its driven mean current and mean passage time agree more closely with experiment than the Langevin reference; overall accuracy is comparable.Source: Colloid experiment · p. 9, Figure 5 · p. 9

S1. FINITE-TIME COMPOSITION AND COLORED NOISE

Finite-time kernels compose when the state is complete and the dynamics are autonomous. Colored-noise dynamics can use the same construction by including variables that carry temporal correlations.

Finite-time composition

  • For a complete Markov state, autonomy makes the second update depend on the intermediate state rather than the time origin, yielding the Chapman–Kolmogorov composition identity.The identity follows by averaging successive updates over their driving histories.Source: S1. FINITE-TIME COMPOSITION AND… · p. 10
  • Repeated draws from the exact fixed-lag kernel reproduce the joint distribution of configurations at sampled times for any chosen initial ensemble.For learned kernels, errors can accumulate, so agreement over one interval does not guarantee accurate long-time dynamics.Source: S1. FINITE-TIME COMPOSITION AND… · p. 10

Colored noise

  • For colored noise, augment the state with variables carrying its temporal correlations; when the joint state is autonomous and Markovian, learn and iterate its kernel using paired states.Each update advances both the field and noise variables, preserving memory between physical intervals.Source: S1. FINITE-TIME COMPOSITION AND… · p. 10

S2. CONDITIONAL DIFFUSION CONSTRUCTION

The conditional diffusion construction learns the distribution of a finite-time increment from paired observations. It diffuses and then samples that increment while keeping the present configuration as a condition.

Training objective

  • Each observed pair uses the present state ϕ as the condition and its increment ∆ϕ as the target; coupled fields must include all variables needed for a Markov description.Increment prediction is convenient for slowly evolving fields, though the next state itself can also be predicted.Source: S2. CONDITIONAL DIFFUSION CONSTRUCTION · p. 10

Score model and reverse sampling

  • The model diffuses only ∆ϕ, supplies the unperturbed condition to the score network, and trains the score by denoising score matching.The objective averages over measured pairs, auxiliary time, and added Gaussian noise; physical dynamics enters through the pairs, not an explicit drift or noise covariance.Source: S2. CONDITIONAL DIFFUSION CONSTRUCTION · p. 10
  • One physical-time increment is sampled by integrating the conditional reverse-time SDE from auxiliary time s = 1 toward s = 0 at fixed ϕ.Sampling begins from a Gaussian with variance Σ2(1); the increment is added to the condition, then the condition is updated for the next physical step.Source: S2. CONDITIONAL DIFFUSION CONSTRUCTION · p. 10

S3. NUMERICAL DETAILS OF THE MODEL B CASE STUDY

The numerical details specify the conserved lattice dynamics, how training pairs are generated, and how the conditional score model samples increments while enforcing conservation.

A. Lattice realization

  • Periodic or no-flux boundaries make net boundary flux vanish, so integrating the continuity equation conserves the spatial mean in every realization.The implementation uses nearest-neighbor current differences and sets λ = T = γ = 1 in the Model B calculations.Source: A. Lattice realization · pp. 10–11

B. Training pairs

  • Training pairs use synthetic Gaussian fields for Fig. 2, canonical ensembles at L = 8, 16, 32 for Figs. 3 and S5, and synthetic domain fields for Fig. 4.The Fig. 4 fields span L = 8, 16, 32, 64; their target increments follow evolution over δt = 10, and pairs use both signs because the dynamics is symmetric under ϕ → −ϕ.Source: B. Training pairs · pp. 11–12

C. Score model and reverse sampling

  • The score network uses periodic convolutions, and reverse sampling uses 2000 Euler–Maruyama steps with independently drawn noise for successive physical intervals.Conservation is imposed during sampling by projecting the residual, score, and noise onto the zero-mean subspace; the generated mean is preserved up to floating-point roundoff.Source: C. Score model and reverse sampling · p. 12

A. Initial ensembles and statistical errors

The simulations compare learned-kernel and SOSRA dynamics from matched equilibrium and Gaussian initial fields, then estimate relaxation times and the dynamic critical exponent across two time grids. Bootstrap uncertainty is calculated by resampling whole trajectories while preserving the grids’ statistical correlation.

A. Initial ensembles and statistical errors

  • 1024 equilibrium and 1024 Gaussian initial fields at L = 32 share matched conserved zero modes, and DM and SOSRA use independent noise.For Fig. 3, two learned kernels start from the same 2048 equilibrium fields; the δt = 0.5 and 10 kernels are iterated to t = 50 and 1000, respectively.Source: A. Initial ensembles and statistical… · p. 12

B. Relaxation times and dynamic critical exponent

  • The effective-decay-rate figure compares the measured rate with a separate large-x tail fit, the logarithmic derivative of Eq. (11), and reference rates for exponential decay and short- and long-time behavior.Error bars and parenthesized z uncertainties in the figure are trajectory-bootstrap standard errors.Source: Figure S3 · p. 13

A. Initial ensembles and statistical errors

B. Relaxation times and dynamic critical exponent

  • Relaxation times are obtained from the ρn decay, using δt = 0.5 through t = 50 and δt = 10 through t = 1000.The e^-1 crossing is linearly interpolated between neighboring grid points; τn ∝ k̂_n^-z is fitted using n = 1–4 at L = 32 and n = 2–8 at L = 64, excluding the unresolved slowest mode.Source: B. Relaxation times and dynamic… · p. 13

C. Dynamic scaling function

The dynamic-scaling analysis jointly fits the resolved modes and compares the inferred decay behavior with rates constrained by equilibrium fluctuations. The measured profile shows an approximate crossover, with fitted parameters depending on mode and fitting window.

C. Dynamic scaling function

  • The proposed scaling form permits an algebraic correction to an exponential tail, f(x) ∼ (ax)^-b e^-λ0x at large x.Earlier studies cited in the passage describe near-Lorentzian critical spectra, approximate Kawasaki-dynamics eigenmodes, and exponential fits in three-dimensional Model B.Source: C. Dynamic scaling function · p. 13
  • Fits with all three parameters free yield initial rates within a few percent of the rate r fixed by static fluctuations for both dynamics.This consistency with the static fluctuations was not imposed during training.Source: C. Dynamic scaling function · p. 14
  • The effective decay rate shows an approximate crossover over the measured range, with fit parameters that depend on mode and fitting window.The comparison includes a joint fit constrained by the short-time sum rule and a separate large-x fit of λ0 + 1/(2x) for x ≥ 2.Source: C. Dynamic scaling function · pp. 13–14

D. Coarsening structure factor

The coarsening structure factor collapses under domain-size rescaling and exhibits a Porod tail, with deviations at box-scale wavelengths. Additional tests examine transfer to larger lattices and whether a faster reverse integrator preserves the learned relaxation spectrum.

D. Coarsening structure factor

  • The structure-factor tail is compared with k̂^-3 because the sign-field walls are lattice steps, giving a Porod tail from inverse domain size to the lattice cutoff.Against continuum k, the tail bends upward near the zone boundary and different times separate there.Source: D. Coarsening structure factor · p. 14
  • The rescaled structure factors collapse onto one curve with a k̂^-3 tail, except for the lowest shells where wavelengths approach the box size.The Porod tail is expected for sharp domain walls; its collapsing amplitude is nontrivial because it requires wall length per unit area to scale as 1/L(t).Source: D. Coarsening structure factor · pp. 14–15

S5. TRAINING ON MULTIPLE SIZES AND TRANSFER TESTS

  • The kernels used for critical scaling at L = 8, 16, 32 are applied at L = 64 without changing their weights, and the reverse-integrator comparison uses 32 NFE per physical transition.DPM-Solver-v3 integrates the probability-flow ODE with solver coefficients estimated from the original L = 32 training pairs; no L = 64 test field enters that calculation.Source: S5. TRAINING ON MULTIPLE SIZES AND… · p. 15, A. Relaxation and reverse sampling · p. 15

A. Relaxation and reverse sampling

  • For the slowest resolved L = 64 mode, τ2 is 213.83 ± 7.96 for SOSRA, 226.82 ± 7.61 for EM, and 221.28 ± 7.29 for DPM-Solver-v3.Across modes n = 2–8, fitted z values are 3.867 ± 0.023, 3.855 ± 0.021, and 3.866 ± 0.020, respectively; the exponents agree within bootstrap errors, and 32 NFE produces no statistically resolved relaxation-spectrum change.Source: A. Relaxation and reverse sampling · pp. 15–16

B. Correlations at the training size and one-step transfer

The supplementary comparisons specify how learned-kernel correlations are evaluated at the training lattice and at an unseen lattice, then describe the computational-cost benchmark conditions.

B. Correlations at the training size and one-step transfer

  • At L = 32, equilibrium correlations compare SOSRA with learned kernels trained at L = 32 or on L = 8, 16, 32; low modes use δt = 10 and faster modes use δt = 0.5.For the faster modes, the comparison uses the mixed-size training data.Source: B. Correlations at the training size… · p. 16
  • The one-step transfer comparison evaluates correlations at L = 32 and L = 64, with the L = 64 initial fields excluded from training.The L = 32 comparison uses 512 initial fields from the training set; both datasets use canonical fields, unlike the Gaussian training fields used for Fig. 2.Source: B. Correlations at the training size… · p. 16

S6. COMPUTATIONAL COST

  • The cost benchmark times evolution of 512 configurations on an L = 64 lattice from t = 0 to 100 using one NVIDIA GH200.Each time is a single synchronized measurement after warmup; loading and compilation are excluded, and the direct solvers use single precision.Source: S6. COMPUTATIONAL COST · p. 16

A. Equilibrium benchmark

The equilibrium benchmark fixes the initial fields and compares methods at t = 100 using calibrated step sizes. Reported observable shifts and runtimes distinguish the tested implementations.

A. Equilibrium benchmark

  • The equilibrium column starts every method from the same equilibrium initial fields at κ = 0.34034 and λ = 1.The learned kernel uses δt = 10 and canonical training pairs, sampled with DPM-Solver-v3 at 32 NFE per transition.Source: A. Equilibrium benchmark · p. 16
  • DPM-Solver-v3 advances ten physical intervals; SOSRA, ShARK, and Diffrax EM use step sizes of 0.00625, 0.004, and 0.0004, respectively.The step sizes are chosen to give comparable action-density accuracy after one physical interval.Source: A. Equilibrium benchmark · p. 16
  • At t = 10, five methods show observable shifts of +0.719%, −0.689%, −0.825%, −0.733%, and +0.775%; completed other EM runs exceed 100 s at t = 100.The five shifts are listed in Table I order; all rows use one NVIDIA GH200.Source: A. Equilibrium benchmark · p. 16, Table S1 · p. 17

B. non-equilibrium benchmark

The non-equilibrium benchmark compares accuracy and wall time against a fused SOSRA reference at t = 100. Under the stated accuracy criteria, DPM-Solver-v3 is faster than the public implementations tested, while fused SOSRA is faster in equilibrium and comparable out of equilibrium.

B. non-equilibrium benchmark

  • The non-equilibrium benchmark uses 512 disordered L = 64 initial fields and a fused SOSRA calculation with h = 0.0001 as the reference at t = 100.Step calibration uses the action density and spatial moments.Source: B. non-equilibrium benchmark · p. 17
  • DPM-Solver-v3 gives ϵ = 0.128%; SOSRA, ShARK, and Diffrax EM give errors from 0.081% to 0.125% with calibrated steps.The individual signed shifts and resulting values are listed in Table S1.Source: B. non-equilibrium benchmark · p. 17
  • StochasticDiffEq.jl and torchsde EM take 256.44 s and 356.89 s, respectively, producing the > 200 s entry in Table I.Their steps use the value calibrated with Diffrax.Source: B. non-equilibrium benchmark · p. 17
  • 3.24 s in equilibrium and 3.26 s out of equilibrium: DPM-Solver-v3 is below the public implementations tested; fused SOSRA takes 1.30 s and 3.28 s, respectively.Fused SOSRA is faster in equilibrium and comparable out of equilibrium.Source: B. non-equilibrium benchmark · p. 17

S7. COLLOID EXPERIMENT

The colloid experiment tests whether kernels learned from short recordings predict later transport. The study describes the experimental system, the learned displacement model, its Langevin reference, and the analysis used to compare predictions with observations.

S7. COLLOID EXPERIMENT

  • After training on the first 10 min, the driven model predicts mean speed and mean passage time over the next 20 min within 1% of measurements; the undriven prediction covers 50 min.Separate diffusion models are trained for undriven and driven motion.Source: S7. COLLOID EXPERIMENT · p. 17

A. Experimental system and observations

  • The quasi-two-dimensional cell has a periodic optical potential; recordings last 60 min without driving and 30 min under driving, with about 1200 complete tracks in each case.The driven cell speed is v0 = 0.053 µm/s, and reflected recordings make mean driven motion point toward negative x.Source: A. Experimental system and observations · p. 17
  • A 10 s training lag resolves within-well relaxation, which takes about 10 s, while undriven escape takes about 10 min and requires many successive steps to predict.Thermal motion explores wells about 3 kBT deep; driving biases barrier crossings and produces a net current.Source: A. Experimental system and observations · p. 17

B. Dynamics learned from short trajectories

  • Each model learns the two-component displacement conditioned on periodic phase, without receiving the potential, diffusion coefficient, or driving force.Both displacement components are learned from measured pairs.Source: B. Dynamics learned from short… · p. 17
  • The later transverse drift in the driven data is absent from both models.Figure S8 marks the training-window end and shows the later drift in the driven data.Source: Figure S8 · p. 18

C. Langevin reference

  • The Langevin reference assumes isotropic white noise and an unchanged periodic potential; it infers the potential from the undriven training density.Its diffusion coefficient is D = 0.04 µm2/s, and its drive term is zero without driving or −v0 under driving.Source: C. Langevin reference · p. 18

D. Observables and uncertainty

  • Data and model trajectories are analyzed on a common 10 s grid, with transport quantities defined over the observation window.Passage times are detected on the sampled grid, so brief excursions may be missed and crossing detection delayed.Source: D. Observables and uncertainty · p. 19
  • Transition paths run between milestones 0.5 and 3.0 µm from the barrier top; returns within one sampling interval are unresolved, lengthening measured path times.Paths are classified by whether they pass the well bottom or barrier top.Source: D. Observables and uncertainty · p. 19

E. Transport and transition paths

The learned dynamics capture key features of colloid transport and driven passage times, while discrepancies and later changes in the recordings mark limits of the predictions. Route-specific path statistics clarify the directional asymmetry under driving.

E. Transport and transition paths

  • The displacement distributions retain peaks separated by the potential period as they broaden, and under driving their envelope moves with particle current.The diffusion model reproduces localization within wells and transport across many periods.Source: E. Transport and transition paths · pp. 19–20
  • Without imposed driving, a weak current persists and the learned model captures its bias, whereas the equilibrium Langevin reference has zero mean current.The measured drift decreases during recording, and the learned model’s mean position eventually exceeds the data; the model also captures directional path asymmetry.Source: E. Transport and transition paths · p. 20
  • Both models predict a mean exit time of about 576 s versus 610(15) s in the data, and an effective diffusion coefficient about 7% too large; the discrepancy’s origin is unresolved.Single-step hopping probabilities and short-time diffusion agree more closely.Source: E. Transport and transition paths · p. 20
  • −0.0284 µm/s and 166.5 s: the driven model predicts mean velocity and passage time, compared with measured −0.0282(3) µm/s and 168(4) s; both agree within 1%.The Langevin reference gives −0.0298 µm/s and 157.3 s, differing by about 6%.Source: E. Transport and transition paths · p. 20
  • About 99% of A →B paths pass the well bottom, versus 2% of B →A paths; mean path times differ by about a factor of five.Once separated by route, forward and backward time distributions are similar in the data and both models; route weights account for the combined-distribution asymmetry.Source: E. Transport and transition paths · p. 20
  • Both models miss the later transverse drift that appears only after the training window; their fixed parameters do not describe changes in experimental conditions.Both models capture increased dispersion under driving, but their effective diffusion coefficients remain below the measured value.Source: E. Transport and transition paths · pp. 20–21
  • Halving the training lag to 5 s leaves the overall comparison similar; predicted driven mean speed changes by less than 1%.Passage-time deviations remain comparable when data and models are evaluated at the same resolution; single-step displacement width and hopping probability match within 2%.Source: E. Transport and transition paths · p. 21

Related papers

Loading 2610.00546v1…