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Memory-Conditioned Diffusion Model for Generalized Langevin Dynamics

Minglei Yang, Sicheng He

arXiv:2609.28371v1stat.MLmath-phmath.MP

TL;DR

Learning non-Markovian dynamics from resolved trajectories requires representing predictive history and conditional next-step variability. This paper combines a recursive multiscale memory bank with conditional diffusion sampling and a distilled neural flow map; its examples demonstrate long-memory retention, predictive compression, and non-Gaussian intermittent-event statistics.

  • Problem

    Resolved variables can depend on unresolved past dynamics, making next-step laws history-dependent rather than determined by the current observation alone.

    Source: §1 · p. 1, §2 · p. 4

  • Method

    A recursive bank of exponential filters conditions a training-free diffusion sampler whose generated conditional samples train an autoregressive neural flow map.

    Source: §1 · p. 3, §3.3 · p. 8

  • Results

    The three examples demonstrate slow-correlation retention, compressed prediction of coupled vector dynamics, and representation of conditional asymmetry and intermittent-event statistics.

    Source: §5 · p. 20

  • Takeaways & Limitations

    The results illustrate complementary roles for compact history representation and expressive conditional distributions in learning non-Markovian dynamics.

    Source: §5 · p. 20

    The method assumes stationary, ergodic observations and fading memory approximable by a modest filter bank; mean-based compression need not preserve non-Gaussian conditional laws.

    Source: §5 · p. 20

Abstract

from arXiv · show

Generalized Langevin equations describe non-Markovian dynamics in which the evolution of resolved variables depends on their past. We propose a memory-conditioned diffusion method for learning stochastic flow maps of these dynamics from observed trajectories, without identifying a memory kernel or reconstructing unresolved variables. A compact, recursively updated bank of exponential filters enables the flow map to retain predictive history over multiple time scales without conditioning on long observation windows. The next-step distribution is conditioned on the current observation and this memory state, whose storage and update costs are independent of the history length for a fixed bank size. Predictive criteria guide the memory budget, with reference-assisted selection in the vector benchmark, and an optional linear projection further reduces the conditioning dimension. A kernel-based score estimator generates conditional samples without training a score network, and these samples are used to train a neural flow map for autoregressive simulation. Three numerical examples assess long-memory retention at small conditioning dimension, predictive compression in coupled vector dynamics, and non-Gaussian conditional distributions and intermittent events. The non-Gaussian example reproduces conditional asymmetry and burst statistics in a stochastic model of the plasma scrape-off layer.

1. Introduction

The paper addresses learning non-Markovian dynamics from resolved trajectories, where history affects future evolution and the latent state or conditional law may be unknown. It introduces a compact memory-conditioned generative flow-map approach and evaluates it across scalar, vector, and scrape-off-layer examples.

1. Introduction

  • Learning from resolved trajectories must retain predictive history, represent the full conditional distribution, and remain stable under autoregressive rollout.When latent state and a closed reduced equation are unavailable, the conditional distribution must be represented from data.Source: §1 · p. 2
  • Explicit reduced-model approaches can yield interpretable equations but require choosing structures for memory and fluctuating terms.Examples include reconstructed memory kernels, correlation-based inference, hidden auxiliary variables, and trajectory-based closures.Source: §1 · p. 2
  • History-window and recurrent approaches represent memory, while stochastic flow maps model unresolved randomness; their history representations and output families pose separate questions.Training-free conditional diffusion can generate labeled samples, but conditioning only on the instantaneous state pools histories with different next-step laws.Source: §1 · p. 2
  • A compact, recursively updated filter bank conditions generative flow maps on long-time history without requiring long observation windows.Predictive criteria select memory scales and size; optional linear reduction compresses the conditioning variable, and fixed-bank storage and update costs do not grow with history length.Source: §1 · p. 3
  • The numerical examples test long-memory retention, predictive compression in coupled vector dynamics, and non-Gaussian conditional distributions with intermittent events.Comparisons include memoryless, raw-history, and recurrent models.Source: §1 · p. 3

2. Problem setting

The paper learns history-conditioned next-step laws from resolved observations without identifying the memory kernel or reconstructing unresolved variables. Its setting assumes stationary, ergodic observations and fading predictive memory, while noting that fading memory alone does not ensure an adequate finite summary.

2. Problem setting

  • The method seeks to reproduce observed dynamics without identifying the memory kernel or reconstructing the unresolved bath state.The GLE formulation includes instantaneous drag, convolution memory, and colored and white-noise forcing; its colored force is Gaussian, unlike the SOL pulse process.Source: §2 · pp. 3–4
  • The learning target is the observed process’s history-conditioned next-step distribution, with no parametric form assumed.A model conditioned only on the current state averages over histories that may imply different predictive laws.Source: §2 · p. 4
  • A recursively computable history summary conditions the generative flow map, which repeatedly samples next states and updates memory.The observations alone enter conditional estimation and flow-map training; reference statistics may be used for validation and explicitly identified memory selection.Source: §2 · pp. 4–5
  • Learning assumes stationarity and ergodicity for estimating a common conditional law, plus fading memory so remote history has diminishing predictive influence.Fading memory motivates a finite history summary but does not guarantee that it accurately determines the next-step distribution.Source: §2 · p. 5

3. Memory-conditioned diffusion and flow-map learning

The method summarizes observed history with recursive exponential filters, estimates conditional samples by diffusion, and distills them into a neural flow map for autoregressive simulation. Predictive selection and optional compression manage the conditioning state, while fixed-bank memory updates are independent of trajectory length.

3. Memory-conditioned diffusion and flow-map learning

  • Training-free conditional diffusion is extended to use a recursively updated history summary alongside the current observation.The score estimator generates conditional samples from trajectory data, and a neural flow map is trained to reproduce them.Source: §3 · p. 5

3.1. A recursive memory model

  • Exponential moving averages of observed increments form a multiscale memory bank, with slow filters retaining longer histories and updates costing O(kd).Rates and bank size are selected using held-out one-step prediction risk and closed-loop autocovariance error; optional linear projection retains predictive memory directions.Source: §3.1 · p. 6

3.2. Conditional sampling by diffusion

  • The conditional diffusion sampler estimates scores from neighboring trajectory pairs and reverses a probability-flow ODE to generate increments conditioned on state and memory.The kernel weights combine condition similarity and compatibility with the diffused state; the memoryless baseline conditions only on the instantaneous state.Source: §3.2 · p. 7

3.3. Neural flow-map training and simulation

  • Diffusion-generated labels train a neural flow map from conditioning states and Gaussian inputs, enabling stochastic increments at the same condition.The map is deployed autoregressively, updating the full memory bank from generated increments; per-step cost is one network evaluation plus O(kd), independent of trajectory length.Source: §3.3 · p. 8

4. Numerical examples

Three numerical examples test memory retention, predictive compression in coupled vector dynamics, and non-Gaussian conditional prediction with intermittent events. The benchmarks evaluate both conditional forecasts and long autoregressive rollouts.

4.1. Linear GLE with multiscale memory

  • The six-filter bank has substantially lower autocovariance error than a six-step raw window, while five predictive coordinates add no measurable loss over tested horizons.The predictive-spectrum drop after the fifth direction agrees with the five hidden variables influencing the observed process.Source: §4.1 · p. 10, Figure 1 · p. 11
  • In long rollouts, the memory model follows the exact autocovariance’s intermediate shoulder and slow tail, whereas raw-history and memoryless models lose or mis-time correlation.Its slow-tail error remains close to the ideal bank reference; raw-history and LSTM errors are near unity.Source: §4.1 · p. 11, Figure 2 · p. 12
  • For the tested conditional forecasts, the memory model follows the exact mean through a slow rebound and has the smallest learned-model error across 32 test histories.Beyond the first several time units, model spreads are similar; at shorter horizons, memoryless and window spreads visibly deviate from the exact forecast.Source: §4.1 · pp. 11–12, Figure 3 · p. 12, Table 1 · p. 13

4.2. A two-dimensional viscoelastic GLE

  • The memory model and LSTM reproduce all four velocity-covariance entries, including negative correlation lobes and subsequent relaxation; memoryless predictions have nearly vanishing cross-correlations.The nonzero lagged cross-covariances reflect memory-induced coupling between components.Source: §4.2 · p. 14, Figure 4 · p. 14
  • Across conditional forecasts, the memory model reproduces joint-density location and spread, while both it and the LSTM match velocity marginals; memoryless marginals are shifted, particularly in vx.The memory model and LSTM also reproduce displacement growth and nonmonotone velocity-change relaxation; the memoryless model overestimates displacement growth and misses the early maximum.Source: §4.2 · pp. 14, 16, Figure 5 · p. 15, Figure 6 · p. 16

4.3. Intermittent scrape-off-layer fluctuations

  • The diffusion model has smaller one-step excess errors than the LSTM in both quiet and active signal-amplitude groups.The LSTM errors are close to the moment-matched Gaussian’s, indicating that its Gaussian output family limits conditional accuracy in this test.Source: §4.3 · p. 17, Table 2 · p. 17
  • From an active test history, the diffusion model follows the reference ridge and upper tail during relaxation, while the LSTM produces a more elliptical cloud and underestimates the upper quantile.The comparison uses 4000 forecast paths per model.Source: §4.3 · p. 18, Figure 8 · p. 19
  • Long rollouts reproduce the scrape-off-layer signal’s skewed stationary density and both correlation scales; the LSTM misses the upper tail.The diffusion model captures burst rise, decay, and broad peak-amplitude distribution, with burst rate close to the reference; its extreme-event probability estimate is lower, with marginally overlapping bootstrap intervals.Source: §4.3 · p. 18, Figure 9 · p. 19, Figure 10 · p. 20

5. Conclusions

The method uses a compact recursive filter bank and conditional diffusion to preserve long-time dependence in generative flow maps. Across three examples, it demonstrates long-correlation retention, predictive compression in coupled dynamics, and representation of non-Gaussian conditional and intermittent-event statistics, subject to assumptions on the observations and memory.

5. Conclusions

  • A compact, recursively updated filter bank preserves long-time dependence without long observation windows, while predictive coordinates can reduce the conditional score estimator’s dimension.For a fixed bank and network size, flow-map evaluation and memory updates have costs independent of trajectory length.Source: §5 · p. 20
  • The three examples show complementary capabilities: retaining slow correlations, supporting conditional sampling in coupled vector dynamics, and representing non-Gaussian conditional and intermittent-event statistics.In the vector example, compression uses reference-assisted bank selection; the other examples use observation-based selection.Source: §5 · p. 20
  • The method assumes stationary, ergodic observations with fading memory that a modest filter bank can approximate.The authors identify richer memory representations and nonlinear predictive reductions for state-dependent and nonstationary dynamics as future work.Source: §5 · p. 20

Appendix A. Additional details on the memory representation

This appendix provides additional details on memory representation, covering memory selection, initialization, predictive reduction, and the filters’ relation to exponential-memory variables.

Appendix A. Additional details on the memory representation

Appendix A.1. Filter conventions, initialization, and rate placement

This subsection details the filter convention, initialization, and rate placement used for the memory representation. It also gives the selection settings for the observation-based examples and notes that Example 2 uses reference-based selection.

Appendix A.1. Filter conventions, initialization, and rate placement

  • The memory recursion uses an unnormalized convention; whitening absorbs the usual factor 1−ρi as a rescaling of component i.Source: Appendix A.1 · p. 21
  • Training begins after a burn-in of ⌈−log δ/(λ1∆t)⌉, with δ set to 0.03, 0.001, and 0.05 in Examples 1–3, respectively; deployment initializes from observed prehistory.Source: Appendix A.1 · p. 21
  • Within each candidate rate band, rates are log-spaced, with the first rate at the band’s geometric center.Source: Appendix A.1 · p. 21
  • Examples 1 and 3 search bank sizes up to 10 and 8, respectively, and select observation-based bands using validation risk and autocovariance criteria; Example 2 uses reference-based selection.The autocovariance thresholds are 0.05 and 0.02 for Examples 1 and 3, respectively.Source: Appendix A.1 · p. 21

Appendix A.2. Predictive rank and predictive coordinates

Predictive reduction identifies memory directions that explain future variation beyond the current state, then uses leading predictive coordinates to reduce conditioning dimension. The appendix defines the linear residual and whitening constructions underlying this reduction and describes its horizon-dependent criterion and example-specific use.

Appendix A.2. Predictive rank and predictive coordinates

  • Predictive coordinates target linear combinations of bank features that explain future variation beyond the current state.Source: Appendix A.2 · p. 22
  • The construction residualizes memory and future variables against the current state, then whitens the memory residual; finite-sample covariance inverses and whitenings use eigenvalue-floored full-rank counterparts.Source: Appendix A.2 · p. 22
  • The singular spectrum of the horizon-wise residual cross-covariances measures each orthogonal whitened memory direction’s contribution to linearly explained future variance.Source: Appendix A.2 · p. 22
  • Predictive rank diagnoses linearly predictive directions in the selected bank, and retaining leading left singular vectors defines reduced predictive memory coordinates.Source: Appendix A.2 · pp. 22–23
  • Candidate predictive dimensions are suggested by sharp spectral drops, while the worst-horizon fraction of discarded memory-explained variance measures projection loss.The horizon range covers the predictive-memory decay range and need not extend to the full closed-loop evaluation horizon.Source: Appendix A.2 · p. 22
  • Compression presents dimension d+dpred to the conditional model while the full bank still updates at cost scaling with kd; the bank need not be autonomous.Example 1 evaluates predictive loss but deploys the full bank; Example 2 deploys a mean-coordinate special case, and Example 3 selects one coordinate sensitive to conditional mean and spread.Source: Appendix A.2 · p. 23

Appendix A.3. Conditional-mean sufficiency of a selected bank

The selected bank is assessed by how much it improves conditional-mean prediction over state-only conditioning. This diagnostic has limits: it does not establish full conditional-law sufficiency or stability of the learned simulator.

Appendix A.3. Conditional-mean sufficiency of a selected bank

  • The held-out empirical risk of the best linear predictor given (x_n, m_n) is used to select the bank in stage A of Section 3.1.It estimates the risk associated with conditioning on the current state and memory.Source: Appendix A.3 · p. 23
  • The population linear-prediction risk upper-bounds the target risk, coinciding with it only when the conditional mean is linear in the conditioning variables.The finite held-out estimate is only a proxy and need not preserve that bound.Source: Appendix A.3 · p. 23
  • When the risks are computable and Vmarkov > Vfull, the conditional-mean sufficiency diagnostic compares bank prediction with the state-only and full-history limits.The construction does not use Vfull, which is unavailable in general.Source: Appendix A.3 · p. 23
  • The diagnostic measures conditional-mean history information, not sufficiency for the full conditional law or convergence of the complete learned simulator.For nested banks, ideal prediction risk cannot increase as the conditioning state grows, although finite-data estimation becomes harder with dimension.Source: Appendix A.3 · p. 23

Appendix A.4. Exact algebraic relation to exponential-memory embeddings

The exponential-filter memory has an exact algebraic relation to discrete auxiliary variables used in exponential-memory embeddings. The relation also clarifies what the filters summarize and which discretization and initialization errors remain.

Appendix A.4. Exact algebraic relation to exponential-memory embeddings

  • For a kernel written as a sum of exponential poles, auxiliary variables are exponentially weighted past-state averages, while the proposed features average past increments; summation by parts relates the two.The convolution interpretation applies when the filter rate matches a kernel pole.Source: Appendix A.4 · p. 24
  • Given X_0, the current state and filter memory determine the discrete auxiliary variable exactly, with residual dependence represented by a term proportional to ρ^n.This is an exact relation, not an approximation.Source: Appendix A.4 · p. 24
  • The discrete auxiliary-variable construction has quadrature error, truncated-prehistory error that decays as ρ_i^n, and rate mismatch when λ_i differs from μ_j.Initializing at zero represents a truncated prehistory; burn-in controls the initialization transient.Source: Appendix A.4 · p. 24
  • For linear-Gaussian models, a linear filter of observed history carries the path’s predictive information about the conditional law of future correlated forcing.The filter can be approximated by elements in the span of the exponential features; that approximation accuracy, rather than the kernel alone, governs the approximation hypothesis error.Source: Appendix A.4 · pp. 24–25

Appendix B. Reverse-ODE discretization and endpoint regularization

The reverse-ODE sampler uses a regularized coefficient schedule and a shifted-node Euler update. These choices introduce coupled numerical approximations, including endpoint blur, so the procedure does not exactly realize the reverse flow.

Appendix B. Reverse-ODE discretization and endpoint regularization

  • The coefficients b and σ^2 diverge as τ approaches 1, so the implementation evaluates them with a regularized schedule.The finite-step sampling scheme is an approximation to the ideal reverse flow.Source: Appendix B · p. 25
  • The shifted-node explicit Euler update evaluates coefficients at the lower backward-grid node to avoid τ = 1, where regularized coefficients are stiff and unregularized ones are singular.The time-dependent coefficients also enter the reverse drift and score estimator.Source: Appendix B · p. 25
  • The regularized schedule uses a Gaussian corruption family, but its initialization is already blurred rather than starting from Z_c.The regularization and shifted-node discretization are coupled approximations and do not exactly realize that family’s reverse flow.Source: Appendix B · p. 25
  • Along a unit direction v, the relative variance increase is 2h + h^2 + h/(κ^2 sV_v), with endpoint blur contributing more in directions where the conditional law is narrow.The integration step and endpoint regularization are tied through h = 1/nODE, so step-doubling changes both effects together.Source: Appendix B · p. 25

Appendix C. Conditional score construction

The conditional score is represented as a weighted average for a Gaussian mixture and estimated from trajectory tuples using kernel-smoothed conditioning weights. Its accuracy depends on conditioning dimension, bandwidth, and available data.

Appendix C. Conditional score construction

  • For 0 < τ < 1, the conditional score is that of a Gaussian mixture whose component weights are determined by the conditioning kernel.The exact score has a weighted-average representation.Source: Appendix C · p. 26
  • Estimator accuracy depends on conditioning dimension, bandwidth, and the amount of available trajectory data.The conditioning weights implement the kernel-based conditional density estimate.Source: Appendix C · p. 26
  • The estimator normalizes the conditioning density, mollifies the Dirac delta with a Gaussian kernel, and replaces integrals with empirical sums over trajectory tuples.The resulting self-normalized weights follow the conditional smoothing construction.Source: Appendix C · p. 26
  • Nearest-neighbor truncation uses J neighbors, while bandwidth selection trades conditioning bias against neighborhood size in temporally dependent data.Neighborhood radius and weight effective sample size monitor truncation adequacy; effective sample size measures weight concentration, not independent observations.Source: Appendix C · p. 26

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