Pith. sign in

REVIEW 3 major objections 4 minor 33 references

Modelisation of chaotic systems with a latent Stochastic Differential Equation

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper claims that a chaotic PDE system can be effectively replaced by a learned stochastic differential equation on a low-dimensional latent space, and that long generative rollouts from unseen initial conditions stay stable and…

desk verdict A useful new latent-SDE generative ROM for chaotic PDEs with strong empirical evaluation; the Gaussian finite-memory kernel assumption is the main soft spot, but the paper deserves a serious referee. read the letter →

arxiv 2608.03438 v1 pith:TZ4HJADG submitted 2026-08-04 physics.flu-dyn

Ismaël Zighed , Nicolas Thome , Patrick Gallinari , Taraneh Sayadi This is my paper · ORCID
classification physics.flu-dyn
keywords stochasticdifferentialequationschaoticdynamicsreduced-ordermodelinglatentspacetransitionkernelKolmogorovflowKuramoto-Sivashinskyequationgenerativeemulation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Chaotic flows are usually emulated with deterministic reduced-order models, but deterministic errors are amplified exponentially and eventually destroy the forecast. The paper argues that after projecting the flow onto a low-dimensional manifold, the future is genuinely uncertain, so the right object to learn is a transition kernel rather than a deterministic map. It models latent dynamics as a discretized Itô SDE with state-dependent drift and diffusion, trained by maximum likelihood, and shows that on a two-dimensional Kolmogorov flow and the one-dimensional Kuramoto–Sivashinsky equation the generated trajectories remain bounded, reproduce the reference attractor manifolds, and match the invariant measure approximated by the DNS data.

What carries the argument

The load-bearing object is the augmented latent state $y_t=(z_t,z_{t-1},\dots,z_{t-\tau+1})$, which makes the non-Markovian latent process Markovian by carrying a finite memory window. The transition is a degenerate Gaussian Markov kernel, $y_{t+1}=y_t+F_\theta(y_t)+G_\theta(y_t)\odot\xi_t$, with stochasticity restricted to the leading $d$ coordinates; this is the Euler–Maruyama discretization of an Itô SDE with unit time step. Drift and diagonal diffusion are produced by one transformer, and the training objective is the exact negative log-likelihood of the next latent state under the Gaussian kernel, including the $\log g_i^2$ term that a Girsanov-based SDE-matching loss would cancel. The autoencoder and the integrator are trained jointly, with multi-step rollouts that feed predicted states back as context.

What would settle it

To settle the claim, encode DNS snapshots and estimate the empirical conditional distribution of $\Delta z = z_{t+1} - z_t$ for a fixed look-back window; if it is clearly non-Gaussian, or if a look-back longer than $\tau=20$ materially changes the best drift and diffusion, then the Gaussian Markov kernel with a twenty-step window is not the true transition kernel.

Watch

Extended reading notes

Core claim

At the center is the claim that a deterministic chaotic PDE can be modeled in a suitable latent space by an SDE: the drift captures the predictable physics, while a diagonal diffusion absorbs both the intrinsic chaos and the information lost in the projection. The paper operationalizes this as a Gaussian Markov transition on an augmented latent state, $z_{t+1}=z_t+f_\theta(y_t)+g_\theta(y_t)\odot\xi_t$, where $y_t$ stacks a look-back window of $\tau=20$ latent snapshots and a single transformer produces both $f_\theta$ and $g_\theta$. The parameters are fit by minimizing the negative log-likelihood of observed transitions, with rollouts of $H$ steps so the model learns from its own predictions. In the reported experiments, generated trajectories from unseen initial conditions stay stable for 2000 rollout steps, reproduce temporal and spatial autocorrelation decay, overlap with the delay-embedding manifolds of kinetic energy and dissipation, and recover the first three moments of the reference densities; the 1$\sigma$ prediction-interval coverage stays near 59–70 percent and the 3$\sigma$ coverage near 91–96 percent.

Load-bearing premise

The entire construction rests on the assumption that, given the last twenty latent snapshots, the next latent state is drawn from a Gaussian distribution with independent coordinates and a shape that depends only on the current window, not on time; if the true latent transitions are far from Gaussian or remember more than twenty snapshots, the learned kernel cannot represent them.

Editorial extensions

If this is right

  • If the central claim is correct, long un-conditioned rollouts of chaotic flows can be produced by sampling a learned transition kernel, so no deterministic closure or repeated correction toward a reference trajectory is needed.
  • Generated trajectories sample the same invariant measure and autocorrelation structure as the DNS, so statistics such as mean, variance, skewness, and energy spectra can be estimated directly from the generative model.
  • The model remains stable well beyond the training window, so it can serve as a generative surrogate for turbulence studies in which individual trajectories are cheap but exact trajectory matching is not the goal.
  • The same construction should transfer to other PDEs with chaotic attractors, provided a suitable latent representation and a sufficient look-back length exist.
  • The explicit $\log g_i^2$ term in the negative log-likelihood gives a principled way to learn state-dependent noise without relying on a separate reference SDE for cancellation.
  • The method provides a concrete probabilistic alternative to deterministic latent ROMs: instead of asking which trajectory follows from an initial condition, it asks which distribution of futures is consistent with the observed past.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: the same setup can expose where the Gaussian Markov assumption fails; if 1$\sigma$ coverage degrades in specific regions of latent space, those regions are where the true transition kernel is non-Gaussian or needs longer memory.
  • Editorial inference: because the diffusion term mixes chaos, projection loss, and model error, comparing the learned $g_\theta$ with the empirical conditional variance of DNS increments at the same look-back window would separate intrinsic stochasticity from approximation error.
  • Editorial inference: testing on systems with a known invariant measure would allow a quantitative check of the generative claim, since the stationary density of the model could be compared directly with the exact density rather than with a ten-trajectory empirical histogram.
  • Editorial inference: the authors attribute the high-wavenumber energy accumulation in the Kolmogorov case to the convolutional decoder; a testable extension is that using a spectral decoder or retaining the spectral regularizer removes the artifact without harming the statistical invariants.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. This paper proposes a probabilistic reduced-order model in which a high-dimensional chaotic PDE is encoded into a low-dimensional latent space and advanced by a discretized Itô SDE with state-dependent drift and diagonal diffusion terms (Eqs. 14–19). To address non-Markovianity, the transition is conditioned on an augmented look-back window of τ=20 latent states; the drift and diffusion are produced by a transformer, and training minimizes a rollout NLL jointly with autoencoder reconstruction (Eq. 26). The method is demonstrated on 2D Kolmogorov flow at Re=90 and 1D Kuramoto–Sivashinsky with ν=0.8, using 10 training trajectories of 800 snapshots; generative rollouts of 2000 snapshots from unseen initial conditions are assessed via PICP, delay embeddings, autocorrelations, marginal densities, and energy spectra.

Significance. The paper is a useful contribution to probabilistic ROMs for chaotic fluids: it treats the SDE as a physical model rather than a mere uncertainty-quantification device, releases code and data, and shows that long autonomous rollouts remain bounded and reproduce invariant manifolds, autocorrelations, spectra, and the first three marginal moments on two standard benchmarks. The trajectory-number ablation in Appendix 7.3 is an honest and informative check of how data diversity affects uncertainty calibration. If the transition-kernel claim is properly validated, the framework would provide a practical stochastic closure for chaotic systems. The current evidence, however, supports the generative/long-term-statistics claim more strongly than the 'true transition kernel' claim; the missing conditional validation and the PICP miscalibration are the main gaps.

major comments (3)
  1. [Section 3, Eq. (19); Section 5] The central modeling assumption is that p(z_{t+1}|y_t) is Gaussian with diagonal covariance and that τ=20 renders (y_t) Markovian. The sentence 'By construction, the process (y_t) is Markovian' is true only when the memory of z_t is exactly ≤τ; for a chaotic latent process this is a truncation hypothesis that is never tested. The validation in Section 5 uses aggregate diagnostics — delay embeddings, autocorrelations, marginal densities, and spectra (Figs. 8–12, 15–17) — which any kernel with the same invariant measure (for instance, one that samples from the marginal distribution) would reproduce. Because the abstract and Section 5 claim consistency 'with the true transition kernel,' this gap is load-bearing. Please add direct conditional diagnostics: standardized residuals (z_{t+1}^{(i)} - z_t^{(i)} - f_i(y_t))/g_i(y_t), Q-Q plots against N(0,1) per latent coordinate, conditional coverage at one-step and short horizons, and, if feasible, a nonparametric comparison with the empirical transition density.
  2. [Tables 1 and 2, Eq. (29)] The reported PICP values are not consistent with the Gaussian model that the method assumes. For the Kolmogorov case, Table 1 gives 1σ coverage 61%/59% and 3σ coverage 93%/92% against nominal 68.3%/99.7%; for KS, Table 2 gives 70%/54% and 96%/95%. The text in Section 5.2 describes this as 'broadly consistent with Gaussian behaviour,' but a 4–8 percentage point shortfall at 3σ indicates that the predictive intervals are systematically too narrow or the innovation distribution has shorter tails than Gaussian. Since PICP is the only quantitative probabilistic diagnostic in the paper, this miscalibration weakens the claim that the diffusion term faithfully represents the 'irreducible uncertainty.' Please report proper scoring rules (CRPS, interval score), post-hoc recalibration, or an additional quantile-based diagnostic, and adjust the wording accordingly.
  3. [Section 3, Eqs. (16)–(17)] The Euler–Maruyama discretization is justified only if Δt is small enough for the transition kernel to be infinitesimally Gaussian, yet Δt=1 is fixed. For the Kolmogorov case, the DNS snapshots are saved every δt=0.12, so one latent transition spans approximately 8.3 solver steps; for KS, Δt=1 spans four integration steps. No evidence is provided that the empirical one-step transition remains approximately Gaussian at this coarse increment, and any discretization error is absorbed into the learned g_θ, confounding the physical interpretation of the diffusion and likely contributing to the PICP undercoverage. Please report a sensitivity analysis with smaller latent time increments (e.g., Δt=1/2 or 1/4 via subsampling) or validate the Gaussianity of the observed transitions at the chosen Δt.
minor comments (4)
  1. [Throughout] There are several typos and language issues that should be corrected in revision: the title uses French 'Modelisation'; Section 3 has 'defiend'; Appendix 7.1 has 'Kurmaoto'; Section 2 has 'passed their Lyapunov time scale.' A careful proofread is needed.
  2. [Section 5.2 and Tables 1–2] The paper makes a comparative claim in the Introduction ('a robust alternative to traditional deterministic methodologies'), but no deterministic ROM baseline (e.g., POD-Galerkin, neural ODE, or LSTM-based emulator) is included. Adding at least one baseline would substantially strengthen the significance assessment.
  3. [Section 5.1, Eq. (29)] Equation (29) is referred to as 'the expression given in 29' but is not displayed with an equation number; please number the equation and update the cross-reference.
  4. [Section 5.2, Table 2] The statement that 'Over the entire dataset, the empirical coverage reaches approximately 66.8% within one standard deviation and 96% within three standard deviations' is consistent with a sample-size-weighted average of the train and test rows, but this should be stated explicitly so the reader does not read it as a test-set result.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the SDE transition kernel is explicitly fitted to DNS transitions and validated against DNS statistics, not derived from its own outputs.

full rationale

The paper's derivation chain is a standard data-driven ROM pipeline: an autoencoder defines the latent space, a transformer parameterizes drift and diffusion of a Gaussian Markov transition, the parameters are fitted by maximum likelihood on DNS transition data (Eqs. 20-23), and the resulting model is then rolled out and compared with DNS statistics, manifolds, autocorrelations, and spectra. There is no step in which a claimed prediction is equivalent by construction to a fitted input. The diffusion term is learned from the data, and the paper explicitly acknowledges in the conclusion that it absorbs aleatoric uncertainty, projection error, and epistemic error; that admission concerns interpretability of a fitted quantity, not circularity. The statement that the augmented latent process is Markovian 'by construction' is a modeling simplification based on a finite look-back window, and the paper is transparent that the underlying latent process may be non-Markovian; this is an assumption open to empirical testing, not a derivation that assumes its conclusion. Self-citations appear (e.g., [6], [7], [29]), but they are contextual and not load-bearing: the need for memory is also justified by the Mori-Zwanzig formalism, and the central claims are validated against DNS data independently of any cited theorem. The reported PICP values and the unvalidated Gaussian kernel raise correctness concerns, but those are not circularity. No equation or fitted parameter is renamed as a prediction, and no load-bearing uniqueness claim is imported from the authors' prior work. The paper is self-contained against the DNS benchmarks it uses, so the honest finding is no significant circularity.

Assumptions & free parameters 7 free parameters · 7 assumptions · 0 invented entities

The central claim rests on a small number of modeling choices: a finite Gaussian Markov transition on an augmented latent state, time-homogeneous drift and diffusion, and a fixed Euler-Maruyama step. Several hyperparameters (latent dimension, window, horizon, loss weights, diffusion bounds) are chosen by hand. No new physical entities are posited; the latent SDE and augmented state are mathematical modeling constructs.

free parameters (7)
  • latent dimension d = 64
    Hand-chosen size of the latent space for both benchmarks; the reduced dynamics and the SDE's capacity depend on it.
  • look-back window tau = 20
    Finite memory length used to make the augmented latent process Markovian; if too short, non-Markovianity is truncated.
  • rollout horizon H = 40
    Number of autoregressive steps in training; chosen to force the model to learn from its own predictions.
  • likelihood weight alpha = 0.01 d^{-1}
    Weight on the latent negative log-likelihood term in the total loss.
  • reconstruction weight beta = 0.01
    Weight on the trajectory reconstruction loss in physical space.
  • energy regularization weight gamma = 0.01
    Weight on the total-energy penalty.
  • log-diffusion bounds = [-6, 2]
    Clipping range for the learned log diffusion; bounds the noise magnitude and affects NLL optimization.
assumptions (7)
  • standard math f and gamma are continuous and Lipschitz, so transition kernels are Gaussian in the infinitesimal sense.
    Invoked in Section 3 before Eq. 14 to justify the Gaussian discrete transition kernel.
  • domain assumption The conditional distribution of z_{t+1} given augmented state y_t is Gaussian with diagonal covariance.
    Eqs. 17-19 define the learned transition as z_{t+1} = z_t + f(y_t) + g(y_t) * xi, which is Gaussian and componentwise independent; chaotic latent dynamics need not satisfy this.
  • domain assumption A finite look-back window of length tau is sufficient for the augmented process to be Markovian.
    Section 3: 'By construction, the process (y_t) is Markovian, even if (z_t) is not.' The Mori-Zwanzig memory is truncated to tau=20.
  • domain assumption The drift and diffusion are time-homogeneous.
    Section 3: 'the coefficients f and gamma are time-homogeneous'.
  • domain assumption Euler-Maruyama with fixed time step Delta t = 1 adequately approximates the continuous SDE transition kernel.
    Section 3: 'a constant time step Delta t = 1 is assumed, interpreted as a unit increment required to be sufficiently small for the infinitesimal Gaussianity approximation to remain valid'.
  • ad hoc to paper Training on 10 trajectories of 800 snapshots is sufficient to learn a representative transition kernel.
    The ablation in Section 7.3 shows monotonic improvement from 1 to 10 trajectories, but no convergence criterion is given; the choice of 10 is practical, not justified by a formal guarantee.
  • ad hoc to paper Projection loss and model error can be absorbed as diagonal diffusion noise without breaking physical consistency.
    Section 3 and the conclusion state that lost information and epistemic error are folded into the learned diffusion, yet the decomposition is not identifiable.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Modelisation of chaotic systems with a latent Stochastic Differential Equation." pith.science (2026). https://pith.science/paper/TZ4HJADG

@misc{pith2026260803438,
  author       = {Pith},
  title        = {Pith review of: Modelisation of chaotic systems with a latent Stochastic Differential Equation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TZ4HJADG}},
  note         = {Machine review of arXiv:2608.03438}
}
read the original abstract

Stochastic Differential Equations (SDEs) have become a cornerstone of scientific machine learning, though they are predominantly utilized as algorithmic tools for uncertainty quantification or distribution matching. In contrast, leveraging SDEs fundamentally to model macroscopic, nonlinear physics as stochastic processes remains largely unexplored. This work introduces a probabilistic, non-intrusive reduced-order model (ROM) for chaotic dynamical systems. We argue that projecting high-dimensional nonlinear dynamics onto a low-dimensional manifold introduces irreducible uncertainty, compounded by the chaotic attractors and multi-admissible futures inherent to turbulent flows. Consequently, a chaotic system governed by a partial differential equation can be effectively modeled by an SDE in a suitable latent space. To this end, a nonlinear autoencoder is employed to map the flow field into a low-dimensional representation, within which the temporal evolution is explicitly governed by an SDE. The predictable component of the dynamics is captured by a learned drift term, while state-dependent stochasticity is absorbed by a diffusion term. We demonstrate that this probabilistic framework successfully propagates highly nonlinear states, offering a robust alternative to traditional deterministic methodologies for chaotic regimes. Ultimately, our model generates new chaotic flow trajectories that remain locally and globally consistent with the true transition kernel learned from Direct Numerical Simulation (DNS) data. Even though these generated trajectories are unique and distinct from the training set, they preserve the underlying statistics and manifolds, validating the strong generative performance and robustness of our methodology.

Figures

Figures reproduced from arXiv: 2608.03438 by the authors.

Figure 1
Figure 1. Data pipeline over a prediction rollout of H steps using [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Snapshots of the Kolmogorov velocity fields [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Numerical integration of 10 independent 1D-KS trajectories over the time horizon [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (19 more)
Figure 4
Figure 4. Figure 4: Reconstruction of the velocity component [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]
Figure 5
Figure 5. Figure 5: Comparison between a reference trajectory ( [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]
Figure 6
Figure 6. Figure 6: Temporal evolution of the velocity components ( [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]
Figure 7
Figure 7. Figure 7: Mean and standard deviation of Uˆ and Vˆ throughout the prediction horizon. Each generated trajectory, although initialized from the same initial condition, is unique and does not ’replicate’ any training path. To formally assess the plausibility of the generated traje…
Figure 8
Figure 8. Figure 8: Local delay embedding at the spatial location [PITH_FULL_IMAGE:figures/full_fig_p017_8.png]
Figure 9
Figure 9. Figure 9: Global delay embedding using a delay parameter [PITH_FULL_IMAGE:figures/full_fig_p017_9.png]
Figure 10
Figure 10. Figure 10: Autocorrelation functions computed over a horizon of [PITH_FULL_IMAGE:figures/full_fig_p018_10.png]
Figure 11
Figure 11. Figure 11: Comparison of the marginal density functions of the velocity components. The left column shows densities estimated [PITH_FULL_IMAGE:figures/full_fig_p019_11.png]
Figure 12
Figure 12. Figure 12: Comparison of the marginal density functions of the dissipation. The reference density is estimated by aggregating [PITH_FULL_IMAGE:figures/full_fig_p020_12.png]
Figure 13
Figure 13. Figure 13: Spectrogram of one generated trajectory averaged over slices of [PITH_FULL_IMAGE:figures/full_fig_p021_13.png]
Figure 14
Figure 14. Figure 14: From left to right : Ground truth, corresponding to the last trajectory from the dataset, mean trajectory prediction [PITH_FULL_IMAGE:figures/full_fig_p022_14.png]
Figure 15
Figure 15. Figure 15: Low-dimensional manifold in the (u, uxx) phase space. The generated stochastic trajectories are projected into the same embedding space for comparison. Figure 16b demonstrates that the learned transition kernel successfully reproduces both the temporal and spatial aut…
Figure 16
Figure 16. Figure 16: Comparison of temporal (a) and spatial (b) autocorrelation functions between the true and generated trajectories. [PITH_FULL_IMAGE:figures/full_fig_p023_16.png]
Figure 17
Figure 17. Figure 17: Individual trajectory densities of uˆ with reference u density. Finally, we evaluate model consistency with respect to physical invariants and spectral properties. Specif￾ically, we compute the spatial kinetic energy spectrum—averaged across temporal slices of 10δt, t…
Figure 18
Figure 18. Figure 18: Comparison of the (a) spatial energy spectra and (b) temporal frequency spectra between the true and generated [PITH_FULL_IMAGE:figures/full_fig_p024_18.png]
Figure 19
Figure 19. Figure 19: Reconstruction of the field u(x, t) at the spatial location x = L 2 over an autoregressive rollout horizon of T = 1200 δt. For each initial condition, an ensemble of 60 stochastic trajectories is generated by the model. The solid line denotes the ensemble mean, while …
Figure 20
Figure 20. Figure 20: Comparison between a reference trajectory [PITH_FULL_IMAGE:figures/full_fig_p030_20.png]
Figure 21
Figure 21. Figure 21: Impact of spectral regularization on the trajectories spectrograms throughout long-rollouts. [PITH_FULL_IMAGE:figures/full_fig_p033_21.png]
Figure 22
Figure 22. Figure 22: Impact of training trajectory diversity ( [PITH_FULL_IMAGE:figures/full_fig_p034_22.png]

Review history (2 revisions) →

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

33 extracted references · 14 canonical work pages

  1. [1]

    Hijazi, G

    S. Hijazi, G. Stabile, A. Mola, G. Rozza, Data-driven pod-galerkin reduced order model for turbulent flows, Journal of Computational Physics 416 (2020) 109513.doi:https://doi.org/10.1016/j.jcp. 2020.109513. URLhttps://www.sciencedirect.com/science/article/pii/S0021999120302874

    arXiv 2020

  2. [2]

    Menier, M

    E. Menier, M. A. Bucci, M. Yagoubi, L. Mathelin, M. Schoenauer, Cd-rom: Complemented deep - reduced order model, Computer Methods in Applied Mechanics and Engineering 410 (2023) 115985. doi:https://doi.org/10.1016/j.cma.2023.115985. URLhttps://www.sciencedirect.com/science/article/pii/S0045782523001081

    arXiv 2023

  3. [3]

    Colanera, L

    A. Colanera, L. Magri, Quantized local reduced-order modeling in time (ql-rom), Computer Methods in Applied Mechanics and Engineering 447 (2025) 118393.doi:https://doi.org/10.1016/j.cma.2025. 118393. URLhttps://www.sciencedirect.com/science/article/pii/S0045782525006656

    doi:10.1016/j.cma.2025 2025

  4. [4]

    S. L. Brunton, M. Budišić, E. Kaiser, J. N. Kutz, Modern koopman theory for dynamical systems, SIAM Review 64 (2) (2022) 229–340.arXiv:https://doi.org/10.1137/21M1401243,doi:10.1137/ 21M1401243. URLhttps://doi.org/10.1137/21M1401243

    doi:10.1137/21m1401243 2022

  5. [5]

    P. J. Schmid, Dynamic mode decomposition of numerical and experimental data, Journal of Fluid Mechanics 656 (2010) 5–28.doi:10.1017/S0022112010001217

    doi:10.1017/s0022112010001217 2010

  6. [6]

    Gupta, P

    P. Gupta, P. Schmid, D. Sipp, T. Sayadi, G. Rigas, Mori–zwanzig latent space koopman clo- sure for nonlinear autoencoder, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 481 (2313) (2025) 20240259.arXiv:https://royalsocietypublishing. org/rspa/article-pdf/doi/10.1098/rspa.2024.0259/2822700/rspa.2024.0259.pdf,doi: 10.109...

    work page arXiv 2025

  7. [7]

    Zighed, N

    I. Zighed, N. Thome, P. Gallinari, T. Sayadi, Uncertainty-aware and parametrized dynamic reduced- order model: Application to unsteady flows, Phys. Rev. Fluids (2025).doi:10.1103/f6ty-t6gl. URLhttps://link.aps.org/doi/10.1103/f6ty-t6gl

    work page doi:10.1103/f6ty-t6gl 2025

  8. [8]

    Özalp, A

    E. Özalp, A. Nóvoa, L. Magri, Real-time forecasting of chaotic dynamics from sparse data and autoen- coders (08 2025).doi:10.48550/arXiv.2508.08729

    work page doi:10.48550/arxiv.2508.08729 2025

Show all 33 references
  1. [9]

    Mohan, D

    A. Mohan, D. Gaitonde, A deep learning based approach to reduced order modeling for turbulent flow control using lstm neural networks (04 2018).doi:10.48550/arXiv.1804.09269

    2018 doi arXiv

  2. [10]

    Solera-Rico, C

    A. Solera-Rico, C. Sanmiguel Vila, M. Gómez-López, Y. Wang, A. Almashjary, S. Dawson, R. Vin- uesa,β-variational autoencoders and transformers for reduced-order modelling of fluid flows, Nature Communications 15 (02 2024).doi:10.1038/s41467-024-45578-4

    2024 doi

  3. [11]

    C. J. G. Rojas, A. Dengel, M. Dias Ribeiro, Reduced-order model for fluid flows via neural ordinary differential equations (2021)

    2021

  4. [12]

    Haller, S

    G. Haller, S. Ponsioen, Nonlinear normal modes and spectral submanifolds: existence, unique- ness and use in model reduction, Nonlinear Dynamics 86 (3) (2016) 1493–1534.doi:10.1007/ s11071-016-2974-z. URLhttps://doi.org/10.1007/s11071-016-2974-z 26

    2016 doi

  5. [13]

    A. Liu, J. Axås, G. Haller, Data-driven modeling and forecasting of chaotic dynamics on inertial manifolds constructed as spectral submanifolds, Chaos: An Interdisciplinary Journal of Nonlinear Sci- ence 34 (3) (2024) 033140.arXiv:https://pubs.aip.org/aip/cha/article-pdf/doi/1...

    2024 doi

  6. [14]

    Mori, Transport, collective motion, and brownian motion*), Progress of Theoretical Physics 4 (3) (1965) 423–455.doi:10.1143/PTP.33.423

    H. Mori, Transport, collective motion, and brownian motion*), Progress of Theoretical Physics 4 (3) (1965) 423–455.doi:10.1143/PTP.33.423

    1965 doi

  7. [15]

    Zwanzig, Nonlinear generalized langevin equations, Journal of Statistical Physics 9 (3) (1973) 215– 220.doi:10.1007/BF01008729

    R. Zwanzig, Nonlinear generalized langevin equations, Journal of Statistical Physics 9 (3) (1973) 215– 220.doi:10.1007/BF01008729

    1973 doi

  8. [16]

    Burrage, P

    K. Burrage, P. Burrage, T. Mitsui, Numerical solutions of stochastic differential equations – imple- mentation and stability issues, Journal of Computational and Applied Mathematics 125 (1) (2000) 171–182, numerical Analysis 2000. Vol. VI: Ordinary Differential Equations and I...

    2000 doi

  9. [17]

    URLhttps://openreview.net/forum?id=hwy0BvBIQQ

    J.M.Benet, A.K.Koupaï, R.D.Regueiro-Espino, N.Baskiotis, P.Gallinari, AdaptiveSDEinterpolants for calibrated probabilistic PDE forecasting (2026). URLhttps://openreview.net/forum?id=hwy0BvBIQQ

    2026

  10. [18]

    L. Kong, J. Sun, C. Zhang, Sde-net: Equipping deep neural networks with uncertainty estimates, Proceedings of Machine Learning Research 119 (2020)

    2020

  11. [19]

    J. Ho, A. Jain, P. Abbeel, Denoising diffusion probabilistic models, Advances in Neural Information Processing Systems (NeurIPS) 33 (2020) 6840–6851

    2020

  12. [20]

    Y. Song, J. Sohl-Dickstein, D. P. Kingma, A. Kumar, S. Ermon, B. Poole, Score-based generative modeling through stochastic differential equations, arXiv preprint arXiv:2011.13456 (2020)

    2020 arXiv

  13. [21]

    Ha, Y.-J

    J.-S. Ha, Y.-J. Park, H.-J. Chae, S.-S. Park, H.-L. Choi, Adaptive path-integral autoencoders: Rep- resentation learning and planning for dynamical systems, Advances in Neural Information Processing Systems 31 (2018)

    2018

  14. [22]

    Course, P

    K. Course, P. B. Nair, State estimation of a physical system with unknown governing equations, Nature 622 (7982) (2023) 261–267

    2023

  15. [23]

    A. F. Ilersich, K. Course, P. B. Nair, Data-driven stochastic reduced-order modeling of parametrized dynamical systems, arXiv preprint arXiv:2601.10690 (2026)

    2026

  16. [24]

    Bartosh, D

    G. Bartosh, D. Vetrov, C. A. Naesseth, Sde matching: Scalable and simulation-free training of latent stochastic differential equations, arXiv preprint arXiv:2502.02472 (2025)

    2025 arXiv

  17. [25]

    Lipman, R

    Y. Lipman, R. T. Chen, H. Ben-Hamu, M. Nickel, M. Le, Flow matching for generative modeling, arXiv preprint arXiv:2210.02747 (2022)

    2022 arXiv

  18. [26]

    Y. Zhu, H. Lei, Effective mori-zwanzig equation for the reduced-order modeling of stochastic systems, Discrete and Continuous Dynamical Systems - S 15 (4) (2022) 959–982.doi:10.3934/dcdss.2021096. URLhttps://www.aimsciences.org/article/id/885db29c-bdff-44a4-a5bd-e33dca8937d9

    2022 doi

  19. [27]

    M. Shen, C. Cheng, Neural sdes as a unified approach to continuous-domain sequence modeling (2025). arXiv:2501.18871. URLhttps://arxiv.org/abs/2501.18871 27

    2025 arXiv

  20. [28]

    Vaswani, N

    A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, L. Kaiser, I. Polosukhin, Attention is all you need (2017). URLhttps://proceedings.neurips.cc/paper_files/paper/2017/file/ 3f5ee243547dee91fbd053c1c4a845aa-Paper.pdf

    2017

  21. [29]

    Zighed, N

    I. Zighed, N. Thome, P. Gallinari, T. Sayadi, Leveraging scale separation and stochastic closure for data-driven prediction of chaotic dynamics, Data-Centric Engineering 7 (2026) e12.doi:10.1017/ dce.2026.10045

    2026

  22. [30]

    Platt, L

    N. Platt, L. Sirovich, N. Fitzmaurice, An investigation of chaotic kolmogorov flows, Physics of Fluids A: Fluid Dynamics 3 (4) (1991) 681–696.arXiv:https://pubs.aip.org/aip/pof/article-pdf/3/ 4/681/12301110/681_1_online.pdf,doi:10.1063/1.858074. URLhttps://doi.org/10.1063/1.858074

    1991 doi

  23. [31]

    Canuto, A

    C. Canuto, A. Quarteroni, M. Y. Hussaini, T. A. Zang, Spectral Methods: Evolution to Complex Geometries and Applications to Fluid Dynamics, 1st Edition, Scientific Computation, Springer Berlin Heidelberg, 2007, published: 30 June 2007, eBook ISBN: 978-3-540-30728-0, Hardcover ...

    2007 doi

  24. [32]

    Pedersen, L

    C. Pedersen, L. Zanna, J. Bruna, Thermalizer: Stable autoregressive neural emulation of spatiotemporal chaos (03 2025).doi:10.48550/arXiv.2503.18731. 28

    2025 doi arXiv

  25. [33]

    Z T 0 ∥u(zt,t)∥ 2 2dt # .(46) This penalty can be expressed purely in terms of the system’s drift and diffusion functions with equation 40: DKL(QSDE∥P SDE) = 1 2 EQξ

    Appendix 7.1. Kuramoto-Sivashinsky - complementary discussions To complement the discussions and statements made in 4.2 we provide here visual support in the Kurmaoto-Sivashinsky benchmark. Figure 19, illustrates how the model rolls out from the initial conditions taken from t...

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.