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REVIEW 4 major objections 6 minor 103 references

Stochastic and Non-local Closure Modeling for Nonlinear Dynamical Systems via Latent Score-based Generative Models

T0 review · 4 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Jointly training the autoencoder and diffusion model yields a stochastic non-local turbulence closure that matches physical-space accuracy while running ensembles about ten times faster.

desk verdict Joint training clearly beats two-phase in their latent closure comparison, but the paper simulates a different closure than the one it defines, and the ~10x speedup is really 7x. read the letter →

arxiv 2506.20771 v2 pith:2YWNU46M submitted 2025-06-25 cs.LG math.DSphysics.comp-ph

Xinghao Dong , Huchen Yang , Jin-Long Wu This is my paper
classification cs.LGmath.DSphysics.comp-ph MSC 68T0776F65
keywords stochasticclosuremodelinglatentdiffusionmodelsscore-basedgenerativejointautoencoder-diffusiontrainingsubgrid-scaletwo-dimensionalturbulencenon-localuncertaintyquantification
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

This paper tries to establish that a data-driven stochastic closure term for nonlinear dynamical systems can be made practical by running a conditional diffusion model in a learned latent space instead of on the full physical field, and that the way the latent space is learned decides whether this works at all. The authors show that the standard recipe of training an autoencoder for compression and only then training the diffusion model in the frozen latent space produces latent distributions that diffusion models sample poorly, so the reduced simulation degrades. Their fix is to train the autoencoder and the diffusion model jointly, with a KL penalty that stops the latent codes from collapsing, so the latent space is shaped by both reconstruction fidelity and how easy the conditional distribution is to generate. On a two-dimensional Kolmogorov flow, the jointly trained model reaches the same class of predictive accuracy as a physical-space diffusion closure (final vorticity relative error 8.4% versus 7.4%) while cutting the cost of a 1000-sample ensemble simulation by roughly an order of magnitude (from about 8,650 seconds to about 1,240 seconds). If this holds, ensemble-based uncertainty quantification for turbulent flows becomes affordable at a scale that was previously impractical.

What carries the argument

The load-bearing object is the joint training objective for the latent conditional diffusion model, $\mathcal{L}_{\mathrm{joint}} = \mathcal{L}_{\mathrm{AE}} + \lambda_{\mathrm{score}}\mathcal{L}_{\mathrm{score}} + \lambda_{\mathrm{KL}}\mathcal{L}_{\mathrm{KL}}$, where $\mathcal{L}_{\mathrm{AE}}$ weights reconstruction of the closure term $H$ more heavily than reconstruction of the conditioning field $\omega$, $\mathcal{L}_{\mathrm{score}}$ is the denoising score-matching loss that trains the conditional score network $s_\theta(\tau, z_H, z_\omega)$ to approximate $\nabla_{z_H}\log p(z_H \mid z_\omega)$, and $\mathcal{L}_{\mathrm{KL}}$ penalizes the divergence between the encoded distribution and a standard Gaussian. The KL term is what prevents latent space collapse, the degenerate outcome in which the encoder maps all inputs to a low-variance distribution and the diffusion objective is minimized trivially. The score network is built from Fourier neural operators, the encoders are convolutional with residual and self-attention blocks, and sampling integrates the reverse variance-exploding SDE in the $16\times 16$ latent space with an adaptive time schedule, so the expensive iterative denoising loop runs on one-sixteenth of the spatial resolution.

What would settle it

Re-run the 2-D Kolmogorov experiment at higher Reynolds number (for example $\nu = 5\times 10^{-4}$) and compare the memoryless joint L-CDM against a version conditioned on the previous several vorticity snapshots: if the memory-augmented closure is markedly more accurate, the instantaneous-conditioning assumption is falsified. A cheaper check is to hold out the forcing and see whether the closure error stays in the range of Table 5, since a model that only reproduces the training conditional distribution would degrade there.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is a failure mode and its cure. Latent diffusion models are normally built in two stages: an autoencoder is trained for reconstruction, and a diffusion model is then trained in the resulting latent space. The paper demonstrates on the 2-D Kolmogorov flow that the reconstruction-optimal latent space is hostile to diffusion sampling: the separately trained autoencoder compresses the $64\times 64$ fields to $16\times 16$ latent codes with tiny reconstruction error (relative error 1.4% for the closure field), yet the conditional diffusion model trained in that space generates physical-space closures with relative error 0.29, more than double the 0.13 of the physical-space baseline. When the autoencoder and the diffusion model are optimized together through a loss that combines reconstructing the closure field, denoising score matching, and a KL term that prevents the latent distribution from collapsing, the same diffusion architecture recovers baseline-level accuracy (relative error 0.11) while keeping the latent-space speed advantage. The interpretation is that joint training lets the encoder discover a latent geometry in which the conditional score function $s_\theta(\tau, z_H, z_\omega)$ is easy to learn, and the paper supports this with t-SNE and energy-spectrum comparisons showing that jointly trained samples align with the ground-truth distribution, plus full 20-second vorticity simulations in which the joint closure stays accurate at roughly an order-of-magnitude lower ensemble cost.

Load-bearing premise

Everything rests on the assumption that the missing physics is fully determined by the current resolved flow field, since the closure is drawn from a distribution conditioned only on the instantaneous vorticity, with no dependence on the flow's history or on unobserved variables, so in flows where unresolved dynamics carry long temporal correlations or hidden degrees of freedom the learned distribution would be misspecified and the reported accuracy would not transfer.

Editorial extensions

If this is right

  • The measured cost of a 1000-member closure ensemble for the 2-D Kolmogorov flow drops from about 2.64 seconds of pure generation for the physical-space model to about 0.49 seconds for the latent model, and the full ensemble simulation from roughly 8,650 to 1,240 seconds, the ~10x acceleration the paper reports in Section 3.3.
  • Because the framework is written for the general partially known system $\partial V/\partial t = \mathcal{M}(V) + \mathcal{U}(V)$ with the closure drawn from $p(H|\omega)$, the same joint-trained latent pipeline applies to other PDE systems with stochastic, non-local unknown terms, not only to the tested vorticity equation.
  • Each closure evaluation is a full-field random draw, so a 1000-member ensemble delivers both a mean correction and a spread that serves as a subgrid uncertainty estimate without extra machinery.
  • The score network's Fourier neural operator backbone keeps the latent model in the same architectural family as the resolution-invariant physical-space closure of [29], so the latent version inherits the route toward cross-discretization operation that the authors name as future work.

Reading between the lines

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

  • I read the two-phase failure as a geometric effect that likely generalizes beyond closure modeling: reconstruction-optimal encoders can arrange the target distribution so that its conditional score is hard to approximate, so any latent diffusion application with continuous fields as conditioning, such as super-resolution or spatiotemporal forecasting, could exhibit the same gap, and the Table 3 co
  • The memoryless assumption is the natural next pressure point: conditioning the sampler on a short history of $\omega$ would reveal how much temporal correlation the subgrid term actually carries, and I expect the gap between memoryless and memory-augmented closures to widen as the Reynolds number grows or forcing becomes more intermittent.
  • With the cost barrier lowered, a follow-up the authors do not report becomes feasible: checking whether the 1000-sample spread of generated closures brackets the true error of the resolved simulation across many initial conditions, turning the speedup into a quantitative statement about calibration of the model's uncertainty estimates.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 6 minor

Summary. The paper proposes a latent conditional diffusion model (L-CDM) for stochastic closure modeling of nonlinear dynamical systems, focusing on a 2D Kolmogorov flow. A convolutional autoencoder compresses 64x64 resolved-vorticity and closure fields to 16x16 latent representations, and a Fourier-neural-operator-based score model is trained in that latent space. The authors contrast conventional two-phase training (autoencoder first, diffusion second) with a joint training objective that combines reconstruction, score matching, and KL regularization. They report that jointly trained L-CDM matches the generation accuracy of a physical-space conditional diffusion model (P-CDM) while being substantially faster, and they integrate the model into ensemble simulations of the vorticity equation. The paper claims roughly 10x acceleration in ensemble simulation with comparable predictive accuracy.

Significance. If the claims held as stated, the paper would make a useful practical contribution: it demonstrates that latent-space diffusion can serve as a computationally cheaper stochastic closure while preserving accuracy, and it provides evidence that joint training of the autoencoder and score model improves generative fidelity relative to the conventional two-phase pipeline. The internal comparison in Tables 3 and 4 is coherent and the improvement of joint L-CDM over two-phase L-CDM is consistent across generation metrics. The paper also reports detailed architecture and hyperparameter choices. However, the central numerical claim is currently weakened by a direct inconsistency between the closure term defined in Section 3 and the closure term implemented in Appendix A.2, and the reported speedup in Table 5 (6.99x) does not match the repeated '~10x' claim. The absence of code or data, and the use of a single flow regime, further limit verification. The core idea is defensible, but the manuscript needs substantial clarification and revision before the claims can be accepted.

major comments (4)
  1. [Section 3.3 / Appendix A.2] There is a direct contradiction in what the learned model replaces. Section 3 defines the unknown closure as H(x,t) = -u·∇ω + βξ (Eq. 26) and Table 5 reports errors for this closure. Appendix A.2 states that the ground-truth correction term is G†(x,t) = ν∇²ω + 2βξ, and Eq. (A.7) rewrites the Crank-Nicolson update so that the surrogate model G replaces G†. Since Eq. (A.6) still computes the exact advection F_n = u·∇ω, the simulation may actually be validating a latent model for viscous-plus-stochastic forcing, not for the advection-plus-stochastic term that the paper claims to learn. This changes the difficulty of the learning problem and the meaning of the results. Please clarify which definition was used in the reported experiments, and release the solver code or data so the reader can determine which closure was actually sampled.
  2. [Section 3.3 / Table 5 / Abstract / Conclusion] The paper repeatedly claims a '~10x acceleration' (abstract, Section 3.3, Conclusion), but Table 5 reports ensemble simulation costs of 1236.87 s for joint L-CDM versus 8652.49 s for P-CDM, which is a factor of 6.99, not 10. In addition, the Introduction's bullet list claims 'orders-of-magnitude faster inference,' yet the generation times in Section 3.1 show only 5.5x acceleration (2.64 s vs. 0.48 s). The reported numbers do not support the stated magnitude of the speedup; please correct the claims or provide a consistent measurement that supports them.
  3. [Section 2.2.2 / Tables 3 and 4] The improvement of joint L-CDM over two-phase L-CDM is attributed to joint optimization, but the joint objective in Eq. (22) also introduces KL regularization (λ_KL) that is absent from the two-phase baseline. The paper does not provide an ablation that isolates the effect of joint optimization from the effect of latent-space regularization. A two-phase pipeline with a KL-regularized autoencoder would clarify whether the observed gain is due to the joint training itself or simply to preventing latent-space collapse. This distinction is load-bearing for the paper's main methodological claim.
  4. [Section 3 / Reproducibility] The experimental evidence is confined to one flow regime, and no code or data are provided. Given the closure-definition ambiguity raised above, the absence of an explicit solver specification makes the central result impossible to verify independently. At minimum, the authors should provide the numerical implementation (or a precise pseudocode of the integration loop) and, ideally, a second test case (e.g., different Reynolds number or forcing) to support the general claim of stochastic and non-local closure modeling.
minor comments (6)
  1. [Eq. (11) and Algorithm 1] The expression for the covariance Σ(τ) is ambiguous: it should read (σ^{2τ} - 1)/(2 ln σ) I, consistent with the perturbation used in Algorithm 1. The current typography '1/2 logσ' can be misread as (1/2) log σ times (σ^{2τ} - 1).
  2. [Table 5 footnote] The time-cost definitions are not fully clear: the footnote says per-sample cost is the average wall-clock time per trajectory and ensemble cost is the total time to generate and average 1000 samples, but it does not state whether the ensemble simulation runs one trajectory with 1000 samples at each evaluation or 1000 independent trajectories. Please specify explicitly, along with the number of score evaluations per trajectory.
  3. [Section 3.3] The text states that closure terms are generated every 5 physical timesteps, but it does not state the total number of closure evaluations per 20-second simulation. This number is needed to interpret the reported wall-clock times and to compare per-sample versus ensemble costs.
  4. [Figure 9 / Eq. (31)] Eq. (31) defines an energy spectrum E(k,t) from the Fourier transform of the closure field H, but Figure 9 shows energy spectra of the resolved vorticity. Please define the spectrum used in Figure 9 explicitly or clarify how Eq. (31) is adapted to the vorticity field.
  5. [Abstract and Introduction] Minor copyediting: the abstract contains the duplicated phrase 'for many real-world applications', and the phrase 'in physical spaces' should be singular 'in physical space'. The Introduction also states 'prohibitive computational inference cost limits practical applications for many real-world applications', which repeats the same wording.
  6. [Section 3.2 / Figure 5] The t-SNE caption states that the left panel shows 'Ten distinct clusters corresponding to different test trajectories', but the visual appears to contain more than ten distinguishable clusters. Please clarify how the ten trajectories are represented and how the colors are assigned.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the joint latent-diffusion closure is evaluated by out-of-sample rollout, and self-citations to the authors' prior work are not load-bearing.

full rationale

The paper's derivation chain is an empirical training-and-rollout loop rather than a formal reduction: the conditional closure distribution p(H|omega) is learned from paired snapshots, and the central claims in Section 3.3 are tested by integrating the trained models into a numerical solver on held-out trajectories (Table 5 reports per-sample and ensemble-mean errors over 1000 independent simulations). No fitted parameter is renamed as a prediction, and the reported 10x acceleration is a measured wall-clock comparison of the same sampling procedure in latent versus physical space. The only self-citations are to the authors' prior work [29], used to motivate the P-CDM baseline ('we implement a physical-space conditional diffusion model (P-CDM) following the methodology established in [29]') and to reference an adaptive time-stepping scheme and the score-matching equivalence; none of these citations supplies the paper's central conclusion, which rests on the paper's own Tables 3-5. There is an internal inconsistency worth noting as a correctness concern, not as circularity: Section 3 defines the unknown target as H = -u dot grad omega + beta xi, while Appendix A.2 states 'Recall that in Section 3, we have the ground truth of the correction term G-dagger(x,t) = nu Laplacian omega + 2 beta xi' and rewrites the solver update with G-dagger; this ambiguity affects which quantity was actually learned and simulated, but it does not make the empirical comparison circular. No load-bearing uniqueness theorem or ansatz is imported from the authors' prior work, and the numerical results are self-contained comparisons among models trained and evaluated under the same protocol.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central claim relies on standard score-matching theory, a Markovian-in-state closure assumption, representativeness of the training data, and several manually chosen loss weights. No new physical entities are introduced; the latent representation is a learned coordinate system, not a new physical object.

free parameters (3)
  • Loss weights lambda_H, lambda_omega, lambda_score, lambda_KL = 10, 0.1, 0.1, 0.01
    Selected via grid search on the validation set (Appendix B); the central accuracy comparison depends on these values.
  • VE SDE scale sigma and final time T = not reported
    The variance-exploding SDE requires sigma and T; simulation uses T=0.1 with 10 reverse steps, but sigma and the step count for Table 3 timings are not stated.
  • Latent resolution 16x16 = 16x16 (from 64x64 fields)
    Architecture choice for dimensionality reduction; affects both reconstruction error and diffusion difficulty, but is not tuned per metric.
assumptions (4)
  • standard math Equivalence between explicit score matching and denoising score matching (Eq. 13)
    Invoked from references [29,94] and used to define the training objective; not proved in this paper.
  • domain assumption The closure term is determined by the instantaneous resolved state V, so p(H|omega) needs no memory
    Section 2 Eq. (3)-(4) models U as a function of current V only; if subgrid closures are path-dependent, the conditional distribution is misspecified.
  • domain assumption Training data from 20-40 s of 90 trajectories at 64x64 are representative of the target regime
    Section 3; no evidence of generalization to other Reynolds numbers, resolutions, or forcing regimes.
  • ad hoc to paper KL regularization on the latent space prevents collapse and preserves diffusion stochasticity
    Section 2.2.2, Eq. (22)-(24); motivated by prior literature but no proof that the chosen strength suffices.

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Cite this review

Pith. "Pith review of Stochastic and Non-local Closure Modeling for Nonlinear Dynamical Systems via Latent Score-based Generative Models." pith.science (2026). https://pith.science/paper/2YWNU46M

@misc{pith2026250620771,
  author       = {Pith},
  title        = {Pith review of: Stochastic and Non-local Closure Modeling for Nonlinear Dynamical Systems via Latent Score-based Generative Models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2YWNU46M}},
  note         = {Machine review of arXiv:2506.20771}
}
read the original abstract

We propose a latent score-based generative AI framework for learning stochastic, non-local closure models and constitutive laws in nonlinear dynamical systems of computational mechanics. This work addresses a key challenge of modeling complex multiscale dynamical systems without a clear scale separation, for which numerically resolving all scales is prohibitively expensive, e.g., for engineering turbulent flows. While classical closure modeling methods leverage domain knowledge to approximate subgrid-scale phenomena, their deterministic and local assumptions can be too restrictive in regimes lacking a clear scale separation. Recent developments of diffusion-based stochastic models have shown promise in the context of closure modeling, but their prohibitive computational inference cost limits practical applications in many real-world settings. This work addresses this limitation by jointly training convolutional autoencoders with conditional diffusion models in latent space, significantly reducing the dimensionality of the sampling process while preserving essential physical characteristics. Numerical results demonstrate that the joint training approach helps discover a proper latent space that not only guarantees small reconstruction errors but also ensures good performance of the diffusion model in the latent space. When integrated into numerical simulations, the proposed stochastic modeling framework via latent conditional diffusion models achieves significant computational acceleration while maintaining comparable predictive accuracy to standard diffusion models in physical space.

Figures

Figures reproduced from arXiv: 2506.20771 by the authors.

Figure 1
Figure 1. A schematic diagram of the proposed stochastic closure framework via latent conditional diffusion models. High-dimensional spatiotemporal fields (i.e., U and V ) are first encoded into latent spaces. A conditional diffusion model constructed via Fourier neural operators is then trained to capture the underlying probability distribution p(z U |z V ), where z U and z V denote the latent representations of the closure … view at source ↗
Figure 2
Figure 2. Physical-space conditional generation at 64 × 64 resolution. Top row: Ground-truth closure terms H(x, t). Middle row: Generated samples Hˆ (x, t) from the P-CDM. Bottom row: Absolute error fields |H − Hˆ | [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Generated samples from the latent space conditional diffusion model using conventional two-phase training. First row: Encoded ground truth latent representations z H. Second row: Generated latent samples zˆ H. Third row: Ground truth closure terms H. Fourth row: Decoded closure terms Hˆ . Fifth row: Absolute error fields |H − Hˆ |. 15 [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: Generation results with jointly trained models. First row: Encoded ground truth z H. Second row: Generated latent samples zˆ H. Third row: Ground truth closure terms H. Fourth row: Decoded samples Hˆ . Fifth row: Absolute error fields |H − Hˆ |. Joint training produces…
Figure 5
Figure 5. Figure 5: t-SNE visualization of (H, ω) pairs from ground truth and generated samples. Left: Ten distinct clusters corresponding to different test trajectories. Right: One representative cluster corresponding to one test trajectory. low wavenumbers corresponding to large-scale c…
Figure 6
Figure 6. Figure 6: Energy spectrum analysis of different models. Left: Mean energy spectra of the ground-truth closure term H and of closure fields generated by P-CDM, two-phase L-CDM, and joint L-CDM. Right: Mean spectra of latent embeddings z H: encoder outputs from joint-trained (blue…
Figure 7
Figure 7. Figure 7: Temporal evolution of vorticity fields over a 20-second simulation period. First row: Ground truth vorticity ω from high-fidelity simulation. Second row: Simulated vorticity using joint L-CDM ensemble-mean closure. Third row: Simulated vorticity with closure terms negl…
Figure 8
Figure 8. Figure 8: Temporal evolution of simulation errors for different closure modeling strategies. Left: Mean squared error (DMSE). Right: Relative Frobenius norm error (DRE). All closure models significantly outperform the uncorrected simulation, with ensemble strategies showing cons…
Figure 9
Figure 9. Figure 9: further compares the energy spectrum of the resolved vorticity among the ground truth simulation and the reduced-order model (ROM) with closure H generated with P-CDM and Joint L-CDM at various times between 30s and 50s. It is evident that the simulations incorporating…

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Reference graph

Works this paper leans on

103 extracted references · 62 canonical work pages

  1. [1]

    P. Moin, J. Kim, Tackling turbulence with supercomputers, Scientific American 276 (1997) 62–68

    1997

  2. [2]

    P. Moin, K. Mahesh, Direct numerical simulation: A tool in turbulence research, Annual review of fluid mechanics 30 (1998) 539–578

    1998

  3. [3]

    Duraisamy, G

    K. Duraisamy, G. Iaccarino, H. Xiao, Turbulence modeling in the age of data, Annual review of fluid mechanics 51 (2019) 357–377

    2019

  4. [4]

    G. A. Holzapfel, Nonlinear solid mechanics: a continuum approach for engineering science, 2002. 23

    2002

  5. [5]

    J. N. Fuhg, G. Anantha Padmanabha, N. Bouklas, B. Bahmani, W. Sun, N. N. Vlassis, M. Flaschel, P. Carrara, L. De Lorenzis, A review on data-driven constitutive laws for solids, Archives of Computational Methods in Engineering (2024) 1–43

    2024

  6. [6]

    Majda, X

    A. Majda, X. Wang, Nonlinear dynamics and statistical theories for basic geophysical flows, Cambridge University Press, 2006

    2006

  7. [7]

    Schneider, S

    T. Schneider, S. Lan, A. Stuart, J. Teixeira, Earth system modeling 2.0: A blueprint for models that learn from observations and targeted high-resolution simulations, Geo- physical Research Letters 44 (2017) 12–396

    2017

  8. [8]

    J. F. Donges, Y. Zou, N. Marwan, J. Kurths, Complex networks in climate dynamics: Comparinglinearandnonlinearnetworkconstructionmethods, TheEuropeanPhysical Journal Special Topics 174 (2009) 157–179

    2009

Show all 103 references
  1. [9]

    C.-Y. Lai, P. Hassanzadeh, A. Sheshadri, M. Sonnewald, R. Ferrari, V. Balaji, Machine learning for climate physics and simulations, Annual Review of Condensed Matter Physics 16 (2024)

    2024

  2. [10]

    Schneider, L

    T. Schneider, L. R. Leung, R. C. Wills, Opinion: Optimizing climate models with process-knowledge, resolution, and AI, EGUsphere 2024 (2024) 1–26

    2024

  3. [11]

    Jost, Dynamical systems: examples of complex behaviour, Springer Science & Busi- ness Media, 2005

    J. Jost, Dynamical systems: examples of complex behaviour, Springer Science & Busi- ness Media, 2005

    2005

  4. [12]

    E. M. Izhikevich, Dynamical systems in neuroscience, MIT press, 2007

    2007

  5. [13]

    B. E. Launder, D. B. Spalding, The numerical computation of turbulent flows, in: Numerical prediction of flow, heat transfer, turbulence and combustion, Elsevier, 1983, pp. 96–116

    1983

  6. [14]

    D. C. Wilcox, et al., Turbulence modeling for CFD, volume 2, DCW industries La Canada, CA, 1998

    1998

  7. [15]

    Smagorinsky, General circulation experiments with the primitive equations: I

    J. Smagorinsky, General circulation experiments with the primitive equations: I. the basic experiment, Monthly weather review 91 (1963) 99–164

    1963

  8. [16]

    J. W. Deardorff, A numerical study of three-dimensional turbulent channel flow at large Reynolds numbers, Journal of Fluid Mechanics 41 (1970) 453–480

    1970

  9. [17]

    Arnold, Stochastic differential equations: Theory and applications, (No Title) (1974)

    L. Arnold, Stochastic differential equations: Theory and applications, (No Title) (1974)

    1974

  10. [18]

    A. Y. Klimenko, R. W. Bilger, Conditional moment closure for turbulent combustion, Progress in energy and combustion science 25 (1999) 595–687

    1999

  11. [19]

    Part I: Model description and testing, Journal of Applied Meteorology and Climatology 46 (2007) 1383–1395

    J.E.Pleim, Acombinedlocalandnonlocalclosuremodelfortheatmosphericboundary layer. Part I: Model description and testing, Journal of Applied Meteorology and Climatology 46 (2007) 1383–1395. 24

    2007

  12. [20]

    A. J. Chorin, F. Lu, Discrete approach to stochastic parametrization and dimension reduction in nonlinear dynamics, Proceedings of the National Academy of Sciences 112 (2015) 9804–9809

    2015

  13. [21]

    X.-H. Zhou, J. Han, H. Xiao, Learning nonlocal constitutive models with neural net- works, Computer Methods in Applied Mechanics and Engineering 384 (2021) 113927

    2021

  14. [22]

    Chen, Stochastic methods for modeling and predicting complex dynamical systems, Springer, 2023

    N. Chen, Stochastic methods for modeling and predicting complex dynamical systems, Springer, 2023

    2023

  15. [23]

    J.-L. Wu, M. E. Levine, T. Schneider, A. Stuart, Learning about structural errors in models of complex dynamical systems, Journal of Computational Physics 513 (2024) 113157

    2024

  16. [24]

    Chen, J.-L

    C. Chen, J.-L. Wu, Neural dynamical operator: Continuous spatial-temporal model with gradient-based and derivative-free optimization methods, Journal of Computa- tional Physics 520 (2025) 113480

    2025

  17. [25]

    J. Sun, Z. Zhang, H. Schaeffer, Lemon: Learning to learn multi-operator networks, arXiv preprint arXiv:2408.16168 (2024)

    2024 arXiv

  18. [26]

    H. You, Y. Yu, S. Silling, M. D’Elia, Nonlocal operator learning for homogenized models: From high-fidelity simulations to constitutive laws, Journal of Peridynamics and Nonlocal Modeling 6 (2024) 709–724

    2024

  19. [27]

    Y. Yu, N. Liu, F. Lu, T. Gao, S. Jafarzadeh, S. A. Silling, Nonlocal attention operator: Materializing hidden knowledge towards interpretable physics discovery, Advances in Neural Information Processing Systems 37 (2024) 113797–113822

    2024

  20. [28]

    X.-H. Zhou, J. Han, M. I. Zafar, E. M. Wolf, C. R. Schrock, C. J. Roy, H. Xiao, Neural operator-based super-fidelity: A warm-start approach for accelerating steady- state simulations, Journal of Computational Physics 529 (2025) 113871

    2025

  21. [29]

    X. Dong, C. Chen, J.-L. Wu, Data-driven stochastic closure modeling via conditional diffusion model and neural operator, Journal of Computational Physics (2025) 114005

    2025

  22. [30]

    Sanderse, P

    B. Sanderse, P. Stinis, R. Maulik, S. E. Ahmed, Scientific machine learning for closure models in multiscale problems: A review, 2025. URL:https://www.aimsciences. org/article/id/6705073e8dfec03d48097326. doi:10.3934/fods.2024043

    2025

  23. [31]

    LeCun, Y

    Y. LeCun, Y. Bengio, G. Hinton, Deep learning, nature 521 (2015) 436–444

    2015

  24. [32]

    Goodfellow, Y

    I. Goodfellow, Y. Bengio, A. Courville, Y. Bengio, Deep learning, volume 1, MIT press Cambridge, 2016

    2016

  25. [33]

    Cristianini, J

    N. Cristianini, J. Shawe-Taylor, An introduction to support vector machines and other kernel-based learning methods, Cambridge university press, 2000

    2000

  26. [34]

    L. P. Kaelbling, M. L. Littman, A. W. Moore, Reinforcement learning: A survey, Journal of artificial intelligence research 4 (1996) 237–285. 25

    1996

  27. [35]

    Baker, F

    N. Baker, F. Alexander, T. Bremer, A. Hagberg, Y. Kevrekidis, H. Najm, M. Parashar, A. Patra, J. Sethian, S. Wild, et al., Workshop report on basic research needs for sci- entific machine learning: Core technologies for artificial intelligence, Technical Report, USDOE Office o...

    2019

  28. [36]

    K. J. Bergen, P. A. Johnson, M. V. de Hoop, G. C. Beroza, Machine learning for data-driven discovery in solid earth geoscience, Science 363 (2019) eaau0323

    2019

  29. [37]

    Carleo, I

    G. Carleo, I. Cirac, K. Cranmer, L. Daudet, M. Schuld, N. Tishby, L. Vogt-Maranto, L. Zdeborová, Machine learning and the physical sciences, Reviews of Modern Physics 91 (2019) 045002

    2019

  30. [38]

    H. Wang, T. Fu, Y. Du, W. Gao, K. Huang, Z. Liu, P. Chandak, S. Liu, P. Van Katwyk, A.Deac, etal., Scientificdiscoveryintheageofartificialintelligence, Nature620(2023) 47–60

    2023

  31. [39]

    Maulik, O

    R. Maulik, O. San, A. Rasheed, P. Vedula, Subgrid modelling for two-dimensional turbulence using neural networks, Journal of Fluid Mechanics 858 (2019) 122–144

    2019

  32. [40]

    S. Rasp, M. S. Pritchard, P. Gentine, Deep learning to represent subgrid processes in climate models, Proceedings of the national academy of sciences 115 (2018) 9684–9689

    2018

  33. [41]

    S. H. Rudy, S. L. Brunton, J. L. Proctor, J. N. Kutz, Data-driven discovery of partial differential equations, Science advances 3 (2017) e1602614

    2017

  34. [42]

    Champion, B

    K. Champion, B. Lusch, J. N. Kutz, S. L. Brunton, Data-driven discovery of coordi- nates and governing equations, Proceedings of the National Academy of Sciences 116 (2019) 22445–22451

    2019

  35. [43]

    M. L. Gao, J. P. Williams, J. N. Kutz, Sparse identification of nonlinear dynamics and Koopman operators with shallow recurrent decoder networks, arXiv preprint arXiv:2501.13329 (2025)

    2025 arXiv

  36. [44]

    Wang, J.-L

    J.-X. Wang, J.-L. Wu, H. Xiao, Physics-informed machine learning approach for recon- structing Reynolds stress modeling discrepancies based on DNS data, Physical Review Fluids 2 (2017) 034603

    2017

  37. [45]

    Karpatne, G

    A. Karpatne, G. Atluri, J. H. Faghmous, M. Steinbach, A. Banerjee, A. Ganguly, S. Shekhar, N. Samatova, V. Kumar, Theory-guided data science: A new paradigm for scientific discovery from data, IEEE Transactions on knowledge and data engineering 29 (2017) 2318–2331

    2017

  38. [46]

    J.-L. Wu, H. Xiao, E. Paterson, Physics-informed machine learning approach for augmenting turbulence models: A comprehensive framework, Physical Review Fluids 3 (2018) 074602

    2018

  39. [47]

    Raissi, P

    M. Raissi, P. Perdikaris, G. E. Karniadakis, Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, Journal of Computational physics 378 (2019) 686–707. 26

    2019

  40. [48]

    L. Sun, H. Gao, S. Pan, J.-X. Wang, Surrogate modeling for fluid flows based on physics-constrained deep learning without simulation data, Computer Methods in Applied Mechanics and Engineering 361 (2020) 112732

    2020

  41. [49]

    R. Wang, R. Yu, Physics-guided deep learning for dynamical systems: A survey, arXiv preprint arXiv:2107.01272 (2021)

    2021 arXiv

  42. [50]

    Kashinath, M

    K. Kashinath, M. Mustafa, A. Albert, J. Wu, C. Jiang, S. Esmaeilzadeh, K. Aziz- zadenesheli, R. Wang, A. Chattopadhyay, A. Singh, et al., Physics-informed machine learning: Case studies for weather and climate modelling, Philosophical Transactions of the Royal Society A 379 (2...

    2021

  43. [51]

    G. E. Karniadakis, I. G. Kevrekidis, L. Lu, P. Perdikaris, S. Wang, L. Yang, Physics- informed machine learning, Nature Reviews Physics 3 (2021) 422–440

    2021

  44. [52]

    Willard, X

    J. Willard, X. Jia, S. Xu, M. Steinbach, V. Kumar, Integrating scientific knowledge with machine learning for engineering and environmental systems, ACM Computing Surveys 55 (2022) 1–37

    2022

  45. [53]

    P. J. Baddoo, B. Herrmann, B. J. McKeon, J. Nathan Kutz, S. L. Brunton, Physics- informed dynamic mode decomposition, Proceedings of the Royal Society A 479 (2023) 20220576

    2023

  46. [54]

    Sharma, W

    P. Sharma, W. T. Chung, B. Akoush, M. Ihme, A review of physics-informed machine learning in fluid mechanics, Energies 16 (2023) 2343

    2023

  47. [55]

    R. Yu, R. Wang, Learning dynamical systems from data: An introduction to physics- guided deep learning, Proceedings of the National Academy of Sciences 121 (2024) e2311808121

    2024

  48. [56]

    A. Beck, D. Flad, C.-D. Munz, Deep neural networks for data-driven LES closure models, Journal of Computational Physics 398 (2019) 108910

    2019

  49. [57]

    Rezende, S

    D. Rezende, S. Mohamed, Variational inference with normalizing flows, in: Interna- tional conference on machine learning, PMLR, 2015, pp. 1530–1538

    2015

  50. [58]

    J. Ho, A. Jain, P. Abbeel, Denoising diffusion probabilistic models, Advances in neural information processing systems 33 (2020) 6840–6851

    2020

  51. [59]

    Y. Song, S. Ermon, Generative modeling by estimating gradients of the data distribu- tion, Advances in neural information processing systems 32 (2019)

    2019

  52. [60]

    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

  53. [61]

    Rombach, A

    R. Rombach, A. Blattmann, D. Lorenz, P. Esser, B. Ommer, High-resolution image synthesis with latent diffusion models, in: Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, 2022, pp. 10684–10695. 27

    2022

  54. [62]

    Z. Kong, W. Ping, J. Huang, K. Zhao, B. Catanzaro, Diffwave: A versatile diffusion model for audio synthesis, arXiv preprint arXiv:2009.09761 (2020)

    2020 arXiv

  55. [63]

    Saharia, W

    C. Saharia, W. Chan, S. Saxena, L. Li, J. Whang, E. L. Denton, K. Ghasemipour, R. Gontijo Lopes, B. Karagol Ayan, T. Salimans, et al., Photorealistic text-to-image diffusion models with deep language understanding, Advances in neural information processing systems 35 (2022) 36...

    2022

  56. [64]

    Dhariwal, A

    P. Dhariwal, A. Nichol, Diffusion models beat GANs on image synthesis, Advances in neural information processing systems 34 (2021) 8780–8794

    2021

  57. [65]

    A. Q. Nichol, P. Dhariwal, Improved denoising diffusion probabilistic models, in: International conference on machine learning, PMLR, 2021, pp. 8162–8171

    2021

  58. [66]

    Stinis, T

    P. Stinis, T. Hagge, A. M. Tartakovsky, E. Yeung, Enforcing constraints for interpola- tion and extrapolation in generative adversarial networks, Journal of Computational Physics 397 (2019) 108844

    2019

  59. [67]

    D. J. Gagne, H. M. Christensen, A. C. Subramanian, A. H. Monahan, Machine learn- ing for stochastic parameterization: Generative adversarial networks in the Lorenz’96 model, Journal of Advances in Modeling Earth Systems 12 (2020) e2019MS001896

    2020

  60. [68]

    L. Yang, D. Zhang, G. E. Karniadakis, Physics-informed generative adversarial net- works for stochastic differential equations, SIAM Journal on Scientific Computing 42 (2020) A292–A317

    2020

  61. [69]

    J.-L. Wu, K. Kashinath, A. Albert, D. Chirila, H. Xiao, et al., Enforcing statistical constraints in generative adversarial networks for modeling chaotic dynamical systems, Journal of Computational Physics 406 (2020) 109209

    2020

  62. [70]

    Stinis, C

    P. Stinis, C. Daskalakis, P. J. Atzberger, Sdyn-gans: Adversarial learning methods for multistep generative models for general order stochastic dynamics, Journal of Computational Physics 519 (2024) 113442

    2024

  63. [71]

    X. Fan, D. Akhare, J.-X. Wang, Neural differentiable modeling with diffusion-based super-resolution for two-dimensional spatiotemporal turbulence, Computer Methods in Applied Mechanics and Engineering 433 (2025) 117478

    2025

  64. [72]

    M. Xu, L. Yu, Y. Song, C. Shi, S. Ermon, J. Tang, Geodiff: A geometric diffusion model for molecular conformation generation, arXiv preprint arXiv:2203.02923 (2022)

    2022 arXiv

  65. [73]

    Y. Song, L. Shen, L. Xing, S. Ermon, Solving inverse problems in medical imaging with score-based generative models, arXiv preprint arXiv:2111.08005 (2021)

    2021 arXiv

  66. [74]

    L. Li, R. Carver, I. Lopez-Gomez, F. Sha, J. Anderson, Generative emulation of weather forecast ensembles with diffusion models, Science Advances 10 (2024) eadk4489. 28

    2024

  67. [75]

    X.-Y. Liu, M. H. Parikh, X. Fan, P. Du, Q. Wang, Y.-F. Chen, J.-X. Wang, Confild- inlet: Synthetic turbulence inflow using generative latent diffusion models with neural fields, arXiv preprint arXiv:2411.14378 (2024)

    2024 arXiv

  68. [76]

    T. Li, A. S. Lanotte, M. Buzzicotti, F. Bonaccorso, L. Biferale, Multi-scale recon- struction of turbulent rotating flows with generative diffusion models, Atmosphere 15 (2023) 60

    2023

  69. [77]

    D. Shu, Z. Li, A. B. Farimani, A physics-informed diffusion model for high-fidelity flow field reconstruction, Journal of Computational Physics 478 (2023) 111972

    2023

  70. [78]

    H. Wen, Y. Lin, Y. Xia, H. Wan, Q. Wen, R. Zimmermann, Y. Liang, Diffstg: Proba- bilistic spatio-temporal graph forecasting with denoising diffusion models, in: Proceed- ings of the 31st ACM International Conference on Advances in Geographic Information Systems, 2023, pp. 1–12

    2023

  71. [79]

    Rasul, C

    K. Rasul, C. Seward, I. Schuster, R. Vollgraf, Autoregressive denoising diffusion models for multivariate probabilistic time series forecasting, in: International conference on machine learning, PMLR, 2021, pp. 8857–8868

    2021

  72. [80]

    Tashiro, J

    Y. Tashiro, J. Song, Y. Song, S. Ermon, Csdi: Conditional score-based diffusion models for probabilistic time series imputation, Advances in neural information processing systems 34 (2021) 24804–24816

    2021

  73. [81]

    Zhuang, S

    Y. Zhuang, S. Cheng, K. Duraisamy, Spatially-aware diffusion models with cross- attention for global field reconstruction with sparse observations, Computer Methods in Applied Mechanics and Engineering 435 (2025) 117623

    2025

  74. [82]

    Jacobsen, Y

    C. Jacobsen, Y. Zhuang, K. Duraisamy, Cocogen: Physically consistent and condi- tioned score-based generative models for forward and inverse problems, SIAM Journal on Scientific Computing 47 (2025) C399–C425

    2025

  75. [83]

    Z. Li, N. Kovachki, K. Azizzadenesheli, B. Liu, K. Bhattacharya, A. Stuart, A. Anand- kumar, Fourier neural operator for parametric partial differential equations, arXiv preprint arXiv:2010.08895 (2020)

    2020 arXiv

  76. [84]

    C. Lu, Y. Zhou, F. Bao, J. Chen, C. Li, J. Zhu, Dpm-solver: A fast ode solver for diffusion probabilistic model sampling in around 10 steps, Advances in Neural Information Processing Systems 35 (2022) 5775–5787

    2022

  77. [85]

    L. Liu, Y. Ren, Z. Lin, Z. Zhao, Pseudo numerical methods for diffusion models on manifolds, arXiv preprint arXiv:2202.09778 (2022)

    2022 arXiv

  78. [86]

    Y. Song, P. Dhariwal, M. Chen, I. Sutskever, Consistency models, in: Proceedings of the 40th International Conference on Machine Learning, ICML’23, JMLR.org, 2023

    2023

  79. [87]

    Salimans, J

    T. Salimans, J. Ho, Progressive distillation for fast sampling of diffusion models, in: International Conference on Learning Representations, 2022. 29

    2022

  80. [88]

    Luhman, T

    E. Luhman, T. Luhman, Knowledge distillation in iterative generative models for improved sampling speed, arXiv preprint arXiv:2101.02388 (2021)

    2021 arXiv

  81. [89]

    Karras, M

    T. Karras, M. Aittala, T. Aila, S. Laine, Elucidating the design space of diffusion- basedgenerativemodels, AdvancesinNeuralInformationProcessingSystems35(2022) 26565–26577

    2022

  82. [90]

    Z. Kong, W. Ping, On fast sampling of diffusion probabilistic models, arXiv preprint arXiv:2106.00132 (2021)

    2021 arXiv

  83. [91]

    Vahdat, K

    A. Vahdat, K. Kreis, J. Kautz, Score-based generative modeling in latent space, Ad- vances in neural information processing systems 34 (2021) 11287–11302

    2021

  84. [92]

    Esser, R

    P. Esser, R. Rombach, B. Ommer, Taming transformers for high-resolution image syn- thesis, in: Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, 2021, pp. 12873–12883

    2021

  85. [93]

    Kingma, T

    D. Kingma, T. Salimans, B. Poole, J. Ho, Variational diffusion models, Advances in neural information processing systems 34 (2021) 21696–21707

    2021

  86. [94]

    Vincent, A connection between score matching and denoising autoencoders, Neural Computation 23 (2011) 1661–1674

    P. Vincent, A connection between score matching and denoising autoencoders, Neural Computation 23 (2011) 1661–1674

    2011

  87. [95]

    H. Liu, Z. Chen, Y. Yuan, X. Mei, X. Liu, D. Mandic, W. Wang, M. D. Plumbley, Audioldm: Text-to-Audio generation with latent diffusion models, in: International Conference on Machine Learning, PMLR, 2023, pp. 21450–21474

    2023

  88. [96]

    Takagi, S

    Y. Takagi, S. Nishimoto, High-resolution image reconstruction with latent diffusion models from human brain activity, in: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2023, pp. 14453–14463

    2023

  89. [97]

    Podell, Z

    D. Podell, Z. English, K. Lacey, A. Blattmann, T. Dockhorn, J. Müller, J. Penna, R. Rombach, SDXL: Improving latent diffusion models for high-resolution image syn- thesis, in: The Twelfth International Conference on Learning Representations, 2024. URL:https://openreview.net/fo...

    2024

  90. [98]

    Y. Li, Y. Cheng, M. van der Schaar, A study of posterior stability for time-series latent diffusion, arXiv preprint arXiv:2405.14021 (2024)

    2024 arXiv

  91. [99]

    Nguyen, C.-H

    B. Nguyen, C.-H. Lai, Y. Takida, N. Murata, T. Uesaka, S. Ermon, Y. Mitsufuji, Im- proving vector-quantized image modeling with latent consistency-matching diffusion, in: ICLR 2025 Workshop on Deep Generative Model in Machine Learning: Theory, Principle and Efficacy, 2025. URL...

    2025

  92. [100]

    Dieleman, L

    S. Dieleman, L. Sartran, A. Roshannai, N. Savinov, Y. Ganin, P. H. Richemond, A. Doucet, R. Strudel, C. Dyer, C. Durkan, et al., Continuous diffusion for categorical data, arXiv preprint arXiv:2211.15089 (2022). 30

    2022 arXiv

  93. [101]

    4664–4683

    Z.Gao, J.Guo, X.Tan, Y.Zhu, F.Zhang, J.Bian, L.Xu, Empoweringdiffusionmodels on the embedding space for text generation, in: NAACL-HLT, 2024, pp. 4664–4683. URL:https://doi.org/10.18653/v1/2024.naacl-long.261

    2024 doi

  94. [102]

    Vahdat, J

    A. Vahdat, J. Kautz, Nvae: A deep hierarchical variational autoencoder, Advances in neural information processing systems 33 (2020) 19667–19679

    2020

  95. [103]

    Tancik, P

    M. Tancik, P. Srinivasan, B. Mildenhall, S. Fridovich-Keil, N. Raghavan, U. Singhal, R. Ramamoorthi, J. Barron, R. Ng, Fourier features let networks learn high frequency functions in low dimensional domains, Advances in neural information processing systems 33 (2020) 7537–7547...

    2020

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