Pith. sign in

REVIEW 2 major objections 6 minor 48 references

Adjoint-based online learning of two-layer quasi-geostrophic baroclinic turbulence

T0 review · 2 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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

Pith's one-line read Online training, with the ocean model inside the learning loop, produces more accurate and stable ML subgrid closures than offline regression; the full adjoint-based version eliminates grid-scale energy accumulation and stays stable…

desk verdict Plausible and mostly well-supported extension of online learning to two-layer QG, but the headline stability-without-constraint claim rests on unshown runs and a removal policy that could bias them. read the letter →

arxiv 2411.14106 v1 pith:N7DMYMJR submitted 2024-11-21 physics.ao-ph cs.LGphysics.flu-dyn

Fei Er Yan , Hugo Frezat , Julien Le Sommer , Julian Mak , Karl Otness This is my paper · ORCID
classification physics.ao-phcs.LGphysics.flu-dyn
keywords onlinelearningadjointsubgridparameterizationbaroclinicturbulencequasi-geostrophicmodelmachineoceaneddydifferentiable
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 argues that machine-learned subgrid parameterizations for ocean turbulence should be trained with the fluid model inside the training loop, rather than as a static regression on precomputed data. In a two-layer quasi-geostrophic baroclinic turbulence test, the full adjoint-based online procedure produces CNN closures that are more accurate and more stable than offline-trained ones. These online-trained closures match the high-resolution kinetic energy spectrum without the grid-scale energy accumulation, show better distribution agreement for the predicted subgrid forcing, and remain stable even when the zero-mean potential-vorticity hard constraint is removed. An approximately online variant that truncates the back-propagation inherits some but not all of these benefits and does not require a differentiable model. If the claims hold, online training offers a reliable and computationally modest route to building ML closures for ocean and climate models, since a training window of only ten time steps suffices.

What carries the argument

The argument turns on a hybrid dynamical model: a low-resolution two-layer quasi-geostrophic solver that supports algorithmic differentiation, coupled to convolutional neural networks that predict the sub-grid forcing $S_q$, defined as the difference between the coarse-grained advective potential-vorticity tendency and the advective tendency computed from the coarse-grained fields. Training minimizes, over a rolling window of $K$ time steps, the squared error between predicted and diagnosed $S_q$ with separate losses per layer, and the online gradient is obtained by back-propagating through the differentiable solver via the adjoint. All networks are trained with a zero-mean (zero net PV tendency) constraint; the full online model is shown to be the one that does not need this constraint for stability.

What would settle it

Re-run the eddy-regime experiment with a larger number of seeds (e.g., 30) per training strategy, explicitly counting failed training runs and integrating them into the averaged scores; if offline or approximately online models do not show higher failure rates, or if the full online model crashes for some seeds, the claimed stability advantage is not robust. A sharper test: train the full online model without the zero-mean PV hard constraint at longer windows (e.g., K=20, 40); if the grid-scale kick-back reappears or the model becomes unstable, the attribution of stability to the online procedure rather than to window size or the constraint would be undercut.

Watch

Extended reading notes

Core claim

The paper's central claim is that the training procedure itself, not the network architecture or the data, determines the prognostic skill and stability of a learned subgrid closure. In the two-layer quasi-geostrophic 'eddy' benchmark, the full adjoint-based online approach (loss defined over a trajectory of the hybrid fluid–ML model, back-propagated end-to-end through a differentiable solver) removes the grid-scale kinetic energy kick-back that offline models robustly display, makes the predicted subgrid potential-vorticity forcing closer in distribution to the high-resolution truth, and stays stable when the zero-mean potential-vorticity constraint is dropped—while offline and approximately online models accumulate top-layer potential vorticity and blow up. The full online model also retains some skill out-of-sample when explicit small-scale dissipation is switched off. These benefits are achieved with a training window of just ten time steps, so the additional cost over offline training is small.

Load-bearing premise

The stability comparison assumes that removing failed ensemble members from the score calculations does not bias the results; if offline or approximately online models fail more often and are excluded, the reported stability advantage of full online training could be inflated.

Editorial extensions

If this is right

  • If the result carries to operational ocean models, ML subgrid closures can be trained to be stable in prognostic use without ad hoc constraint terms, removing a major obstacle to deployment.
  • The full online approach removes the grid-scale kinetic energy kick-back that marks unresolved energy accumulation, implying longer and more faithful integrations at coarse resolution.
  • A training window of ten time steps is enough to obtain these benefits, so the computational overhead of online training is modest and feasible for larger models.
  • The approximately online variant offers a practical route for existing non-differentiable model codes, inheriting improved energy spectra and fluxes though not the stability without the hard constraint.
  • Because the full online model keeps skill when explicit small-scale dissipation is removed, it learns a more complete representation of sub-grid processes, requiring less dissipative tuning afterward.

Reading between the lines

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

  • One testable extension is whether online training also removes the need for other constraints (e.g., energy conservation) in more realistic primitive-equation models, since the stability appears to arise from the trajectory-based loss rather than from any specific constraint term.
  • The paper's comparison could be extended to derivative-free online methods, such as ensemble Kalman inversion; if such methods match the full-online stability, the requirement of a differentiable model would become less binding.
  • The observation that the approximate online model needs the zero-mean constraint while the full model does not suggests that truncating back-propagation removes precisely the gradient information that prevents top-layer potential-vorticity accumulation; this could be diagnosed by comparing learned sensitivity maps.
  • The finding that offline training cannot tune away the kick-back while online training removes it hints that the stability property is structural to trajectory-based training rather than a hyper-parameter effect; testing across other closure architectures in the same benchmark would clarify this.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 6 minor

Summary. An empirical comparison of offline, full adjoint-based online, and approximately online training for a CNN subgrid parameterization in a two-layer quasi-geostrophic model, following the Ross et al. (2023) benchmark in the eddy-dominated regime. Ten-member ensembles are trained using the same architecture and zero-mean PV hard constraint, then evaluated a priori and a posteriori via Q-Q distribution plots, kinetic energy spectra, spectral energy budgets, and similarity scores. The paper reports that full online training removes the grid-scale energy kick-back, gives better distribution agreement and spectral fluxes, remains stable without the hard constraint, and that the approximate online method inherits some but not all of these benefits.

Significance. The comparison is carefully constructed and largely non-circular: skill is measured on held-out a posteriori statistics (KE spectra, energy fluxes, similarity scores) that are not training objectives. The use of ten-member ensembles, a common architecture, and the public pyqg-JAX/Equinox stack strengthens reproducibility, and the paper provides code and data access. If the stability claims are confirmed, the paper offers concrete evidence that adjoint-based online training is a viable route for ocean eddy closures and quantifies what the approximate online method gives up. The main risk is that the strongest mechanistic conclusion depends on omitted no-constraint experiments, so the empirical core is sound but the headline requires additional reporting.

major comments (2)
  1. [Sec. 3.3, Sec. 5, Sec. 2.3] The central stability claim—that full online training removes the need for the zero-mean PV hard constraint of Eq. (9)—is asserted with '(not shown)' in Sec. 3.3 and repeated in Sec. 5. Section 2.3 states that any ensemble member that fails is removed from score calculations, so in the no-constraint comparison the failure rate is precisely the outcome being measured. The text gives no per-seed survival counts, no blow-up times, and no quantitative q-accumulation values for offline, approximate online, and full online under the no-constraint condition. Without these numbers, the reader cannot rule out selection bias in the reported stability advantage. Please report the per-seed outcomes and the q-accumulation statistics, or explicitly qualify the conclusion as a preliminary observation.
  2. [Sec. 5] Analogous investigations in the jet regime are summarized only as '(not shown)' with claims of 'moderately positive similarity scores' and a resolution dependence. Since the conclusion states online learning is preferable 'over a wider range of conditions,' the jet-regime transfer is part of the paper's scope. Please include at least a compact summary (e.g., a table of similarity scores and stability counts for the jet regime, with and without the hard constraint), or restrict the conclusion to the eddy-dominated regime that is actually documented.
minor comments (6)
  1. [Abstract and key points] 'Others details' should be 'Other details'; the key-points bullet 'with the not requiring a differentiable model' is missing a noun, e.g., 'with the approximate approach not requiring a differentiable model.'
  2. [Sec. 1, Sec. 2.2, Sec. 2.3] Typos: 'a posteori' should be 'a posteriori' (Sec. 1); 'geostrohic' should be 'geostrophic' (Sec. 2.2); 'at least two orders of magnitude layer than' should be 'larger than' (Sec. 2.3).
  3. [Fig. 5] The ensemble standard deviations overlap substantially for many metrics, so the statement that 'it is generally the case that the online models further improve' would be strengthened by a paired significance test or at least a statement of how many of the ten members improve for each metric.
  4. [Eq. (14)] The mixed-loss weight α̃ is introduced but no values or normalization are reported for the mixed-loss experiments; please state the values used or note that they were scanned.
  5. [Sec. 4.1 and Fig. 6] The text says Fig. 6 is shown with a linear scale, but the axis labels are in scientific notation with uneven spacing; please clarify the axis scaling or revise the wording.
  6. [Sec. 2.3] The phrase 'the choice of seeds is uniform across all three ensembles' should be accompanied by the actual seed values or a repository pointer to the exact seeds, since the stability conclusions depend on these seeds.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the headline skill/stability claims are tested against external a posteriori metrics; the main caveat (unshown no-constraint runs) is an evidence gap, not a circular reduction.

full rationale

This is an empirical comparison, not a derivation, and I find no step in which a claimed prediction is equivalent by construction to its input. The loss functions (Eqs. 1–2, 8) target the sub-grid forcing Sq, whereas the headline claims are assessed with a posteriori kinetic-energy spectra (Fig. 3), spectral energy budgets (Fig. 4), and similarity scores (Eqs. 11–13), none of which appears in the training loss; the offline/online/approximate-online difference is therefore not forced by the objective. Self-citations to Frezat et al. (2022), Ouala et al. (2023), List et al. (2022, 2024), and the pyqg-JAX software (Otness et al., 2023; Otness, 2024) supply prior motivation, method definitions, and computational infrastructure, but the benchmark results are evaluated against held-out high-resolution statistics and would stand or fall independently of those works. The strongest caveat is evidentiary rather than circular: Sec. 3.3's claim that, without the hard constraint of Eq. (9), offline and approximately online models accumulate q and blow up while the full online model remains stable is reported only as '(not shown)', and Sec. 2.3's policy of removing failed ensemble members from score calculations would be load-bearing for exactly that comparison if any members had failed. However, for the constrained main results the paper explicitly states that no members crashed (Sec. 3), so the removal policy does not distort the reported headline scores. Missing figures and a potentially selection-prone removal policy for the unshown no-constraint experiments are correctness/evidence concerns, not circular reductions.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The paper's results depend on a benchmark and a set of hand-chosen hyperparameters, not on new physics. The main free parameters are the training window length and data scaling factors; the loss weights alpha and alpha_tilde are mentioned but not used in the headline comparison. No new physical entities are introduced.

free parameters (4)
  • training window size K = 10 time steps (curriculum from K=2 to K=10)
    Default window length chosen after exploring K=2,4,6,10; longer windows give little additional skill but cost more (Sec. 4.1).
  • data scaling factors = 1e13 for Sq, 1e6 for q
    Applied to compensate single-precision arithmetic in pyqg-JAX; authors state empirical testing suggests the choice is not crucial (Sec. 2.3).
  • combined loss weight alpha = not used in main results
    Parameter in Eq. (7); authors chose separate per-layer losses instead, noting subjectivity in choosing alpha (Sec. 2.3).
  • mixed loss weight alpha_tilde = not comprehensively tuned
    Weight in Eq. (14) for Sq plus q loss; only briefly explored with poor results (Sec. 4.3).
assumptions (5)
  • domain assumption The two-layer quasi-geostrophic equations with the specified parameters (Table 1) and spectral discretization faithfully represent baroclinic turbulence in the eddy regime.
    The high-resolution model is treated as the truth for target data and metrics; this is inherited from the benchmark setup of Ross et al. (2023).
  • domain assumption The sub-grid forcing Sq as defined in Eq. (6) and the zero-mean hard constraint (Eq. 9) are sufficient closure terms for the coarse-grained equations.
    The entire parameterization approach assumes that the residual advective PV tendency is the quantity to learn and that enforcing zero domain-mean PV tendency is physically justified.
  • domain assumption The CNN architecture used (from Ross et al., 2023) is expressive enough to represent the target mapping.
    No universality or capacity argument is provided; performance is empirical.
  • standard math pyqg-JAX (v0.8.1) computes the true numerical adjoint of the discretized two-layer QG model via automatic differentiation.
    The paper relies on the correctness of the AD implementation for the full online gradients; no independent verification of the adjoint correctness is given.
  • domain assumption Time windows of 10 steps are long enough to capture the benefits of online training while avoiding gradient instability.
    Authors note longer windows give little gain and this may be setup-specific.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Adjoint-based online learning of two-layer quasi-geostrophic baroclinic turbulence." pith.science (2026). https://pith.science/paper/N7DMYMJR

@misc{pith2026241114106,
  author       = {Pith},
  title        = {Pith review of: Adjoint-based online learning of two-layer quasi-geostrophic baroclinic turbulence},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/N7DMYMJR}},
  note         = {Machine review of arXiv:2411.14106}
}
read the original abstract

For reasons of computational constraint, most global ocean circulation models used for Earth System Modeling still rely on parameterizations of sub-grid processes, and limitations in these parameterizations affect the modeled ocean circulation and impact on predictive skill. An increasingly popular approach is to leverage machine learning approaches for parameterizations, regressing for a map between the resolved state and missing feedbacks in a fluid system as a supervised learning task. However, the learning is often performed in an `offline' fashion, without involving the underlying fluid dynamical model during the training stage. Here, we explore the `online' approach that involves the fluid dynamical model during the training stage for the learning of baroclinic turbulence and its parameterization, with reference to ocean eddy parameterization. Two online approaches are considered: a full adjoint-based online approach, related to traditional adjoint optimization approaches that require a `differentiable' dynamical model, and an approximately online approach that approximates the adjoint calculation and does not require a differentiable dynamical model. The online approaches are found to be generally more skillful and numerically stable than offline approaches. Others details relating to online training, such as window size, machine learning model set up and designs of the loss functions are detailed to aid in further explorations of the online training methodology for Earth System Modeling.

Figures

Figures reproduced from arXiv: 2411.14106 by the authors.

Figure 1
Figure 1. Snapshots of upper-layer perturbation potential vorticity q1 at the end of a ten year simulation. Data from the (a) high resolution model true, (b) low resolution model with no CNN active, (c) offline model, (d) online model (e) approximately online model. resolve the mesoscale eddies. We use a model grid with size 2562 , corresponding to ∆x = 3.9 km (with a time-step of ∆t = 1 hour) as a reference high resolution s… view at source ↗
Figure 2
Figure 2. Quantile-Quantile (Q-Q) plot, with the distribution of diagnosed target (the S q from high resolution filtered onto the coarse grid) on the x axis, against the predicted Sˆq gen￾erated by the CNNs on the y axis (green: offline model; orange: online (approx) model; blue: online model). (a, b) Results from a priori testing, and (c, d) a posteriori testing, for layer 1 and 2 respectively. The red line is the identity l… view at source ↗
Figure 3
Figure 3. Depth-averaged total kinetic energy spectra, normalized by n 2 x × n 2 y. All panels show the data from the target high resolution simulation (black line) and the low resolution simu￾lation with no CNN active (grey line). Results from the (a) the offline model (green line), (b) full online model (blue line), and (c) the approximately online model (orange line). The colored lines are from a ten member ensemble averag… view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Full spectral energy budget for the APE flux (blue line), APE generation from the imposed background shear (orange line), KE flux (green line), bottom friction (red line), dissipa￾tion from the spectral filter (purple line), and input from the CNNs (brown line). (a) Th…
Figure 5
Figure 5. Figure 5: Distribution similarity scores for potential vorticity q, both components of the velocity u and v, the streamfunction ψ, KE and enstrophy, depth-averaged KE flux, APE flux, APE generation, and the bottom drag (labeled as Friction). Subscripts 1 and 2 denote the upper a…
Figure 6
Figure 6. Figure 6: Zoomed in depth-averaged KE spectra (cf [PITH_FULL_IMAGE:figures/full_fig_p017_6.png]
Figure 7
Figure 7. Figure 7: Spectral fluxes for a posteriori calculations in a hybrid model with the explicit small-scale dissipation switched off, using the CNNs reported in Sec. 3. (a) The offline hybrid model. (b) The online hybrid model. (c) The approximately online hybrid model. The colored …
Figure 8
Figure 8. Figure 8: Diagnostics from the full online hybrid model, where the associated CNNs are trained with hybrid models without any explicit small-scale dissipation active. (a) KE spectrum as [PITH_FULL_IMAGE:figures/full_fig_p020_8.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

48 extracted references · 14 canonical work pages

  1. [1]

    , Pritchard, M

    Beucler-et-al21 APACrefauthors Beucler, T. , Pritchard, M. , Rasp, S. , Ott, J. , Baldi, P. \ Gentine, P. APACrefauthors \ 2021 . Enforcing analytic constraints in neural networks emulating physical systems Enforcing analytic constraints in neural networks emulating physical systems . Phys. Rev. Lett. 126 098302 . APACrefDOI doi:10.1103/PhysRevLett.126.09...

    doi:10.1103/physrevlett.126.098302 2021

  2. [2]

    \ Zanna, L

    BoltonZanna19 APACrefauthors Bolton, T. \ Zanna, L. APACrefauthors \ 2019 . Applications of deep learning to ocean data inference and sub-grid parameterisation Applications of deep learning to ocean data inference and sub-grid parameterisation . J. Adv. Model. Earth Syst. 11 376--399 . APACrefDOI doi:10.1029/2018MS001472 APACrefDOI

    doi:10.1029/2018ms001472 2019

  3. [3]

    \ Bretherton, C S

    BrenowitzBretherton19 APACrefauthors Brenowitz, N D. \ Bretherton, C S. APACrefauthors \ 2019 . Spatially extended tests of a neural network parametrization trained by coarse-graining Spatially extended tests of a neural network parametrization trained by coarse-graining . J. Adv. Model. Earth. Syst. 11 2728--2744 . APACrefDOI doi:10.1029/2019MS001711 APACrefDOI

    doi:10.1029/2019ms001711 2019

  4. [4]

    \ Hassanzadeh, P

    ChattopadhyayHassanzadeh23 APACrefauthors Chattopadhyay, A. \ Hassanzadeh, P. APACrefauthors \ 2023 . Long-term instabilities of deep learning-based digital twins of the climate system: The cause and a solution. Long-term instabilities of deep learning-based digital twins of the climate system: The cause and a solution. APACrefURL https://arxiv.org/abs/23...

    arXiv 2023

  5. [5]

    , Rubanova, Y

    Chen-et-al18 APACrefauthors Chen, R T Q. , Rubanova, Y. , Bettencourt, J. \ Duvenaud, D K. APACrefauthors \ 2018 . Neural Ordinary Differential Equations Neural ordinary differential equations . S. Bengio, H. Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi \ R. Garnett\ ( ), Advances in Neural Information Processing Systems Advances in neural informat...

    work page 2018

  6. [6]

    , Barnes, E A

    Connolly-et-al23 APACrefauthors Connolly, C. , Barnes, E A. , Hassanzadeh, P. \ Pritchard, M. APACrefauthors \ 2023 . Using neural networks to learn the Jet Stream forced response from natural variability Using neural networks to learn the Jet Stream forced response from natural variability . Artif. Intell. Earth Syst. 2 e220094 . APACrefDOI doi:10.1175/A...

    work page doi:10.1175/aies-d-22-0094.1 2023

  7. [7]

    , Campin, J M

    Forget-et-al15a APACrefauthors Forget, G. , Campin, J M. , Heimbach, P. , Hill, C N. , Ponte, R M. \ Wunsch, C. APACrefauthors \ 2015 . ECCO version 4: an integrated framework for non-linear inverse modeling and global ocean state estimation ECCO version 4: an integrated framework for non-linear inverse modeling and global ocean state estimation . Geosci....

    doi:10.5194/gmd-8-3071-2015 2015

  8. [8]

    , Le Sommer , J

    Frezat-et-al22 APACrefauthors Frezat, H. , Le Sommer , J. , Fablet, R. , Balarac, G. \ Lguensat, R. APACrefauthors \ 2022 . A posteriori learning for quasi-geostrophic turbulence parametrization A posteriori learning for quasi-geostrophic turbulence parametrization . J. Adv. Model. Earth Syst. 14 11 e2022MS003124 . APACrefDOI doi:10.1029/2022MS003124 APACrefDOI

    doi:10.1029/2022ms003124 2022

Show all 48 references
  1. [9]

    , White, A

    Gelbrecht-et-al23 APACrefauthors Gelbrecht, M. , White, A. , Bathiany, S. \ Boers, N. APACrefauthors \ 2023 . Differentiable programming for Earth system modeling Differentiable programming for Earth system modeling . Geosci. Model Dev. 16 3123--3125 . APACrefDOI doi:10.5194/g...

    2023 doi

  2. [10]

    , Dzamba, M

    Greydanus-et-al19 APACrefauthors Greydanus, S. , Dzamba, M. \ Yosinski, J. APACrefauthors \ 2019 . Hamiltonian Neural Networks Hamiltonian neural networks . H. Wallach, H. Larochelle, A. Beygelzimer, F. d Alch\' e -Buc, E. Fox \ R. Garnett\ ( ), Advances in Neural Information ...

    2019

  3. [11]

    \ Zanna, L

    GuillauminZanna21 APACrefauthors Guillaumin, A P. \ Zanna, L. APACrefauthors \ 2021 . Stochastic-deep learning parameterization of ocean momentum forcing Stochastic-deep learning parameterization of ocean momentum forcing . J. Adv. Model. Earth Syst. 13 e2021MS002534 . APACref...

    2021 doi

  4. [12]

    APACrefauthors \ 2003

    Gunzburger-control APACrefauthors Gunzburger, M D. APACrefauthors \ 2003 . Perspectives in flow control and optimization Perspectives in flow control and optimization . SIAM

    2003

  5. [13]

    APACrefauthors \ 2013

    Hallberg13 APACrefauthors Hallberg, R. APACrefauthors \ 2013 . Using a resolution function to regulate parameterizations of oceanic mesoscale eddy effects Using a resolution function to regulate parameterizations of oceanic mesoscale eddy effects . Ocean Modell. 72 92--103 . A...

    2013 doi

  6. [14]

    Han-et-al19 APACrefauthors Han, J. , M, C. , Ma, Z. \ E, W. APACrefauthors \ 2019 . Uniformly accurate machine learning-based hydrodynamic models for kinetic equations Uniformly accurate machine learning-based hydrodynamic models for kinetic equations . Proc. Natl Acad. Sci. U...

    2019 doi

  7. [15]

    , Fukumori, I

    Heimbach-et-al19 APACrefauthors Heimbach, P. , Fukumori, I. , Hill, C N. , Ponte, R M. , Stammer, D. , Campin, J. Zhang, H. APACrefauthors \ 2019 . Putting it all together: Adding value to the global ocean and climate observing systems with complete self-consistent ocean state...

    2019

  8. [16]

    , Law, K J H

    Iglesias-et-al13 APACrefauthors Iglesias, M A. , Law, K J H. \ Stuart, A M. APACrefauthors \ 2013 . Ensemble Kalman methods for inverse problems Ensemble Kalman methods for inverse problems . Inverse Problems 29 4 045001 . APACrefDOI doi:10.1088/0266-5611/29/4/045001 APACrefDOI

    2013 doi

  9. [17]

    APACrefauthors \ 2021

    IPCC-AR6 APACrefauthors IPCC. APACrefauthors \ 2021 . Climate Change 2021: The Physical Science Basis. Contribution of Working Group I to the Sixth Assessment Report of the Intergovernmental Panel on Climate Change Climate Change 2021: The Physical Science Basis. Contribution ...

    2021 doi

  10. [18]

    \ Held, I M

    JansenHeld14 APACrefauthors Jansen, M F. \ Held, I M. APACrefauthors \ 2014 . Parameterizing subgrid-scale eddy effects using energetically consistent backscatter Parameterizing subgrid-scale eddy effects using energetically consistent backscatter . Ocean Modell. 80 36--48 . A...

    2014 doi

  11. [19]

    APACrefauthors \ 2002

    Kalnay-DA APACrefauthors Kalnay, E. APACrefauthors \ 2002 . Atmospheric Modeling, Data Assimilation and Predictability Atmospheric Modeling, Data Assimilation and Predictability \ ( 142). Cambridge University Press . APACrefDOI doi:10.1017/CBO9780511802270 APACrefDOI

    2002 doi

  12. [20]

    , Mustafa, M

    Kashinath-et-al21 APACrefauthors Kashinath, K. , Mustafa, M. , Albert, A. , Wu, J. , Jiang, C. , Esmaeilzadeh, S. Prabhat APACrefauthors \ 2021 . Physics-informed machine learning: case studies for weather and climate modelling Physics-informed machine learning: case studies f...

    2021

  13. [21]

    \ Garcia, C

    KidgerGarcia21 APACrefauthors Kidger, P. \ Garcia, C. APACrefauthors \ 2021 . Equinox: neural networks in JAX via callable PyTrees and filtered transformations Equinox: neural networks in JAX via callable PyTrees and filtered transformations . Differentiable Programming worksh...

    2021 doi arXiv

  14. [22]

    , Yuval, J

    Kochkov-et-al24 APACrefauthors Kochkov, D. , Yuval, J. , Langmore, I. , Norgaard, P. , Smith, J. , Mooers, G. Hoyer, S. APACrefauthors \ 2024 . Neural general circulation models for weather and climate Neural general circulation models for weather and climate . Nature 632 1060...

    2024 doi

  15. [23]

    \ Stuart, A M

    KovachkiStuart19 APACrefauthors Kovachki, N B. \ Stuart, A M. APACrefauthors \ 2019 . Ensemble Kalman inversion: a derivative-free technique for machine learning tasks Ensemble Kalman inversion: a derivative-free technique for machine learning tasks . Inverse Problems 35 9 095...

    2019 doi

  16. [24]

    , Thiffeault, J

    Lin-et-al11 APACrefauthors Lin, Z. , Thiffeault, J. \ Doering, C R. APACrefauthors \ 2011 . Optimal stirring strategies for passive scalar mixing Optimal stirring strategies for passive scalar mixing . J. Fluid Mech. 675 465--476 . APACrefDOI doi:10.1017/S0022112011000292 APACrefDOI

    2011 doi

  17. [25]

    , Jones, R

    Ling-et-al16b APACrefauthors Ling, J. , Jones, R. \ Templeton, J. APACrefauthors \ 2016 . Machine learning strategies for systems with invariance properties Machine learning strategies for systems with invariance properties . J. Comp. Phys. 318 22--35 . APACrefDOI doi:10.1016/...

    2016 doi

  18. [26]

    , Chen, L W

    List-et-al24 APACrefauthors List, B. , Chen, L W. , Bali, K. \ Thuerey, N. APACrefauthors \ 2024 . How Temporal Unrolling Supports Neural Physics Simulators. How temporal unrolling supports neural physics simulators. APACrefURL https://arxiv.org/abs/2402.12971 APACrefURL

    2024 arXiv

  19. [27]

    , Chen, L W

    List-et-al22 APACrefauthors List, B. , Chen, L W. \ Thuerey, N. APACrefauthors \ 2022 . Learned turbulence modelling with differentiable fluid solvers: physics-based loss functions and optimisation horizons Learned turbulence modelling with differentiable fluid solvers: physic...

    2022 doi

  20. [28]

    , Christopoulos, C

    LopezGomez-et-al22 APACrefauthors Lopez-Gomez , I. , Christopoulos, C. , Langeland Ervik , H L. , Dunbar, O R A. , Cohen, Y. \ Schneider, T. APACrefauthors \ 2022 . Training physics-based machine-learning parameterizations with gradient-free ensemble Kalman methods Training ph...

    2022 doi

  21. [29]

    , Goldberg, D N

    Maddison-et-al19 APACrefauthors Maddison, J R. , Goldberg, D N. \ Goddard, B D. APACrefauthors \ 2019 . Automated calculation of higher order partial differential equation constrained derivative information Automated calculation of higher order partial differential equation co...

    2019 doi

  22. [30]

    , Pritchard, M

    Mooers-et-al21 APACrefauthors Mooers, G. , Pritchard, M. , Beucler, T. , Ott, J. , Yacalis, G. , Baldi, P. \ Gentine, P. APACrefauthors \ 2021 . Assessing the potential of deep learning for emulating cloud superparameterization in climate models with real-geography boundary co...

    2021 doi

  23. [31]

    APACrefauthors \ 1971

    Orszag71b APACrefauthors Orszag, S A. APACrefauthors \ 1971 . On the elimination of aliasing in finite-difference schemes by filtering high-wavenumber components On the elimination of aliasing in finite-difference schemes by filtering high-wavenumber components . J. Atmos. Sci...

    1971 doi

  24. [32]

    APACrefauthors \ 2024

    Otness24 APACrefauthors Otness, K. APACrefauthors \ 2024 . pyQG-JAX v0.8.1 [software]. pyQG-JAX v0.8.1 [software]. APACrefDOI doi:10.5281/zenodo.10719906 APACrefDOI

    2024 doi

  25. [33]

    , Zanna, L

    Otness-et-al23 APACrefauthors Otness, K. , Zanna, L. \ Bruna, J. APACrefauthors \ 2023 . Data-driven multiscale modeling of subgrid parameterizations in climate models. Data-driven multiscale modeling of subgrid parameterizations in climate models. APACrefURL https://arxiv.org...

    2023

  26. [34]

    , Chapron, B

    Ouala-et-al23 APACrefauthors Ouala, S. , Chapron, B. , Collard, F. , Gaultier, L. \ Fablet, R. APACrefauthors \ 2023 . Online Calibration of Deep Learning Sub-Models for Hybrid Numerical Modeling Systems. Online calibration of deep learning sub-models for hybrid numerical mode...

    2023 arXiv

  27. [35]

    , Zanna, L

    Perezhogin-et-al23 APACrefauthors Perezhogin, P. , Zanna, L. \ Fernandez-Granda , C. APACrefauthors \ 2023 . Generative data-driven approaches for stochastic subgrid parameterizations in an idealized ocean model Generative data-driven approaches for stochastic subgrid paramete...

    2023 doi

  28. [36]

    Ross-et-al23 APACrefauthors Ross, A. , Li, Z. , Perezhogin, P. , Fernandez-Granda , C. \ Zanna, L. APACrefauthors \ 2023 . Benchmarking of machine learning ocean subgrid parameterizations in an idealized model Benchmarking of machine learning ocean subgrid parameterizations in...

    2023 doi

  29. [37]

    APACrefauthors \ 1980

    Salmon80 APACrefauthors Salmon, R. APACrefauthors \ 1980 . Baroclinic instability and geostrophic turbulence Baroclinic instability and geostrophic turbulence . Geophys. Astrophys. Fluid Dyn. 15 167-211

    1980

  30. [38]

    , Reichl, B G

    Sane-et-al23 APACrefauthors Sane, A. , Reichl, B G. , Adcroft, A. \ Zanna, L. APACrefauthors \ 2023 . Parameterizing vertical mixing coefficients in the ocean surface boundary layer using neural networks Parameterizing vertical mixing coefficients in the ocean surface boundary...

    2023 doi

  31. [39]

    , Bolibar, J

    Sapienza-et-al24 APACrefauthors Sapienza, F. , Bolibar, J. , Schäfer, F. , Groenke, B. , Pal, A. , Boussange, V. Rackauckas, C. APACrefauthors \ 2024 . Differentiable Programming for Differential Equations: A Review. Differentiable programming for differential equations: A rev...

    2024

  32. [40]

    , Stuart, A M

    Schneider-et-al22 APACrefauthors Schneider, T. , Stuart, A M. \ Wu, J. APACrefauthors \ 2022 . Ensemble Kalman inversion for sparse learning of dynamical systems from time-averaged data Ensemble Kalman inversion for sparse learning of dynamical systems from time-averaged data ...

    2022

  33. [41]

    , Chekroun, M D

    Srinivasan-et-al24 APACrefauthors Srinivasan, K. , Chekroun, M D. \ McWilliams , J C. APACrefauthors \ 2024 . Turbulence closure with small, local neural networks: Forced two-dimensional and -plane flows Turbulence closure with small, local neural networks: Forced two-dimensio...

    2024 doi

  34. [42]

    , Hassanzadeh, P

    Sun-et-al23 APACrefauthors Sun, Y Q. , Hassanzadeh, P. , Alexander, M J. \ Kruse, C G. APACrefauthors \ 2023 . Quantifying 3D gravity wave drag in a library of tropical convection-permitting simulations for data-driven parameterizations Quantifying 3D gravity wave drag in a li...

    2023 doi

  35. [43]

    APACrefauthors \ 2012

    Thiffeault12 APACrefauthors Thiffeault, J L. APACrefauthors \ 2012 . Using multiscale norms to quantify mixing and transport Using multiscale norms to quantify mixing and transport . Nonlinearity 84 R1--R44 . APACrefDOI doi:10.1088/0951-7715/25/2/R1 APACrefDOI

    2012 doi

  36. [44]

    APACrefauthors \ 2008

    Villani-OT APACrefauthors Villani, C. APACrefauthors \ 2008 . Optimal Transport: Old and New Optimal Transport: Old and New . Springer

    2008

  37. [45]

    \ Mak, J

    YanMak24 APACrefauthors Yan, F E. \ Mak, J. APACrefauthors \ 2024 . Data collection for online learning in two-layer quasi-geostrophic model [Dataset]. Data collection for online learning in two-layer quasi-geostrophic model [dataset]. APACrefDOI doi:10.5281/zenodo.14197522 APACrefDOI

    2024 doi

  38. [46]

    , Mak, J

    Yan-et-al23 APACrefauthors Yan, F E. , Mak, J. \ Wang, Y. APACrefauthors \ 2023 . On the choice of training data for machine learning of geostrophic mesoscale turbulence On the choice of training data for machine learning of geostrophic mesoscale turbulence . J. Adv. Model. Ea...

    2023 doi

  39. [47]

    \ O'Gorman , P A

    YuvalOGorman20 APACrefauthors Yuval, J. \ O'Gorman , P A. APACrefauthors \ 2020 . Stable machine-learning parameterization of subgrid processes for climate modeling at a range of resolutions Stable machine-learning parameterization of subgrid processes for climate modeling at ...

    2020 doi

  40. [48]

    \ Bolton, T

    ZannaBolton20 APACrefauthors Zanna, L. \ Bolton, T. APACrefauthors \ 2021 . Data-driven equation discovery of ocean mesoscale closures Data-driven equation discovery of ocean mesoscale closures . Geophys. Res. Lett. 47 e2020GL088376 . APACrefDOI doi:10.1029/2020GL088376 APACrefDOI

    2021 doi

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.