Calibration of a neural network ocean closure for improved mean state and variability

Alistair Adcroft; Laure Zanna; Pavel Perezhogin

arxiv: 2604.06398 · v2 · pith:LJ6JEZDOnew · submitted 2026-04-07 · ⚛️ physics.ao-ph · cs.LG· physics.comp-ph

Calibration of a neural network ocean closure for improved mean state and variability

Pavel Perezhogin , Alistair Adcroft , Laure Zanna This is my paper

Pith reviewed 2026-05-21 10:22 UTC · model grok-4.3

classification ⚛️ physics.ao-ph cs.LGphysics.comp-ph

keywords ocean modelingmesoscale parameterizationneural networkensemble Kalman inversioncoarse resolutioncalibrationeddy closureidealized ocean models

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The pith

Calibrating a neural network ocean eddy parameterization with ensemble Kalman inversion reduces errors in mean state and variability by factors of 1.7 to 3.3 in coarse-resolution models.

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

The paper sets out to show that parameter tuning can be treated as a systematic calibration problem rather than ad hoc adjustment. Using Ensemble Kalman Inversion, the authors optimize the coefficients of a neural network that represents the effects of unresolved mesoscale eddies in two simplified ocean models run at coarse resolution. A sympathetic reader would care because global ocean models at practical resolutions miss important eddy-driven transports, leading to persistent biases in temperature, salinity, and circulation; a working calibration method could therefore improve forecasts and climate projections without requiring finer grids. The work also demonstrates that the inversion remains stable even when the target statistics are noisy due to the chaotic nature of the flow, and introduces a shortcut that avoids running each simulation to full statistical equilibrium.

Core claim

By treating the neural network weights as parameters to be inferred, Ensemble Kalman Inversion applied to time-averaged diagnostics from two idealized ocean configurations produces a parameterization that cuts errors in the mean position of fluid interfaces and in their temporal variability by factors between 1.7 and 3.3 relative to an unparameterized control.

What carries the argument

A neural network parameterization of mesoscale eddy fluxes, whose internal coefficients are adjusted by Ensemble Kalman Inversion to match target statistics from higher-resolution reference runs.

If this is right

Systematic calibration can replace manual tuning for eddy closures in ocean models.
The calibrated model exhibits substantially lower biases in both time-mean fields and variability metrics.
EKI remains effective despite noise in the time-averaged targets caused by ocean chaos.
An efficient protocol exists that avoids full equilibration by selecting a suitable initial condition.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If the same procedure works in global models with realistic boundaries and forcing, it could reduce the need for higher-resolution ensembles in climate studies.
Similar calibration approaches might be applied to other subgrid closures such as those for convection or boundary layers.
Testing the calibrated network in coupled atmosphere-ocean simulations would reveal whether the improvements persist under interactive forcing.
Extending the method to online learning, where parameters adjust during a single long integration, could further lower computational cost.

Load-bearing premise

That the error reductions seen in two idealized configurations will carry over when the same parameterization is inserted into full global ocean models that include realistic forcing, coastlines, and coupling to the atmosphere.

What would settle it

Running the calibrated neural-network closure inside a global ocean model at coarse resolution and checking whether the bias reductions in sea-surface height, temperature, and eddy kinetic energy remain comparable to those observed in the idealized tests.

Figures

Figures reproduced from arXiv: 2604.06398 by Alistair Adcroft, Laure Zanna, Pavel Perezhogin.

**Figure 1.** Figure 1: (a) Idealized wind-driven ocean model GFDL MOM6 in a double-gyre configuration. (b) The eddy kinetic energy (EKE) spectrum as a function of isotropic horizontal wavenumber in the upper fluid layer and domain 5◦E−15◦E × 35◦N−45◦N. The percentages show the integral over the spectrum relative to the high-resolution simulation. Panel (c) shows how the Ensemble Kalman Inversion calibration algorithm interacts… view at source ↗

**Figure 2.** Figure 2: Calibration of the eddy parameterization in Double Gyre configuration. The upper row shows time-averaged sea surface height (SSH), the second row shows the temporal standard deviation of SSH. On these panels, all simulations are 100 years long and results are averaged over 90 years. (a,d) is a coarse (1/2 ◦ ) unparameterized model, (b,e) is the coarse model with calibrated eANN backscatter parameterization… view at source ↗

**Figure 3.** Figure 3: Evaluation of calibrated parameterizations in 30000-day simulations in configuration NeverWorld2. (a) Zonally- and time-averaged vertical coordinate of internal fluid interfaces. Lower row shows temporal standard deviation of sea surface height for simulations with: (b) parameterization trained offline, (c) parameterization calibrated in Double Gyre with manually adjusted coefficient γ, (d) parameterizati… view at source ↗

read the original abstract

Global ocean models exhibit biases in the mean state and variability, particularly at coarse resolution, where mesoscale eddies are unresolved. To address these biases, parameterization coefficients are typically tuned ad hoc. Here, we formulate parameter tuning as a calibration problem using Ensemble Kalman Inversion (EKI). We optimize parameters of a neural network parameterization of mesoscale eddies in two idealized ocean models at coarse resolution. The calibrated parameterization reduces errors by factors of 1.7-3.3 in the time-averaged fluid interfaces and their variability compared to the unparameterized model, depending on the metric and configuration. The EKI method is robust to noise in time-averaged statistics arising from chaotic ocean dynamics. Furthermore, we propose an efficient calibration protocol that bypasses integration to statistical equilibrium by carefully choosing an initial condition. These results demonstrate that systematic calibration can substantially improve coarse-resolution ocean simulations and provide a practical pathway for reducing biases in global ocean models.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Calibrated NN eddy closure cuts errors 1.7-3.3x in two idealized coarse models via EKI, but gains are shown only in simplified setups with no global test yet.

read the letter

Colleague, the main thing to know is that this paper tunes a neural network parameterization of mesoscale eddies with Ensemble Kalman Inversion and reports error reductions of 1.7-3.3 times in time-averaged fluid interfaces and variability versus the unparameterized baseline, all in two idealized coarse-resolution ocean configurations. They also show the calibration stays stable despite noise from chaotic dynamics and introduce an initial-condition choice that avoids running to full statistical equilibrium. That combination is the concrete new piece: a systematic calibration route instead of the usual hand tuning, with a practical speedup. The work does well at delivering specific quantitative gains in those controlled cases and at treating the tuning as a proper inverse problem that accounts for the statistics they actually have. The approach looks technically reasonable for the setups they chose. The soft spot is the narrow domain. All the reported improvements come from idealized configurations, so there is no direct evidence yet that the same NN closure reduces biases once realistic wind stress, bathymetry, lateral boundaries, or atmosphere coupling are added. The claim that this opens a pathway for global models therefore rests on an untested extrapolation about how well the learned closure travels when the background flow regime changes. That is the main limitation at present. This paper is aimed at ocean and climate modelers who work on eddy closures at coarse resolution and are willing to try machine-learning tools. A reader looking for concrete calibration methods would get usable ideas from the EKI protocol and the initial-condition shortcut. It has enough new application and measurable results to deserve a serious referee, even though reviewers will likely press on the generalization question and ask for more verification details. I would recommend sending it to peer review.

Referee Report

1 major / 2 minor

Summary. The paper formulates parameter tuning of a neural network mesoscale eddy parameterization as an Ensemble Kalman Inversion (EKI) calibration problem. It demonstrates the approach in two coarse-resolution idealized ocean models, reporting error reductions by factors of 1.7-3.3 in time-averaged fluid interfaces and their variability relative to the unparameterized baseline. The work also claims robustness of EKI to chaotic noise in time-averaged statistics and proposes an efficient calibration protocol that avoids full integration to statistical equilibrium.

Significance. The use of EKI for systematic calibration of an NN closure, together with the reported error reductions in idealized cases and the noise-robustness result, represents a concrete step toward reducing ad-hoc tuning in ocean models. If the improvements hold under more realistic conditions, the method could meaningfully improve mean-state and variability biases in global ocean simulations used for climate studies.

major comments (1)

The central claim that the calibrated NN provides 'a practical pathway for reducing biases in global ocean models' is not directly supported by the presented evidence. All quantitative results (error reduction factors of 1.7-3.3) are obtained exclusively in two idealized configurations; realistic wind stress, bathymetry, lateral boundaries, and atmosphere coupling are omitted. These omissions can alter the eddy-mean interactions that the NN learns, so the generalization step remains an untested extrapolation.

minor comments (2)

The neural network architecture (layer count, widths, activation functions) and the precise form of the closure (e.g., how the NN output is injected into the momentum or tracer equations) should be stated explicitly, preferably with a diagram or pseudocode, to allow independent reproduction.
The efficient calibration protocol that bypasses statistical equilibrium is described only at a high level; the precise choice of initial condition and the quantitative criterion used to confirm that equilibrium is not required should be detailed in the methods section.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their constructive comments on our manuscript. We respond to the major comment below.

read point-by-point responses

Referee: The central claim that the calibrated NN provides 'a practical pathway for reducing biases in global ocean models' is not directly supported by the presented evidence. All quantitative results (error reduction factors of 1.7-3.3) are obtained exclusively in two idealized configurations; realistic wind stress, bathymetry, lateral boundaries, and atmosphere coupling are omitted. These omissions can alter the eddy-mean interactions that the NN learns, so the generalization step remains an untested extrapolation.

Authors: We agree that the quantitative demonstrations and error reductions are obtained exclusively in two idealized configurations, and that direct evidence for global ocean models with realistic wind stress, bathymetry, lateral boundaries, and atmosphere coupling is not provided. The idealized setups were deliberately chosen to isolate mesoscale eddy dynamics and enable a controlled, computationally feasible test of the EKI calibration procedure. We have revised the abstract and conclusions to change the phrasing from 'provide a practical pathway' to 'suggest a practical pathway' and have added a dedicated paragraph in the discussion section that explicitly acknowledges the idealized nature of the experiments, notes that eddy-mean interactions may differ under realistic forcing, and outlines the additional steps required to extend the approach to global models. These changes ensure the claims accurately reflect the scope of the presented evidence while preserving the motivation for the method as a systematic alternative to ad-hoc tuning. revision: partial

Circularity Check

0 steps flagged

No circularity: calibration outcomes are measured against external reference statistics

full rationale

The paper presents a calibration procedure that uses Ensemble Kalman Inversion to tune neural network parameters so that coarse-resolution ocean simulations better match time-averaged statistics drawn from higher-resolution truth runs. The reported error reductions (factors of 1.7-3.3) are direct numerical outcomes of this optimization in two specific idealized configurations; they are not obtained by re-expressing the inputs, by renaming a fitted quantity as a prediction, or by any self-citation chain that would render the result tautological. Because the reference data and the unparameterized baseline are independent of the calibration algorithm itself, the derivation remains self-contained and externally falsifiable.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that a neural network can represent mesoscale eddy effects once its parameters are calibrated, plus the practical assumption that EKI can extract useful updates from noisy time-averaged statistics in chaotic flows.

free parameters (1)

Neural network parameters
Weights and biases of the neural network are adjusted by EKI to match target statistics from reference simulations.

axioms (1)

domain assumption A neural network of the chosen architecture can serve as an effective closure for unresolved mesoscale eddies when its parameters are optimized.
Invoked when the authors formulate the parameterization and apply EKI to its coefficients.

pith-pipeline@v0.9.0 · 5694 in / 1295 out tokens · 60534 ms · 2026-05-21T10:22:44.836079+00:00 · methodology

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discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We solve the optimization problem using Ensemble Kalman Inversion (ETKI) ... L(θ)=||y−gj||22

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contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

[1]

Bachman, S. D. (2024). An eigenvalue-based framework for constraining anisotropic eddy viscosity.Journal of Advances in Modeling Earth Systems,16(8), e2024MS004375

work page 2024
[2]

Cesa, G., Lang, L., & Weiler, M. (2022). A program to build E(N)-equivariant steerable CNNs. InInternational conference on learning representations (iclr)

work page 2022
[3]

Connolly, A., Cheng, Y., Walters, R., Wang, R., Yu, R., & Gentine, P. (2025). Deep learning turbulence closures generalize best with physics-based methods.Authorea Preprints. doi: https://doi.org/10.22541/essoar.173869578.80400701/v1

work page doi:10.22541/essoar.173869578.80400701/v1 2025
[4]

M., & Hallberg, R

Griffies, S. M., & Hallberg, R. W. (2000). Biharmonic friction with a Smagorinsky-like viscosity for use in large-scale eddy-permitting ocean models.Monthly Weather Review,128(8), 2935–2946. doi: https://doi.org/10.1175/1520-0493(2000)128⟨2935:BFWASL⟩2.0.CO;2

work page doi:10.1175/1520-0493(2000)128 2000
[5]

M., Winton, M., Anderson, W

Griffies, S. M., Winton, M., Anderson, W. G., Benson, R., Delworth, T. L., Dufour, C. O., . . . others (2015). Impacts on ocean heat from transient mesoscale eddies in a hierarchy of climate models.Journal of Climate,28(3), 952–977. doi: https://doi.org/10.1175/ JCLI-D-14-00353.1

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Guan, Y., Subel, A., Chattopadhyay, A., & Hassanzadeh, P. (2022). Learning physics- constrained subgrid-scale closures in the small-data regime for stable and accurate LES. Physica D: Nonlinear Phenomena, 133568. doi: https://doi.org/10.1016/j.physd.2022 .133568

work page doi:10.1016/j.physd.2022 2022
[7]

Pawar, S., San, O., Rasheed, A., & Vedula, P. (2023). Frame invariant neural network closures for Kraichnan turbulence.Physica A: Statistical Mechanics and its Applications,609, 128327. doi: https://doi.org/10.1016/j.physa.2022.128327 April 9, 2026, 12:10am X - 18:

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[8]

Perezhogin, P., Adcroft, A., & Zanna, L. (2025). Generalizable neural-network parameterization of mesoscale eddies in idealized and global ocean models.Geophysical Research Letters, 52(19), e2025GL117046. doi: https://doi.org/10.1029/2025GL117046

work page doi:10.1029/2025gl117046 2025
[9]

Ross, A., Li, Z., Perezhogin, P., Fernandez-Granda, C., & Zanna, L. (2023). Benchmarking of machine learning ocean subgrid parameterizations in an idealized model.Journal of Advances in Modeling Earth Systems,15(1), e2022MS003258. doi: https://doi.org/10.1029/ 2022MS003258

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[10]

D., & McWilliams, J

Smith, R. D., & McWilliams, J. C. (2003). Anisotropic horizontal viscosity for ocean models. Ocean Modelling,5(2), 129–156

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Vallis, G. K. (2017).Atmospheric and oceanic fluid dynamics. Cambridge University Press. doi: https://doi.org/10.1017/9781107588417

work page doi:10.1017/9781107588417 2017
[12]

Weiler, M., & Cesa, G. (2019). General E(2)-Equivariant Steerable CNNs. InConference on neural information processing systems (neurips)

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[13]

Zanna, L., & Bolton, T. (2020). Data-driven equation discovery of ocean mesoscale clo- sures.Geophysical Research Letters,47(17), e2020GL088376. doi: https://doi.org/10.1029/ 2020GL088376 April 9, 2026, 12:10am

work page 2020

[1] [1]

Bachman, S. D. (2024). An eigenvalue-based framework for constraining anisotropic eddy viscosity.Journal of Advances in Modeling Earth Systems,16(8), e2024MS004375

work page 2024

[2] [2]

Cesa, G., Lang, L., & Weiler, M. (2022). A program to build E(N)-equivariant steerable CNNs. InInternational conference on learning representations (iclr)

work page 2022

[3] [3]

Connolly, A., Cheng, Y., Walters, R., Wang, R., Yu, R., & Gentine, P. (2025). Deep learning turbulence closures generalize best with physics-based methods.Authorea Preprints. doi: https://doi.org/10.22541/essoar.173869578.80400701/v1

work page doi:10.22541/essoar.173869578.80400701/v1 2025

[4] [4]

M., & Hallberg, R

Griffies, S. M., & Hallberg, R. W. (2000). Biharmonic friction with a Smagorinsky-like viscosity for use in large-scale eddy-permitting ocean models.Monthly Weather Review,128(8), 2935–2946. doi: https://doi.org/10.1175/1520-0493(2000)128⟨2935:BFWASL⟩2.0.CO;2

work page doi:10.1175/1520-0493(2000)128 2000

[5] [5]

M., Winton, M., Anderson, W

Griffies, S. M., Winton, M., Anderson, W. G., Benson, R., Delworth, T. L., Dufour, C. O., . . . others (2015). Impacts on ocean heat from transient mesoscale eddies in a hierarchy of climate models.Journal of Climate,28(3), 952–977. doi: https://doi.org/10.1175/ JCLI-D-14-00353.1

work page 2015

[6] [6]

Guan, Y., Subel, A., Chattopadhyay, A., & Hassanzadeh, P. (2022). Learning physics- constrained subgrid-scale closures in the small-data regime for stable and accurate LES. Physica D: Nonlinear Phenomena, 133568. doi: https://doi.org/10.1016/j.physd.2022 .133568

work page doi:10.1016/j.physd.2022 2022

[7] [7]

Pawar, S., San, O., Rasheed, A., & Vedula, P. (2023). Frame invariant neural network closures for Kraichnan turbulence.Physica A: Statistical Mechanics and its Applications,609, 128327. doi: https://doi.org/10.1016/j.physa.2022.128327 April 9, 2026, 12:10am X - 18:

work page doi:10.1016/j.physa.2022.128327 2023

[8] [8]

Perezhogin, P., Adcroft, A., & Zanna, L. (2025). Generalizable neural-network parameterization of mesoscale eddies in idealized and global ocean models.Geophysical Research Letters, 52(19), e2025GL117046. doi: https://doi.org/10.1029/2025GL117046

work page doi:10.1029/2025gl117046 2025

[9] [9]

Ross, A., Li, Z., Perezhogin, P., Fernandez-Granda, C., & Zanna, L. (2023). Benchmarking of machine learning ocean subgrid parameterizations in an idealized model.Journal of Advances in Modeling Earth Systems,15(1), e2022MS003258. doi: https://doi.org/10.1029/ 2022MS003258

work page 2023

[10] [10]

D., & McWilliams, J

Smith, R. D., & McWilliams, J. C. (2003). Anisotropic horizontal viscosity for ocean models. Ocean Modelling,5(2), 129–156

work page 2003

[11] [11]

Vallis, G. K. (2017).Atmospheric and oceanic fluid dynamics. Cambridge University Press. doi: https://doi.org/10.1017/9781107588417

work page doi:10.1017/9781107588417 2017

[12] [12]

Weiler, M., & Cesa, G. (2019). General E(2)-Equivariant Steerable CNNs. InConference on neural information processing systems (neurips)

work page 2019

[13] [13]

Zanna, L., & Bolton, T. (2020). Data-driven equation discovery of ocean mesoscale clo- sures.Geophysical Research Letters,47(17), e2020GL088376. doi: https://doi.org/10.1029/ 2020GL088376 April 9, 2026, 12:10am

work page 2020