Calibrating subgrid parametrizations of single-column ocean models via simulation-based inference

Luben M. C. Cabezas; Sacha Wendling; Aurèle Gallard; Gabriel Mouttapa; Julien Le Sommer; Pedro L. C. Rodrigues

physics.ao-phSep 3, 2026

Calibrating subgrid parametrizations of single-column ocean models via simulation-based inference

Authors: Luben M. C. Cabezas, Sacha Wendling, Aurèle Gallard, Gabriel Mouttapa, Julien Le Sommer, Pedro L. C. Rodrigues

Organizations: DEs, ICMC Federal Univ. of São Carlos, Univ. of São Paulo, Brazil · Univ. Grenoble Alpes, Inria, CNRS, Grenoble INP, LJK, France · Univ. Grenoble Alpes, Inria, CNRS, IRD, Grenoble INP, INRAE, IGE, France

Abstract

Subgrid parametrizations of vertical mixing in ocean models depend on free coefficients that cannot be measured directly and must be calibrated against high-fidelity references such as large-eddy simulations (LES). Existing approaches return point estimates and leave the associated uncertainty unquantified, a limitation when the inverse problem is ill-posed or when distinct parameter configurations fit the data comparably well. Simulation-based inference (SBI) addresses exactly this: given a prior and access to the simulator, it approximates the full posterior over parameters without requiring a tractable likelihood, at a cost set by the number of simulator evaluations. We apply it to tunax, a JAX-based single-column ocean model, to calibrate the coefficients of its $k$ -- $\varepsilon$ closure. A blockwise PCA summary statistic compresses the simulator output along the depth axis while preserving its forcing--horizon--variable structure, making inference tractable at modest budgets. We compare neural posterior estimation and its sequential variants against a recent training-free approach built on a tabular foundation model. The latter recovers informative posteriors from a few hundred simulator calls, outperforming the trained estimators at every budget considered.

Figures & tables

Figure 1: RMSE between LES profiles and the tunax output evaluated at each posterior mean. Markers and error bars show the mean and SEM across 5 realizations, and the dashed line denotes the default tunax configuration. The gray marker denotes full-profile gradient calibration; its position counts

1000

simulator calls, although differentiation makes these calls more costly than ordinary forward simulations. NPE-PFN-MEAN reaches lower RMSE rapidly at small budgets and performs best among the posterior methods from moderate budgets onward. All methods plateau at budgets of around 1000 simulations, yet none approach the gradient descent optimum.

Figure 2: Prior and posterior predictive final profiles. Colored curves and shaded regions show the predictive medians and central 90% intervals, while black curves denote the LES profiles. SNPE adjusts the predictive medians but retains uncertainty bands broadly similar to the prior. In contrast, NPE-PFN produces substantially narrower intervals, particularly for

u

and

v

, indicating a stronger information gain in output space and close tracking of LES profiles.

Figure 3: Prior–posterior comparison in physical parameter space for

c_{\epsilon 1}

c_{\epsilon 2}

c_{\epsilon 3}^{+}

, and

c_{2}

, selected from the strongest aggregate Spearman correlations across methods detailed in Appendix B.2 . Diagonal panels show marginal KDEs and lower panels show bivariate density contours; dashed lines and diamonds denote

\boldsymbol{\theta}_{\mathrm{GD}}

from the full-profile gradient calibration, which generally lies away from the main posterior concentration. SNPE remains broadly similar to the prior, whereas NPE-PFN produces clearer shifts and more concentrated densities, indicating a stronger posterior update.

Appendix figures & tables13 assets

Supplementary material from the paper’s appendix.

Appendix

Parameter	Default value	Parameter	Default value
$c_{1}$	$5.0000$	$c_{b1}$	$5.9500$
$c_{2}$	$0.8000$	$c_{b2}$	$0.6000$
$c_{3}$	$1.9680$	$c_{b3}$	$1.0000$
$c_{4}$	$1.1360$	$c_{b5}$	$0.3333$
$c_{6}$	$0.4000$	$c_{bb}$	$0.7200$
$c_{\mu}^{0}$	$0.5477$	$\sigma_{k}$	$1.0000$

Appendix

Table 1: Customary tunax parameter values defining the physical-space prior medians.

Figure 4: Prior predictive final profiles for buoyancy

b

. Columns show forcing regimes and rows show 6, 24, and 72 hours. At 24 and 72 hours, the central 90% prior predictive interval generally encompasses the LES profiles. At 6 hours, mild departures occur over some depths and forcing regimes.

Figure 5: Prior predictive final profiles for the along-wind velocity

u

. The central 90% prior predictive interval encompasses the LES profiles over most retained forcing and depth combinations, with localized departures in some cases. Free convection is not retained because it has no imposed wind and therefore no informative

u

response.

Figure 6: Prior predictive final profiles for the transverse velocity

v

. Across the retained rotating and wind-forced configurations, the central 90% prior predictive interval encompasses most of the LES profiles. Free convection and the two zero-Coriolis configurations are not retained because their transverse profiles provide negligible information for calibration.

Figure 7: Prior predictive final profiles for the passive tracer

p_{t}

. The central 90% prior predictive interval generally encompasses a larger portion of the LES profiles at 24 and 72 hours. More visible departures occur during the 6-hour transient, showing that the prior predictive coverage of the tracer varies throughout the simulated evolution.

Figure 8: Blockwise PCA dimensions selected using a 90% explained-variance threshold. Panels denote physical variables, rows denote time horizons, and columns denote forcing regimes; cells report the selected dimension and retained variance, while gray cells indicate masked combinations. Overall, $b$ generally requires a larger PCA basis than $p_{t}$ , supporting an adaptive block-specific truncation.

Figure 9: Empirical covariance structure in the 159-dimensional PCA-summary space: (a) covariance magnitudes, showing nearly zero values for $b$ and the largest values for $p_{t}$ ; (b) standardized within- and cross-variable correlations; and (c) eigenspectra before and after adding the $10^{-4}$ diagonal nugget. The regularization lifts the low-variance directions dominated by $b$ while preserving the dominant covariance structure.

Figure 10: Physical-space RMSE between the posterior mean and

\boldsymbol{\theta}_{\mathrm{GD}}

, reported as mean and SEM across seeds. The dashed line denotes the customary tunax parameters; the starred TSNPE point at

B=50

uses a 1%-trimmed mean, and values above

2.2

are clipped. NPE-PFN yields lower RMSE across budgets, while NPE, SNPE, and TSNPE generally fluctuate slightly above the default-reference distance.

Figure 11: Posterior parameter differences relative to

\boldsymbol{\theta}_{\mathrm{GD}}

for each method’s lowest-profile-RMSE seed at budget

3000

. NPE, SNPE, and TSNPE largely overlap, whereas NPE-PFN shows the clearest shifts for

c_{\epsilon 2}

and

c_{3}

Figure 12: Posterior Spearman correlation matrices in physical parameter space for the selected runs at a simulation budget of $3000$ . All panels share the same color scale. Pairwise correlations are generally weak, with the positive association between $c_{\varepsilon 1}$ and $c_{\varepsilon 2}$ providing the clearest relationship consistently recovered across methods. These matrices motivate the four-parameter subset examined in the posterior pairplots.

Parameter pair	NPE	SNPE	TSNPE	NPE-PFN	$R_{ij}$
$(c_{\varepsilon 1},c_{\varepsilon 2})$	0.215	0.105	0.183	0.236	0.185
$(c_{\varepsilon 2},c_{\varepsilon 3}^{+})$	0.063	0.060	0.106	0.040	0.067
$(c_{2},c_{\mu}^{0})$	-0.074	-0.091	-0.091	-0.007	0.066
$(c_{2},c_{\varepsilon 1})$	-0.065	-0.087	-0.071	-0.027	0.062
$(c_{\varepsilon 1},c_{\varepsilon 3}^{+})$	0.092	0.057	0.062	0.021	0.058
$(\sigma_{\varepsilon},c_{\varepsilon 2})$	-0.120	-0.068	-0.035	-0.006	0.057

Appendix

Table 2: Parameter pairs with the largest mean absolute posterior Spearman correlation across methods. Method-specific coefficients retain their signs, while

R_{ij}

summarizes the average association strength.

Figure 13: Prior and posterior predictive final profiles under strong wind and weak cooling at 6 hours. Colored curves and bands denote predictive medians and central 90% intervals, and black curves denote the LES profiles. SNPE largely retains the prior predictive spread, whereas NPE-PFN produces sharper intervals, particularly for

u

and

v

, together with clearer median adjustments for

b

v

, and

p_{t}

Figure 14: Prior and posterior predictive final profiles under strong wind at 24 hours. NPE-PFN preserves the behavior observed at 6 hours, producing the clearest uncertainty contraction for the velocity profiles and predictive medians generally closer to the LES reference. SNPE improves the median profiles but retains uncertainty bands similar to the prior.

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Abstract

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