NUTS: Eddy-Robust Reconstruction of Surface Ocean Nutrients via Two-Scale Modeling
Authors: Hao Zheng
Shiyu Liang
Yuting Zheng
Chaofan Sun
LEI BAI
Enhui Liao
Venue: NeurIPS 2025 poster ()
Keywords: Oceanic Nutrients Reconstruction
Advection-Diffusion
Primary area: applications

Abstract: Reconstructing ocean surface nutrients from sparse observations is critical for understanding long-term biogeochemical cycles.  Most prior work focuses on reconstructing atmospheric fields and treats the reconstruction problem as image inpainting, assuming smooth, single-scale dynamics. In contrast, nutrient transport follows advection–diffusion dynamics under nonstationary, multiscale ocean flow. This mismatch leads to instability, as small errors in unresolved eddies can propagate through time and distort nutrient predictions.
To address this, we introduce NUTS, a two-scale reconstruction model that decouples large-scale transport and mesoscale variability. The homogenized solver captures stable, coarse-scale advection under filtered flow. A refinement module then restores mesoscale detail conditioned on the residual eddy field.
NUTS is stable, interpretable, and robust to mesoscale perturbations, with theoretical guarantees from homogenization theory. NUTS outperforms all data-driven baselines in global reconstruction and achieves site-wise accuracy comparable to  numerical models. On real observations, NUTS reduces NRMSE by 79.9% for phosphate and 19.3% for nitrate over the best baseline. Ablation studies validate the effectiveness of each module.

NUTS: Eddy-Robust Reconstruction of Surface Ocean Nutrients via Two-Scale Modeling
Hao Zheng!"
Shiyu Liang" Yuting Zheng' Chaofan Sun' Lei Bai² Enhui Liao Shanghai Jiao Tong University, China Shanghai Artificial Intelligence Laboratory, China (hubert zheng, 1sy18602808513, zhengyt058, scf024, shliao/@sjju. edu, cn
baisanshi Saraiih@gmma...oo
Abstract
Reconstructing ocean surface nutrients from sparse observations critical for understanding long-term biogeochemical cycles. Most prior work focuses constructing atmospheric fields and treats the reconstruction problem as image inpainting, assuming smooth, single-scale dynamics. In contrast, nutrient transport follows cdlvection-diffusse dynamics under nonstationary, multiscale ocean flow. This mismatch leads instability, as small errors unresolved eddies can propa- gate through time and distort nutrient predictions. To address this, we introduce NUTS, two-scale reconstruction model that decouples large-scale transport and mesoscale variability. The homogenized solver captures stable, coarse-scale ad- vection under filtered flow. A refinement module then restores mesoscale detail conditioned on the residual eddy field. NUTS is stable, interpretable, and robust to mesoscale perturbations, with theoretical guarantees from homogenization theory. NUTS outperforms all data aata-driven baselines in global reconstruction and achieves site-wise accuracy comparable to numerical models. On real observations, NUTS reduces NRMSE by 79.9% for phosphate and 19.3% for nitrate over the best baseline. Ablation studies validate the effectiveness of each module.
Introduction
Reconstructing historical nutrient concentrations the surface ocean is essential for understanding long-term biogeochemical cycles, ecosystem variability, and anthropogenic influence Stüeken et al. [2024]. However, nutrient observations are extremely sparse, especially before the bio-Argo era when data came from irregular ship-based campaigns. Even today, nutrient data remain far less available than satellite-maasured variables like sea surface temperature (SST) or chlorophyll Mishonov et al.
Mishonoo
2024 Locarnini et (2018).
Recent deep learning advances have driven progress in forecasting and reconstructing high- dimensional atmospheric fields. Transformer-based models Pathak et al.) (2022 Bi et al. (2033
Lam et al. (2023) achieve state-of-the-ar short-term forecasts by capturing temporal dependencies in
data-rich regimes with complete initial conditions. In contrast, climate field reconstruction operates
in sparse settings and is often framed as a spatial in-painting task [Bochow et al.) [2025) Plésiat et al.
2024 Kadow al. 2020). Early models (ReeeebeRnebrrerr
al. [2015]. Dosovitskiy
[2020] Gao
et al. 2022 focus on spatial correlations, while recent hybrids Sabrids[Lie Li al. 2020). Wang et al. 2025],
Beauchamp al. [2023] add physical constraints for greater consistency. However, these methods
are mainly designed for smooth, single-scale wind fields with well-resolved large-scale structure.
'Equal contribution. Corresponding author.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Reconstructing ocean nutrients demands a fundamentally different approach. Unlike atmospheric fields, nutrient transport follows irrvecionndiffussee dynamics driven by a nonstationary, multiscale velocity field. Surface currents consist of slowly evolving large-scale mean flow overlaid with rapidly fluctuating mesoscale eddies. These eddies- vortices spanning 10-100 km- on-govern most lateral nutrient transport Vallis [017]] McWilliams (2016). , [Chelton et al. [201T], yet are poorly resolved in numerical circulation models due to limited resolution and inherent uncertainty. As a result, reconstruction models that rely directly such flow fields are fragile: even small perturbations
in the eddy component can degrade predictions.
Robust nutrient reconstruction presents a core modeling dilemma. Filtering the input velocity field improves stability by suppressing figuecry eddy perturbations. However, it also removes fine-scale structures essential for capturing local nutrient gradients. Retaining all scales introduces instability; over-filtering sacrifices resolution. A principled solution must separate scales preserving large-scale transport while reintroducing mesoscale variability in controlled manner.
We propose NUTS, novel and robust two-scale model that, for the first time, resolves the reconstruc- tion challenge through a structured decomposition. At its core is a homogenized cndvection-diffssiin solver that models nutrient transport under the filtered large-scale flow. By replacing unresolved mesoscale variability with an effective diffusion term, this formulation captures the net impact of fine-scale dynamics without tracking unstable eddy fluctuations. The coarse module leverages this framework to propagate nutrient fields with stability and physical consistency. To recover fine-scale structure, the refinement module models localized redistribution conditioned on the residual mesoscale flow and the coarse prediction. This coarse-to-red torre-ed- architecture preserves large-scale transport patterns while restoring spatial detail in dynamically active regions. NUTS robust to mesoscale per- turbations, respects scale separation, and generalizes effectively under sparse observational coverage. We establish accuracy and stability guarantees under standard assumptions from homogenization theory, and empirically demonstrate that NUTS consistently outperforms prior baselines on both simulated and real-world datasets. Our contributions are as follows:
We formulate nutrient reconstruction as spatiotemporal ddvection-diffusiio problem and reveal the vulnerability of naive methods to mesoscale perturbations.
We propose NUTS, two-scale model that combines a homogenized PDE solver with adaptive diffusion and a refinement module conditioned on normalized eddy flow. We provide theoretical justification of its effectiveness under standard homogenization assumptions.
We empirically demonstrate that NUTS outperforms all data-driven baselines in global nutrient reconstruction on both simulated and real-world datasets, achieving site-wise accuracy comparable physics-based numerical models. On the WOD dataset of real observations, NUTS reduces NRMSE by 79.9% for phosphate and 19.3% for nitrate relative to the best baseline.
Ablation studies highlight the contribution of each component and offer empirical guidance for designing robust reconstruction architectures.
2 Related Work
This section outlines key related work and provide comprehensive review with extended background and references in Appendix(M [A]
Nutrient Data. Ocean nutrient data are typically derived from observational datasets and simulation- based products. Raw observational archive, such as WOD[Mishonov et al. [2024], provide high-quality and in-situ measurements but suffer from sparse and uneven distribution. In contrast, simulation- based products like the CMEMS Global Ocean Biogcochemical Hindcast (GOBH) Perruche (2018). ECCO-Darwin Carroll et al. 22200]. J coverage data product with coupled physical and biogeochemical dynamics, but require extensive calibration. Furthermore, most of these simulation- based products do not incorporate biogeochemical data assimilation and often employ simplified parameterizaton of biogeochemical processes, resulting in regional biases and uncertainties in nutrient fields.
Griffies et al. [2012] can offer global
Reconstruction Approaches. Traditional methods such as optimal interpolation Conkright et al. (2002). J, 3D/4D- Var Courtier et al. [1994], ensemble Kalman filters Nerger and Gregg 20008) J. and variational inverse models Brasseur and Haus] [199T) rely on data assimilation and inverse modeling to integrate sparse observations with physical dynamics, but are often limited by computational cost

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(a) Ocean Flow (w")
Mean Flow (w") 115 188 12PE
(c) Eddy Flow (v')
188
125E
128E
135T 138
1289 L35T 1387
113E 1886 H2 128' 1357 18 H8
Density Flow Velocity (mis)
0.30
85NN
0.20
0.15
0.10
Kuroshio Current
Eddy Flow
2PN
0.04
Ocean Flow Density
Mean Flow Density
Eddy Flow Density
0.02
0.02
0.04
0.08
0.02
0.04
0.02 0.04
Radial Frequency
Radial Frequency
Radial Frequency
Figure I: Decomposition of surface ocean flow into mean eddy components. Full velocity field containing both large-scale and mesoscule structures. (b) Mcan flow obtained low-puss filtering, capturing large-scale structures such the Kurushio Current. (c) Eddy flow cumputed the residual, representing high-frequency mesoscale variability. Bottom panels display the radial frequency spectra corresponding to each flow component, with energy concentrated at low frequencics for the mean flow and at higher frequencies for the eddy flow, illustrating effective scale separation.
and data sparsity. Recent advances in deep learning offer alternative solutions for spatiotemporal reconstruction. CNN-based models (e.g.. (-oeeBBoneeerrger al.] (2015) and transformers (e.g.. ViT Dosovitky et al. [2020), Farthformer Gao et al. [2022) capture spatial structures but lack physical grounding. Physics-informed approaches such as neural operators [i et al. [2020), Wang et al. 2025). implicit neural representations [uo et al.] [2024], and 4DVarNer [eaacchap et al.
22023 embed governing equations or physical constraints into the learning process to improve physical consistency but are limited in capacity. Foundation models (e.g., Prithvi [Schmude et al. (2024), AtmoRep[esse et al. (2023) ) show promise in meteorology but remain untested marine biogeochemistry. General-purposs impainting methods using GANs Zhao et al. [202T] and diffusion models Lugmayr et al. 2022 perform well vision tasks but lack physical constraints and robustness
to sparse data.
Methodology
Notations. Let $2 denote the unit surface in R3, parameterized by latitude-longitude coordinates
(0) E Ω i- 2: ]] x [-m, x]) For time-dependent function (p(0, o, define Bip The divergence and spherical Laplacian operators denoted V and V respectively.
Problem Setup. The nutrient concentration yo follow the advection advection-diffusion diffusion equation: Lw, ew.n [(p] =
è (wip) 2 = s, where 7) = n7(0Φ) denotes the lnve-invariant diffusion coefficient and s(, Φ, t) represents the external source and sink terms. These terms account for biological uptake and remineralization through photosynthesis, respiration, and demineralization, as well as physical downwelling and upwelling. Given sparse nutrient measurements on 2 x C Q [0, 7] and perturbed ocean flow estimates w, our goal is reconstructing nutrient concentrations by solving the constrained PDE: L [yl = subject to 2l = f'zxT, where / represents observed nutrient concentrations and f denotes its restriction to the subset Z T.
3.1 A Naive Spatial-Temporr Reconstruction Model
In this subsection, we introduce naive spatiotemporal reconstruction model and discuss its advantage over image-ing soageeinpainttig--- based methods. We then demonstrate its sensitivity to mesoscale perturbations in the eddy component of the ocean velocity field.
Naive Model. Given an interval [to: fi]. the naive model first uses a data-driven initializer Fo to estimate the initial nutrient field (xx, 4n) through the velocity field w, auxiliary variables 4. and sparse observations f. The estimate is then propagated by solving the advection-diffusin equation L 8, where both the diffusion coefficient y and source true field. Prior work Schiesser [2012) has demonstrated that this propagation can be implemented via the method of lines (MOL), which discretizes the PDE into a system of first-order ODEs at spatial locations {x: $(xx, t) = Φ(xx, to) + fth [-V (wy) + 2 s] (xx; r)dr, where forward solution can be solved approximately using numerical solvers such Kunge-Kutta, TentaaTeeeeee Veque 2007]. During the model training, all components, i.c.. Fo. and 77 are jointly optimized minimize
are learned to match the
3

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Physice-tooee based Operations
Intermediate Variables
Helesty
Eddy
Refinement Components
Maan Flow
Intarizor
Coarse Field
Rletna Modilo
Qutpur
Natter
w
Fh
FlX
Ady Dr. with Ettccirs Citckon V P-LYY
I Homo PCC
Suutze Modillo 5
Suurce
Ver
Solver
Tarm
Oster Frai
Coarse Module
Figure 2: Overview of NUTS. NUTS two-scale model that combines data-driven initializer, homogenized PDE solver with learned effective diffusion, and refinement module to reconstruct ocean nutrients under sparse observations. The velocity field is decorpposd inlo mean and eddy compunents for scale separation. A learnable source module captures unresolved inputs. Trainable modules are marked with fire icons. Cat denotes channcl concatenation and @ denotes clement-wise addition.
the mean squared error between the prediction X and the ground-truth yp, i.e., min LMSE & ||se - 111
Advantages over Existing Image In-painting Approach. (1) Physical consistency and mass conservation. The naive model evolves nutrient fields through an advection-diffusion PDE, ensuring temporally consistent reconstructions that follow physical transport processes and conserve mass. In contrast, image in painting methods rely purely on spatial interpolation, lacking temporal dynamics and physical grounding. (2) Effective of sparse observations. By jointly learning the initializer, source term, and diffusion coefficient, the naive model directly integrates observational data to constrain transport dynamics, leading more data-consistent estimates in sparsely sampled regions.
Sensitivity to Mesoscale Perturbations. The naive reconstruction approach evolves nutrient esti- mates using velocity fields from numerical circulation models, which accurately capture large-scale mean currents but often misrepresent mesoscale eddies due limited resolution and structural uncer- tainties. Mesoscale eddies are small-scale (10-100 km), high-energy structures that play dominant role nutrient transport. As illustrated in Figure T the true velocity field w can decomposed into smooth mean component w and rapidly fluctuating eddy component v*. When the MOL uses a perturbed velocity field W, = w + 6. structural errors ò in the eddy component introduce an additional transport term into the conectiondiiffsion dynamics:
(xx,, t1) = ((xx, to) +
(w" (p) + V (X),
V (di)) (xk. T) dr. (I)
advection-ddffusion
from perturbation
The perturbation term scales with | which be large, mesoscale eddies typically carry more energy than the mean flow (Figure [T. By Equation (I). such perturbations induce significant transport errors that accumulate and propagate over time. This underscores the need for reconstruction models that are robust to mesoscale flow inaccuracies.
3.2 NUTS: Eddy-Robust Nutrient Reconstruction via Two-Scale Modeling
We introduce NUTS, a principled two-scale model that reconstructs surface occan nutrients from sparse observations and noisy velocity inputs (see Figure 2 Unlike prior approaches, NUTS separates nutrient transport into stable mean dynamics and unstable mesoscale variability. applies homogenized PDE solver for large-scale propagation and refinement module for controlled recovery of fine-scale structure. This decomposition improves robustness and generalization in multiscale ocean flows. All architectural details are provided in Appendix B
Coarse Module Part I: Robust Initializer: The coarse stage begins by estimating the nutrient field pxx;, (o) at the start of the reconstruction interval. To suppress mesoscale noise, we apply a Fourier- based low-pass spatial filter to the input velocity field and extract the mean flow w This filtered flow, along with sparse nutrient observations and auxiliary variables, is encoded by a spatiotemporal transformer that captures long-range dependencies across space and time. The initializer is designed to be robust to flow perturbations and produces stable starting point for physical propagation.
Coarse Module Part II: Homogenized PDE Solver. To evolve the field forward, NUTS applies homogenized udvection-diffusion equation:
(xx,, t1) ((xe, + /'
(KV
(xk, T) dr.
(2)
mean advection
effective diffusion

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This formulation replaces unresolved mesoscale effects with an effective diffusion tensor K (x), which is predicted by a hypemetwork conditioned on S. We discretize the system using the method of lines and numerically integrate it over time. This structured PDE solver ensures stable and physically grounded transport under filtered dynamics.
Refinement Module. The refinement stage corrects residual errors and restores mesoscale vari- ability. takes input the coarse prediction sb, mean flow w", normalized eddy velocity (w- - w w - w lloo, sparse observations, and static covariates. These inputs are tok- enized and passed through a vision transformer that produces the refined estimate Oxxx, t), i.e. ((xx t) = R (p, w', v, $. f] (x. t). Refinement is performed independently at each timestep and learns localized spatial redistribution driven by eddy structures.
Source Term and Conservation Loss. To account for unresolved sources and sinks, we introduce a learnable correction term S(x), where S is parameterized by ResNet. The final prediction is PPimal = & + 8. To enforce physical realism, we define total nutrient mass as M M(+](t) = Ex (xxx, (),
and penalize mass drift through the conservation loss:
Coons. = L.a [M] M( [M] ( M[ dr.
The final training objective is: Lioral = ||:Pnnal - -012 + XCcons. which governs the optimization of all learnable components in NUTS.
Why Two-Scale Modeling Works. The key challenge in reconstructing ocean nutrient fields lies in the dual nature of the underlying dynamics: large-scale currents govern basin-wide transport, while mesoscale eddies induce localized variability and dominate error sensitivity. NUTS addresses this by explicitly separating these two regimes. The coarse module filters out unstable mesoscale fluctuations and models stable transport via homogenized PDE with learnable diffusion. This prevents error accumulation from uncertain eddy inputs. The refinement module then selectively reintroduces mesoscale information- not as direct forcing, but as spatial corrections conditioned on the residual flow. This two-stage architecture mirrors the physical structure of ocean transport and enables both robustness and resolution in a way that single-scale models cannot.
Core Insight:
Ad Advantages over the Naive Approach. NUTS preserves the physical grounding of the naive model, including advection diffussio transport and the effective use of sparse observations. But it adds two critical improvements: (1) Scale-aware architecture. By decoupling mean and eddy-driven dynamics, NUTS reconstructs both broad circulation and localized nutrient features with greater fidelity. (2) Built-in robustness. Homogenization shields the system from mesoscale perturbation errors, while spatial refinement restores resolution without destabilizing temporal evolution.
Context and Relation to Prior Work. While prior hybrid models such as FNO Li et al. 2020)
4DVarNet Beauchamp al. [202]], and GraphCast[aan Lam al. [2023] embed physical priors into data- driven forecasting pipelines, they typically rely on direct PDE application or sarr-toosolyy strategies that do not explicitly separate stable and unstable components. In contrast, NUTS reformulates the transport equation itself: it applies homogenization to eliminate mesoscale instability at the PDE level and delegates high-frequency recovery to a separate spatial refinement module. This scale-aware decomposition is essential for robustness in noisy flow regimes.
3.3 Theoretical Analysis: Effectiveness of NUTS under Eddy Perturbations
We adopt a standard multiscale formulation for ocean velocity Pavliotis and Stuart 2008 modeling w" wx(x, t) = w (x, + v'v y; r), where & < 1 characterizes the scale separation between slow large-scale transport and fast mesoscale variability, and y = x/ē, 7 l/e² are fast space-time variables that resolve high- nigh-frequency eddy dynamics. The mean flow w governs large-scale advec- tion, while v captures mesoscale eddies with rapid, oscillatory fluctuations. This parabolic scaling is standard in homogenization theory for advection-diffusion systems Pardoux and Veretennikov (2005). ensuring that mesoscale variability mixes locally without inducing net large-scale transport. We further assume that both v and the perturbation 8 satisfy the same structural form: periodic and mean-zere zero in the fast variables T). This assumption is classical in homogenization theory Jikov et al. (012) and reflects the physical behavior of mesoscale eddies eddies-bighly highly energetic but oscillatory and net-zero under space-time time averaging.
5

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Theorem 1 (Informal: Accuracy and Robustness under Eddy Perturbations), Suppose that both the true velocity field and the perturbation satisfy the periodic, mean-zew zero eddy flow assumption. Then, under mild regularity conditions, the NUTS prediction & differs from the true solution s" by at most (e)) independent of the perturbation strength |6||
Remark: The formal statement and proof of this result are provided in Appendix C
Interpretation. This theorem establishes two key properties of NUTS. First, the error is O(e) and independent of the perturbation strength ||o]|oo. ensuring robustness: fast, high-amplitude eddy perturbations have negligible impact on the coarse-scale reconstruction. Second, the result guarantees accuracy when the true eddy field varies small spatial and temporal scales (8 < 1). This is nontrivial, as the eddy field enters the dynamics with magnitude 1/e; despite being mean-zero, its local influence is large. The bound confirms that the homogenized model captures the correct large-scale behavior, justifying the use of coarse dynamics in this regime.
4 Experiment
In this section, we answer the following research questions:
RQ1. How does NUTS perform in reconstructing global surface ocean nutrient concentrations compared to existing baselines, using both simulated and real-world observations?
RQ2. Does the proposed two-scale modeling framework enhance robustness to mesoscale perturba- tions? How do filtering strategies and diffusion implementations influence this robustness?
RQ3. How do individual design sroooces-suliee as model architecture, auxiliary inputs, conservation loss, and reconstruction interval- affect reconstruction accuracy?
4.1 Experimental Setup
We present the experimental setup, including datasets, baselines and evaluation metrics. Implementa- tion and training details are in Appendix D Code and data available at URL
Data. We conduct experiments using two datasets for global surface nutrient reconstruction. Simulation Dataset. To sup- port high-quality long-term reconstruction, we release two data products generated by the numerical biogechcaiiog model CMMM6-COBALT|[LIE Liu et a 2022], referred to as M0M6 (Daily) and M0M6 (Monthly). The simulations were conducted on 1000 CPU cores of AMD EPYC 9654 96-Core Processors
Table 1: NRMSES (4) MOM6
(Monthly) and GOBH (Monthly) data com- pared to real observations from WOD.
Data Source
Nitrate
Phosphate
M0M6 GOBII
0.463
0.301 1,335
1.444
over 1 11-day period, spanning 1959 2022 global nominal resolution of 0.5% (576 720). The model output is subsequently regridded to a uniform 0.5" grid (360 x 720) using bilinear in- terpolation. Each data product includes surface nitrate and phosphate concentrations, along with auxiliary variables such as temperature, salinity, and horizontal velocities (u. v). Compared to GOBH (Monthly) Peeruche [2018). our M0M6 (Monthly) data product show improved agreement with in-situ observations from WOD, achieving approximately 60% lower NRMSE on a 0.5 x 0.5% grid (Table D. Additional details are provided in Appendix D. Real Observations. We use in-situ nutrient measurements from the World Ocean Database (WOD) Mishonov et al.] (202)), which contains nitrate and phosphate records from 1959 to 2022. These observations are extremely sparse, covering only 0.16% % of the full spatio-temporal grid. All measurements are regridded to match the spatial and temporal resolution the MOM6 data product.
Tasks. We evaluate the model on two resolution-speecfii nutrient reconstruction tasks. Daily Average Recon- struction. Sparse observations are simulated by ran- domly sampling nutrient values from the MOM6 (Daily) dataset at sparsity levels of 0.1 %6 1 1,, and 10%. The 0.1% level reflects the sparsity of real-world observations, while 10% aligns with settings used in prior work Luo et al. [2024). The model reconstructs full daily nitrate and phosphate fields using these samples together with M0M6 daily flow and auxiliary variables. Monthly Average Recon- struction. Real-world nutrient measurements from WOD and monthly flow and auxiliary variables from MOM6 are used to reconstruct complete monthly averages of nutrient fields. Dataset partitions summarized in Table 2 with sampling details in Appendix D.4
Table 2: Overview of dataset divisions by year.
Task
Train
Validation
Test
Daily Avg. 2019. 2020
2021
2022
Montily Avg.
1959-1998
1999-2010
2011-2022

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Table 3: NRMSE (J) of different models for reconstructing (1) global daily average nutrient concentrations from the MOM6 simulation under sampling ratios 0.1%, 1%, and 10%, and (2) global monthly average concentrations from WOD observations. Params denotes the number of model parameters. The numbers after = standard errors under 3 trials.
M0M6 (Daily)
WOD (Monthy)
Methods
Params
Phusphate
Nitrate
Phosphate
0.1%
1%
10%
0.1%
1%
10%
Kriging(Esp.)
0.535 022
0.262
0.184
0.642=0.020
0.268=0.025
0.256-0.012
1.275 100
1.495 091
(pig
0.537.2.00 019
0.276 022
0.192
0.64920.017
0.39920.018
0.272_0.021
1.270_20.066
1.517 057
4D-Nar
0.3M
0.151
0.
0.154+0.012
0.0000
0.168
0.170-0.007
0,187
0.203++0.0 009
Marble
0.6M
0.397:00.05
0.227::0.044
0.232:0.058
2..41
0.222.0.064
=.297=0.047
0.363
20.05
0.326. 056
FNO
4.8M
0.251 015 0.227 010
0.22910.014
0.261
0.256_0.013
0.257_0.0r
0.244.0.015
0.276 017
U-Net
31.0M
0.151 cos
0.148.20 .13
0.149
0.169
007
0.166=0.012
0.167 =0.17
0.174=0.012
0.187 008
VIT
77.7M
0.257+0.052
0.242+-0.0
.044
0.359+0.048
0.31140.040
0.256-0.044
0.256
0.263+0.084
0.260++..00 002
AtmoRep
0.7B
0.196::0.110
0.194-2:0.0
0.192:20.100
0.19020.000
0.219=0.011
0.218=0.015
0.206:20.013
0.260 013
Prithvi
2.3B
0.21610.005 055
0.19720.00 o4g
0.208
0.279gn.ocs
0.274_p.057
0.275_0.0a6
0.22220.042
0.338g 048
NUTS
125.6M
0.01420.000 002
0.015 001
0.022 002
0.143 o0a
0.136 ona
0.142.20
004
0.035 002
0.151
Promotion
90.7%
89.9%
85.2%
14.9%
18.1%
11.8%
79.9%
19.3%
Baselines. We compare our model against wide range of baselines grouped into six categories: (1) Kriging interpolation with exponential and spherical variogram models; (2) CNN- based model
U-Net Ronnebbrgrr et al. [2015]; ; (3) transformer-basee based model ViT Dosovviivt et al. [2020]: : (4)
neural operator Fourier Neural Operator (FNO) Li et al. [2020]: (5) implicit representation method
Marble model Wang et al. [025]; : (6) foundation models pretrained on climate data, including Prithvi WxC Schmude et al. [2024] and AtmoRep SappoRepLeessie Lessig et al.
[0233: ; (7) physics-guided hybrid
assimilation model such as 4D 4er [Beauchamp et al. 2023 All baselines except Marble and
Kriging reconstruct each frame independently using static inputs observations, auxiliary variables, and velocity fields at a single time step. Marble leverages temporal observations but excludes auxiliary variables and flow inputs. Kriging uses only static observations. In contrast, model takes temporal sequences of all inputs and generates spatiotemporal nutrient reconstructions. See Appendix B.2 D.5 for details.
Metrics. We use Normalized Root Mean Squared Error (NRMSE) to evaluate model performance, which ensures scale independence Shcherbakov et al. [2013). We first calculate the latitude- weighted RMSE between the reconstructed values and the corresponding ground-truth, while NRMSE is obtained by normalizing RMSE using the mean of the ground-truth.
4.2 Main Results (RQ1)
We compare the reconstruction performance of our model on simulation and observation data as summarized in Table Jnd Figure
Obs 1: NUTS achieves the lowest NRMSE across all daily and monthly reconstruction tasks. We evaluate performance under varying observation sparsity across simulated and real-world datasets. As shown in Table 3, NUTS consistently outperforms all baselines. On the daily task with 0.1% sparsity, it reduces NRMSE by 90.7% for phosphate and 14.9% for nitrate compared to U-Net. On the monthly WOD dataset, it achieves 79.9% and 19.3% improvement, respectively. The gain is more substantial for phosphate, which exhibits smoother temporal variation and is easier to model dynam- ically. Figure supporrss this, showing the spatial distribution of the phosphate-to-nitee ratio of coefficients of variation (CVs), where each CV is defined as the temporal standard deviation divided by the mean concentration. Lower values indicate weaker phosphate fluctuations, which NUTS captures more reliably.
10" Phos. CV Nitr. CV
10'
Figure 3: Spatial distribution the phosphate-to-nitee ratio of coeffi- cients of variation.
Among baselines, U-Net and 4D- VarNet perform best. U-Net extracts multiscale features via skip-connected encoders Ronneberger et al. (2015), while 4D- VarNet enforces physical consis- tency through advection- multiscale modeling and physics-based levels. These gains are especially pronounced under low observation density, where auxiliary physical variables become essential for accurate reconstruction. Baselines that lack such inputs- as Kriging (Exp.) and (Sph.)-xhhin large accuracy drops. In contrast, NUTS remains robust by
design Beauchamp et al. [2023]. NUTS combines both principles -iiaaii-yiieling consistent improvements across sparsity
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Table 4: Model Analysis (NRMSE J). (a) Comparison of different low-pass filter types; (b) Evaluation of cutoff ratios for frequency filtering; (e) Comparison of iivectionddiffusion implementations, including advection-only, fixed diffusion matrix, and learned ditfusion network. Unless otherwise specified, the target nutrient is nitrate, the low-pass filter is Fourier-based with a cutofl ratio of 1. and the diffusion module is implemented using 6-layer ResNet.
(a) Filter Type.
(b) Filter Cutoff Ratio.
(c) Implementation of Advection Diffusion.
filter
Daily
Monthly
param.
Daily
Monthly
case advection-only
Daily
Monthly
Fourier
0.136
0.151
0.1
0.136
0.151
0.138
0.176
Wavelet
0.144
0.197
0.2
0.145
0.194
adv. + diffusion matrix
0.142
0.165
Gaussian
0.143
0.169
0.5
0.145
0.189
adv. + diffusion network
0.136
0.151
Moving Avg.
0.154
0.167
1.0
0.171
0.189
leveraging oceanographic drivers like sea surface temperature, as further confirmed in our ablation study in Section
Obs 2: In reconstructing real observation site records, our model outperforms data-driven baselines and matches the performance of traditional numerical methods. We evaluate the site-wise reconstruction accu- racy by training on 75% of WOD sites and testing on the remaining 25%. As shown in Figure 4 NUTS achieves site-wise NRMSES of 1.32 for phosphate and 2.18 for nitrate, outperforming all data-driven baselines. Its per- formance is comparable to traditional numerical models, including MOM6 and GOBH, demonstrating strong gen- eralization under real-worll sparsity.
Para Driven Model Inversion Model
NEEXSN NHARSE
8.00
5.64
5.77
2.73
3.8t
3,00
NUTSS
NUYS
2.36
2.55
2.56
2.07
2.51
2.48
218
DD-Vartes
Maride
PSO
UNe
V/T
AtmoRty
Nula
MOMs
GOIBI
4D
Figure Site-wise NRMSE (L) of different meth- ods evaluated WOD real observation.
4.3 Component Analysis: Contribution of the Coarse and Refinement Modules (RQ2)
We evaluate the contribution of two key coarse-stage diffusion-aa as well as the refinement module. NUTS is compared against U-Net and three ablated variants, each omitting a specific component while keeping all other settings fixed. The structural details of these variants are summarized in Table and all variants are marameter-matcher with NUTS for a fair comparison. All ablation results reported in this section use nitrate as the reconstruction target. Results for the daily task are reported under a 1 1% sparsity ratio. Full hyperparameter configurations are provided in Appendix D.5]
lowooeents--wwppsss filtering and effective
Obs 3: Our model achieves both robustness and accu- racy; filtering alone improves stability but sacrifices mesoscale information. We assess robustness by perturb- ing the eddy component v* using Fourier-based scaling, generating = -v", and injecting it into the velocity field, As shown in Figure 5 NUTS maintains low NRMSE across all perturbation levels, demonstrating strong re- silience mesoscale variability. Naive-B. which directly propagates the full velocity field without filtering, suffers large errors especially a = +1 highlighting its sensi- tivity to unresolved eddy perturbations. Naive-F improves robustness by suppressing high- high-frequency noise but exhibits degraded accuracy due to the removal of informative mesoscale signals. In contrast, NUTS combines the strengths of both: the coarse stage stabilizes dynamics through filtering, while the refinement stage recovers fine-scale nutrient structure conditioned on residual eddy flow.
Table 5: Overview of Model Ablation Variants. "B". "F" and "F+D" represent the buse model, the base model with filtering, and the base model with both filtering and diflusion, respectively. de- notes inclusion; x denotes exclusion.
Variants
Params Count
Low-pass Filler
Effective Refine Module
Diflusion
Naive-B Naive-J
131.9M
131.9M
Nohhe-(F+D)
131.7M
NUTS
125.6M
Naive-B
Obs 4: Effective diffusion enhances filtered transport, but refinement is essential for recovering mesoscale structure. As shown in Figure 3 Naive-(F+DD- combines flow filtering with the effective diffusion mod- ule achieves lower RMSE than Naive-F and remains robust under mesoscale perturbations. This validates the dynamics to sta-
NRMSE
0.23
ULNot. NUTS
(a
0.20
0.15
use of homogenized advection-diffusee bilize transport and retain partial mesoscale effects. How- ever, despite comparable architecture and parameter count, Naive-(F+D) still underperforms our full model, highlight-
-1.0
-0.5
0.0
0.5
Scaling Factor y
Figure 5: NRMSE of different models under vary- ing mesoscale perturbation levels.
8

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Table Ablation Study (NRMSE 1). (a) Comparison course-stage initializers, including static and dynamic architectures: (b) Analysis of model depth the coarse module; (c) Analysis of model depth in the refinement module; (d) Evaluation source and conservation loss terms; (e) Quantification of the impact auxiliary input variables; (f) Assessment sensitivity temporal interval length. All experiments use the default setting: coarse/refine depth of 12/6, all loss terms and inputs included, und interval length set 4.
(a) Coarse Model Structure.
(b) Depth of Coarse Module.
(c) Depth of Refine Module.
case
Daily
Monthly
depth
Daily
Monthly
depth
Daily
Monthly
2D CNN
0.206
0.161
6 0.185
0.160
2
0.142
0.164
VIT
0.159
0.157
8
0.165
0.153
4
0.146
0.210
3D CNN
0.162
0.153
12
0.136
0.151
6
0.136
0.151
NUTS
0.136
0.151
16
0.148
0.177
8
0.180
0.170
(d) Source and Conservation Loss.
(e) Auxiliary Variables.
(f) Interval Length.
case
Daily
Monthly
removed var.
Daily
Monthly
length
Daily
Monthly
w/ src, w/ cons.
0.136
0.151
temp. salt
1.059
1.014
I 0.159
0.157
w/ sIC, w/o cons.
0.153
0.151
0,170
0.168
2
0.166
0.151
w/o sne, w/ cons.
0.156
0.155
0,164
0.155
4
0.136
0.151
w/o sre, w/o cons.
0.155
0.154
0.142
0,160
8 0.220
0.158
ing the importance of the refinement module in reconstructing fine-scale nutrient variability lost during filtering.
Obs 5: Filter design in the coarse module is critical; Fourier filtering with strong higu-freyeency suppression yields the best performance. We ablate the design of the low-pass filter used in the coarse module of NUTS. Among several options, the Fourier filter achieves the lowest NRMSE on both daily (0.136) and monthly (0.151) tasks, outperforming wavelet, Gaussian, and moving average filters (Table 4ap. This result is consistent with prior work in ocean modeling and geophysical fluid dynamics Abernathey and Marshall 2013), Callies and Ferrari (2033), where spectral -roued)r-based) filtering is widely adopted to separate scale flow from unresolved mesoscale variability. We further vary the cutoff ratio of the Fourier filter, which determines the extent of eighfrequency suppression. Lower retioing more unresolved eddy components-consiae
improve reconstruction accuracy, while higher ratios degrade performance (Table (bb). These results highlight that principled filtering in the coarse module is essential for stabilizing nutrient transport, while fine-scale variability is later recovered by the refinement stage.
Obs 6: Incorporating a learnable diffusion module improves accuracy; state-dependent designs further enhance performance. We ablate the diffusion design in the adveetion-diffusion solver of NUTS. We compare three variants: (1) advection-only. (2) with trainable, time-invariant diffusion matrix K UUT, and (3) the state-dependent formulation used in NUTS, where K GGT and
G = G p is produced by hyper-network conditioned on the coarse prediction ø. As shown in Table 4c both diffusiin-eehannced variants outperform the advection-only baseline on daily and monthly tasks, confirming the benefit of modeling unresolved subgrid dispersion. The state-dependent design used in NUTS further improves accuracy over the fime-invariant variant (0.151 vs. 065 on the monthly task), consistent with the theoretical expectation that effective diffusion depends on the tracer state Mc McWiliiams (2006), McDougall and McIntosh [2001]. The improvement is more substantial in the monthly setting, where longer temporal scales allow diffusion to play more dominant role in shaping nutrient transport.
5 Discussions
Ablation Study (RQ3). We evaluate the impact of architectural selection of both modules, source module, loss design, auxiliary variables and temporal interval length on model performance. Ad- ditional ablation results on spatial and temporal resolution, as well as the conservation loss weight coefficient, are provided in Appendix E.I] All ablation results in this section use nitrate as the target variable. Coarse and Refine Module Structure. We evaluate architecture and depth for both the coarse and refinement modules. As shown in Table 6a] static 2D CNNs underperform due to the lack of temporal modeling, while dynamic architectures 3D CNN and spatiotemporal ViT achieve lower errors. NUTS, which uses spatiotemporal transformer, yields the best NRMSE of 0.151. Depth analysis (Tables 6b 6c) shows that performance peaks with 12 layers in the coarse module and 6 layers in the refinement module. Shallower models underfit, while deeper ones degrade due to over-smoothing or training instability. These results highlight the importance of both dynamic structure and moderate depth. Source and Conservation Loss. Incorporating the source module

---

and conservation loss enhances reconstruction accuracy (Table Additionally, the conservation loss contributes to preserving total nutrient mass, detailed in Appendix E.2 Auxiliary Variables. Sea surface temperature is the most influential auxiliary input, with its removal causing the largest increase reconstruction error (Table (6). This highlights its essential role in guiding nutrient reconstruction and is consistent with prior findings in related work such as 4DVarNet [Beauchamp ct al. 2023]. Reconstruction Interval Length. Model performance is sensitive to the choice of reconstruction interval length, with both short and long intervals resulting in higher error relative to intermediate settings (Table 61). In the daily task, 4-step interval yields the lowest NRMSE (0. (0.136), balancing informative temporal context and noise from redundancy uncorrelated variability.
Conclusion and Broader Impact. We present NUTS, two-stage, physics-informed framework for reconstructing global surface ocean nutrients from sparse observations. By combining coarse iivection-diffusiin dynamics with data-driven refinement, NUTS achieves state-of-the-a per- formance on both simulated and real-world datasets. While our experiments focus on nitrate and phosphate, the framework grounded in general transport physics and naturally extends other passive tracers. Preliminary results (Appendix [F show promising generalization, supporting broader applications in envirommental reconstruction, climate monitoring, and Earth system science.
Future Work. Future directions include extending NUTS in three areas: spatial coverage, biogeo- chemical complexity, and air-sea exchange. A 3D extension will capture vertical transport and subsurface gradients. Adding processes like remineralization and nutrient uptake will improve model- ing of regeneration and biological consumption. Air-sea gas exchange will enable reconstruction of gas tracers for carbon and oxygen eycle monitoring.
Acknowledgements
This research is supported by the National Natural Science Foundation of China (No. 62306179), the National Key Research and Development Program of China (2023YFC2808802) Southern Marine Science and Engineering Guangdong Laboratory (Zhuhai) (nos. SML20233P2199, the Ocean Negative Carbon Emissions (ONCE) Program.
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Reviews:
[{"review_id": "FCJk5WVSzX", "reviewer": "Reviewer_9Xop", "summary": "This paper proposes a model named NUTS pioneers a theoretically grounded, two-scale framework for eddy-robust nutrient reconstruction, achieving SOTA accuracy. While computationally heavy and surface-limited, its scale separation strategy and proven robustness offer a template for multiscale geoscientific modeling.", "questions": "1. The refinement module processes each time step independently, which may ignore the temporal dependencies in the eddies, and may limit the coherence of long-term reconstructions.\n2. the overview of NUTS (Figure 2) makes the method look too complicated, with multiple connections between each module, and whether there is suspicion of over-modeling, which will make the work difficult to follow.\n3. I carefully want to see the model complexity, computational complexity, and time complexity of total model and each module.\n4. I am very interested in confirming whether the model code will be open source and to what extent.", "limitations": "Please check the Weaknesses and Questions", "rating": 4, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths\n1. This paper integrates homogenization theory (from multiscale PDEs) into deep learning for ocean sciences. The coarse module ensures stability, while the refinement recovers details without propagating errors.\n2. Formal proof (Theorem 1) guarantees immunity to mesoscale perturbations—addressing a key flaw in prior data-driven/assimilation methods.\n3. The experiments are implemented on simulated (MOM6) and real (WOD) data across daily/monthly scales. it achieves 79.9% NRMSE reduction for phosphate and 19.3% for nitrate vs. best baseline on real-world (WOD) data.\nWeaknesses\n1. The learnable source/sink module (for biological processes) lacks detailed evaluation. Its impact on long-term nutrient cycling is not quantified beyond ablation (Table 6d).\n2. NUTS has 125.6M parameters, exceeding lightweight baselines (e.g., 4DVarNet: 0.3M). Training requires high-resolution flow data and multi-step PDE integration, limiting accessibility for resource-constrained researchers. Its model complexity, computational complexity, and time complexity are also indicators that need to be considered.\n3. Performance relies heavily on Fourier filtering with strict cutoff ratios (Table 4b). Real-world flows may violate assumed scale separation, risking over-smoothing in dynamically complex regions (e.g., boundary currents).", "was_revised": true, "final_justification": "Thanks to the author for the rebuttal, I will keep my positive review score.", "rebuttal": "We sincerely thank the reviewer for the valuable feedback. Below, we provide a point-by-point response and hope the response can effectively address your concerns.\n\n## Weaknesses\n\n> **W1:** The learnable source/sink module (for biological processes) lacks detailed evaluation. Its impact on long-term nutrient cycling is not quantified beyond ablation (Table 6d).\n    \n**Response:** Thank you for your insightful comment. To quantitatively evaluate its long-term effect, we added an additional analysis during the rebuttal phase: we compute the monthly variation rate of total nutrient mass, defined as the ratio of each month's total mass to that of January in the same year, across four distinct years. As shown in **Table T1**, the refinement module maintains variation rates within $\\pm 0.4$\\%, adhering to the mass conservation constraint. Moreover, by introducing the source module, NUTS closely follows the ground-truth (G-T) variation trends. Since the final output of NUTS is defined as the sum of the learned source contribution and the refinement correction, this result highlights the source module’s ability to capture seasonally varying source–sink dynamics and maintain physically consistent nutrient cycling over time. Experiments are conducted on the Monthly dataset with nitrate as the target nutrient.\n\n**Table T1**: Monthly variation rates (\\%) of total nutrient mass from Refinement Module, NUTS and ground-truth (G-T) across two years. The variation rate is defined as the ratio of each month’s total nutrient mass to that of January within the same year.\n\n| Year | Data       | Feb.   | Mar.   | Apr.   | May   | Jun.   | Jul.    | Aug.    | Sep.    | Oct.    | Nov.    | Dec.   |\n| ---- | ---------- | ----- | ----- | ----- | ----- | ----- | ------ | ------ | ------ | ------ | ------ | ----- |\n| 2014 | Refinement | 0.00% | 0.01% | 0.03% | 0.03% | 0.04% | 0.03%  | 0.02%  | 0.00%  | 0.00%  | 0.00%  | 0.01% |\n|      | NUTS       | 0.17% | 2.14% | 5.40% | 3.10% | 3.54% | 6.78%  | 8.59%  | 10.60% | 10.04% | 6.97%  | 2.94% |\n|      | G-T        | 0.11% | 1.38% | 2.61% | 3.31% | 4.55% | 6.60%  | 8.08%  | 7.45%  | 7.28%  | 5.09%  | 0.43% |\n| 2020 | Refinement | 0.02% | 0.02% | 0.03% | 0.03% | 0.04% | 0.04%  | 0.04%  | 0.04%  | 0.03%  | 0.04%  | 0.04% |\n|      | NUTS       | 3.59% | 5.13% | 6.03% | 8.09% | 8.09% | 12.07% | 14.38% | 15.25% | 16.11% | 12.35% | 7.84% |\n|      | G-T        | 1.00% | 2.99% | 4.99% | 6.57% | 7.94% | 10.67% | 13.07% | 14.08% | 13.36% | 11.22% | 7.69% |\n\n> **W2:** NUTS contains 125.6M parameters compared to 0.3M in 4DVarNet, leading to higher model, computational, and time complexity.\n    \n**Response:**  Thank you for raising this concern. While NUTS has 125.6M parameters—significantly larger than 4DVarNet (0.3M)—it achieves a markedly lower reconstruction error on phosphate (NRMSE: 0.035 $\\pm$ 0.002) and demonstrates comparable computational efficiency under matched hardware settings. To ensure fair comparison, we report training and inference times in Appendix D.6 of the supplementary materials (Table 15), where all models were trained and evaluated on two RTX 6000 Ada GPUs with identical batch size (8). Under this setup, NUTS trains in **92s/epoch** and infers in **58s/epoch**, closely matching 4DVarNet's training and inference times (**94s/epoch** and **46s/epoch**, respectively), despite having over **400 $\\times$ more** parameters.\n\nThis underscores a key point: **efficiency is not solely determined by parameter size**. 4DVarNet incurs high cost due to its iterative variational optimization, requiring repeated forward–adjoint PDE solves and offering limited parallelism. In contrast, NUTS employs batched, multi-step PDE rollouts with data-driven initialization, enabling efficient GPU acceleration and scalable inference.\n\nFurthermore, compared to foundation-scale models like Prithvi (2.3B) and AtmoRep (0.7B), NUTS is **10-20 $\\times$ smaller**, faster to train and deploy, and more accurate. It thus strikes a practical balance: accurate and efficient relative to both lightweight and large-scale baselines. Pretrained checkpoints and evaluation scripts are provided to support ease of use.\n\n\n> **W3:** Fourier filtering with strict cutoff ratio may lead to over-smoothing in dynamically complex regions such as boundary currents.\n\n**Response:** We thank the reviewer for this thoughtful observation. While Fourier filtering may suppress high-frequency features in complex regions such as boundary currents, NUTS explicitly addresses this by reintroducing the **normalized filtered component** in the refinement module to recover eddy-driven variability. To ensure physical coherence, we further impose a **mass conservation loss** during refinement, preventing artificial changes in total tracer mass. This design allows NUTS to restore fine-scale detail while maintaining global consistency.\n\n## Questions\n> **Q1:** The refinement module may overlook temporal eddy dependencies, limiting long-term coherence. \n    \n**Response:** We thank the reviewer for this valuable point. Although the refinement module operates independently at each time step, we impose a **mass conservation loss** (Eq. 162–163) to ensure no artificial tracer mass is introduced. Since the homogenized PDE preserves mass at the coarse level, this constraint enforces consistency during refinement and introduces an **implicit temporal linkage** across the refined predictions.\n\nWe agree that capturing temporal dependencies in the eddy component could be beneficial. To test this, during the rebuttal phase, we implemented a spatio-temporal transformer that integrates 4 frames of eddy components into the refinement stage. However, performance worsened—NRMSE on nitrate increased from 0.151 to 0.162 on the Monthly dataset—and computation rose from 990 to 1215 GFLOPs. We suspect accumulated truncation errors in the coarse fields disrupt temporal attention, offsetting potential gains.\n\n> **Q2.1:** Figure 2 makes NUTS appear overly complex, with many inter-module connections that may hinder readability.\n\n**Response:** Thank you for raising this point. In the final version, we will simplify the figure to enhance readability. Specifically, we will merge functionally related components--for example, merging the two \"**Coarse Nutrient Field**\" boxes into a single box labeled \"**Coarse Nutrient Field  $\\bar{\\varphi}(\\mathbf{x}_k, t)$**\". Additionally, we will streamline the data flow by removing some elements, such as the \"**Mass Conservation Loss**\" box, to reduce visual clutter and overlapping connections.\n\n> **Q2.2:** Is there over-modeling in NUTS?\n\n**Response:** We thank the reviewer for this thoughtful question. We believe each component in NUTS is functionally necessary, as confirmed by extensive ablation studies. (1) In the coarse module, spatio-temporal architectures outperform static baselines, and transformer-based models consistently surpass CNNs due to superior capacity (Table 6(a)). (2) In the refinement module, a moderately deep network is essential to recover fine-scale nutrient structure from eddy flows (Figure 5, Table 6(c)). (3) The source module and conservation loss both contribute measurable gains (Table 6(d)), with the conservation loss further enforcing mass conservation (Appendix E.2 of the supplementary materials). These empirical results underscore the necessity of the proposed model components.\n\n> **Q3:** I would like to see the model, computational, and time complexity of the full model and each module.\n\n**Response:** We thank the reviewer for highlighting this important point. **Table T2** provides a detailed breakdown of **parameter count**, **computation (GFLOPs)**, and **runtime per forward pass** (batch size = 8) for each component of NUTS. While the PDE solver has negligible parameter count (0.04M), its compute (181 GFLOPs) and runtime (75ms) are comparable to the coarse and refinement modules due to the cost of multi-step integration. The coarse and refinement modules account for most of the complexity (328 and 300 GFLOPs), yielding a total of 125.6M parameters, 990 GFLOPs, and 413ms per batch. We also refer the reviewer to Appendix D.6 of the supplementary materials (Table 15) for full comparisons against other models. Despite its larger size, NUTS achieves state-of-the-art accuracy while maintaining competitive training and inference efficiency under matched hardware settings.\n\n**Table T2:** Number of parameters, computation complexity and time complexity of NUTS and its components.\n\n| Module            | # Params (M) | Computation (GFLOPs) | Time (ms) |\n| :-----------------: | :------------: | :--------------------: | :---------: |\n| Coarse Module     | 83.2         | 328                  | 105       |\n| PDE Solver        | 0.04         | 181                  | 75        |\n| Refinement Module | 42.1         | 300                  | 207       |\n| Source Model      | 0.04         | 181                  | 26        |\n| **NUTS**   | **125.6**    | **990**              | **413**   |\n\n> **Q4:** Will the code be open-sourced, and if so, to what extent?\n\n**Response:** Thank you for raising this point. We open-sourced the code for training and testing our NUTS, as well as the Daily and Monthly datasets. While we are prohibited to provide external links according to the guidelines, the URL in Sec 4.1 directs to the anonymous code repository containing the code and datasets.\n\n___\nThanks again for your valuable reviews and we hope our response can resolve your concerns. Please don't hesitate to contact us if you have further questions. Looking forward to hearing from you."}, {"review_id": "cIbGbIVTEJ", "reviewer": "Reviewer_9Yz6", "summary": "The authors present NUTS, an ocean nutrient reconstruction model that tackles the reconstruction problem using two different scales. The first scale models large-scale transport (mean flow) while the second scale models mesoscale variability (eddy flows). The first scale is implemented using a coarse module that leverages a homogenized PDE solver, which shields the system from mesoscale perturbation errors. To not lose out on reconstruction accuracy, the second scale performs refinement and learns localized spatial redistribution driven by eddy structures. With respect to results, the authors find NUTS to perform favorably compared to all baselines, outperforming all for global reconstruction and performing on par with numerical models for site-wise accuracy. Finally, the authors also include extensive ablation studies on the various components of their model.", "questions": "**Clarifications / suggestions**\n1. Line 106: what does $x$ contain here? Is it just the $\\theta$ and $\\phi$?\n2. Figure 2: does the source module implement both the source term and the conservation loss?\n3. Line 130: could the authors show why the perturbation term scales with $||\\delta||$?\n4. Line 145 - 146: the authors state \"The initializer is trained to be robust to flow perturbations ...\". How exactly is this done?\n5. Line 163: is it this loss that is driving the updates for all the learnable parameters in Figure 2?\n6. Line 260 - 261: could the authors write out the formula being used for computing the NRMSE? That ways it's very clear exactly what is being computed. Especially the normalization confuses me a bit.\n7. Are the numbers after the +- in Table 3 standard errors? Or something else? Please clarify.\n8. Figure 4: it's not super clear what P and N in the legend are without the context in the surrounding text. Maybe these can just be written out into phosphate and nitrate for improved readability?\n9. While lines 312 - 315 state that naive-B suffers large errors because it directly propagates the full velocity field without filtering, it seems from Figure 5 that NUTS suffers (relatively) equally for gamma tending towards -1. Could the authors comment on this? \n10. The paragraph starting on Line 332 doesn't make any reference to Table 4 even though it's the results in Table 4 that are under discussion. Maybe the authors could add an explicit reference in the same way they do it in the next paragraph?\n11. Table 6: it seems there is result tables for a - f but the caption only has explanations for a - e (some of which don't match their respective table, I think). Please fix.\n12. Table 6 (f): what are the units of the interval length? And is this the length of the prediction or the length of the context? Also, why are the short intervals worse than the intermediate ones? This feels counterintuitive.\n\n**Grammar**\n1. Line 260: change \"obatained\" to \"obtained\"", "limitations": "yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "## Strengths\n\n### Quality\nGenerally, I found the quality of the paper to be very good for several reasons:\n1. The method generally seems principled, drawing on homogenization with PDEs to circumvent the eddy current effects.\n2. The method works well compared to the many baselines being considered.\n3. The authors have done a pretty extensive job checking the importance of several design choices of their model as well as provide ablation studies on things like architecture, loss functions being used, etc.\n\nOne final highlight is lines 327 - 331 combined with Figure 5: I think the authors did a great job here highlighting the importance of the refinement module. The gap between Navie-(F+D) and NUTS is substantial which provides clear and convincing evidence for the need to restore full resolution.\n\n### Significance\nI'm not super well equipped to rate the paper on significance since I'm not super familiar with the literature, but based on the provided related work the proposed method seems like a significant contribution to the area. Specifically, it seems the work directly attacks some of the key pain points like adherence to physical constraints and handling of sparse data.\n\n## Weaknesses\nI did not find the work to have significant weaknesses. However, I did have a bunch of minor places in the paper where I got confused, and have listed some clarifications / suggestions around this in the \"Questions\" section.", "was_revised": true, "final_justification": "While I had raised no significant weaknesses in the initial review, the authors have done a great job clarifying the questions and confusions I brought up. After reading the rebuttal, I maintain confident in my positive score for the paper.", "rebuttal": "We sincerely thank the reviewer for the detailed and constructive feedback. We greatly appreciate the time and effort devoted to evaluating our work. Please find our point-by-point responses below.\n\n# Questions\n\n> **Q1:** Line 106: what does $\\mathbf{x}$ contain here? Is it just the $\\theta$ and $\\phi$ ?\n\n**Response:** Yes, $\\mathbf{x}$ refers only to latitude and longitude: $\\theta$ and $\\phi$. We will revise Lines 90–91 to clarify this as follows: \"*..., parameterized by latitude–longitude coordinates $\\mathbf{x} = (\\theta, \\phi) \\in \\Omega = [-\\frac{\\pi}{2}, \\frac{\\pi}{2}] \\times [-\\pi, \\pi]$*.\"\n\n> **Q2:** Figure 2: does the source module implement both the source term and the conservation loss?\n\n**Response:** Thank you for raising this helpful point. The source module handles the source term exclusively, while the mass conservation loss is implemented separately by computing (1) the inter-frame change in $\\bar{\\varphi}$ and (2) the frame-wise difference between the coarse nutrient field $\\bar{\\varphi}$ and the refined field $\\hat{\\varphi}$. We agree that Figure 2 may be misleading in this regard, and we will revise it by removing the mass conservation block to avoid potential confusion.\n\n> **Q3:** Line 130: could the authors show why the perturbation term scales with $\\Vert \\delta \\Vert$ ?\n\n**Response:** Thank you for the insightful question. Let us write the velocity perturbation as $\\delta = \\|\\delta\\| \\cdot \\hat{\\delta}$, where $\\|\\delta\\|$ is a spatially varying scalar field and $\\hat{\\delta}$ is a unit vector field. Applying the product rule:\n$$\n\\nabla \\cdot (\\delta \\varphi) = \\varphi \\hat{\\delta} \\cdot \\nabla \\|\\delta\\| + \\|\\delta\\| \\left(\\hat{\\delta} \\cdot \\nabla \\varphi +  \\varphi \\nabla \\cdot \\hat{\\delta}\\right).\n$$\nThis decomposition shows that the second term in the perturbation  scale linearly with $\\|\\delta\\|$. We will clarify this point in the revised text.\n\n> **Q4:** Line 145 - 146: the authors state \"The initializer is trained to be robust to flow perturbations ...\". How exactly is this done? \n\n**Response:** Thank you for this thoughtful question. To enhance robustness to flow perturbations, we apply a Fourier filter in the coarse module to remove high-frequency eddy components from the flow field before passing it to the initializer. This ensures that the initializer operates on the large-scale, stable part of the flow and is less sensitive to high-frequency variability. The original phrasing \"is trained to be robust\" reflects that all components of NUTS are trained jointly. To more accurately convey this architectural intent, we will revise Lines 145-146 to read: \"*The initializer is designed to operate on filtered, low-frequency flow fields, ensuring robustness to high-frequency perturbations...*\".\n\n> **Q5:** Line 163: is it this loss that is driving the updates for all the learnable parameters in Figure 2? \n\n**Response:** Yes. All learnable parameters in Figure 2 are optimized via the total objective $\\mathcal{L}_{\\text{total}}$. \n\nFor clarity, we will revise the sentence to: The final training objective is \"*The final training objective is $\\mathcal{L}_{\\text{total}}=...,$ which governs the optimization of all learnable components in NUTS.*\", which governs the optimization of all learnable components in NUTS.\n\nWe hope this revision makes it clearer that $\\mathcal{L}_{\\text{total}}$ governs the training of all learnable components in the model.\n\n> **Q6:** Line 260 - 261: could the authors write out the formula being used for computing the NRMSE? \n\n**Response:** Thank you for raising this point. We provide the full formula for computing NRMSE below and refer the reviewer to Appendix D.2 of the supplementary material for a detailed explanation. Let $y_{thw}$ represents the ground-truth value at a give time $t$, latitude $h$ and longitude $w$, and $u_{thw}$ is the corresponding reconstructed value, the RMSE is calculated as follow:\n$$\n    \\text{RMSE} = \\frac{1}{N} \\sum_{t=1}^{N} \\sqrt{\\frac{1}{HW} \\sum_{h=1}^{H} \\sum_{w=1}^{W} \\alpha(h) (y_{thw} - u_{thw})^2},\n$$\n\nwhere $\\alpha(h)=cos(h)/(\\frac{1}{H}\\sum_{h'}cos(h'))$ is the latitude weight.\nThe NRMSE is calculated by normalizing RMSE with the mean of the ground-truth values:\n$$\n    \\text{NRMSE}=\\frac{RMSE}{\\frac{1}{NHW} \\sum_{t=1}^{N} \\sum_{h=1}^{H} \\sum_{w=1}^{W}y_{thw}}.\n$$\n\nNRMSE quantifies the relative reconstruction error normalized by the mean nutrient concentration, providing a scale-independent measure of accuracy. We hope this clarification is helpful, and please refer to Appendix D.2 of the supplementary materials for further details.\n\n> **Q7:** Are the numbers after the $\\pm$ in Table 3 standard errors? Or something else? Please clarify.\n\n**Response:** Yes, the numbers following the $\\pm$ symbol in Table 3 are standard errors computed over 3 trials. We will revise the caption in the final version to explicitly state this for improved clarity.\n\n> **Q8:** Figure 4: it's not super clear what P and N in the legend are without the context in the surrounding text. Maybe these can just be written out into phosphate and nitrate for improved readability.\n\n**Response:** Thank you for your suggestion. We use abbreviations \"P\" for phosphate and \"N\" for nitrate due to space limitations. We totally agree that writing out the full terms would improve clarity.  We will update the legend accordingly in final version, where space limitations are less restrictive.\n\n> **Q9:** While lines 312 - 315 state that naive-B suffers large errors because it directly propagates the full velocity field without filtering, it seems from Figure 5 that NUTS suffers (relatively) equally for gamma tending towards -1. Could the authors comment on this?\n\n**Response:** Thank you for your insightful observation. In our experiments, the perturbation level is defined as $\\delta = \\gamma \\mathbf{v}^\\*$, where $\\gamma < 0$ corresponds to removing a portion of the eddy components from the flow. For instance, setting $\\gamma = -1$ yields a modified flow $\\mathbf{w}^* = \\bar{\\mathbf{w}} + \\frac{1}{\\epsilon}(1 + \\gamma)\\mathbf{v}^* = \\bar{\\mathbf{w}}$, effectively eliminating the eddy-driven information. This removal leads to degraded performance across models, including NUTS, as the fine-scale variability essential for accurate reconstruction is lost.\n\n> **Q10:** The paragraph starting on Line 332 doesn't make any reference to Table 4 even though it's the results in Table 4 that are under discussion. Maybe the authors could add an explicit reference in the same way they do it in the next paragraph?\n\n**Response:** Thank you for raising this point. We agree that the reference to Table 4 should be made explicit in the paragraph discussing Obs. 5. We will revise Line 334-336 to \"*Among several options, the Fourier filter achieves the lowest NRMSE on both daily (0.136) and monthly (0.151) tasks, outperforming wavelet, Gaussian, and moving average filters (Table 4a).*\" and Line 339-340 to \"*Lower ratios—removing more unresolved eddy components—consistently improve reconstruction accuracy, while higher ratios degrade performance (Table 4b)*\", including direct reference to Table 4.\n\n> **Q11:** Table 6: it seems there is result tables for a - f but the caption only has explanations for a - e (some of which don't match their respective table, I think). Please fix.\n\n**Response:** Thank you for your suggestion. In the caption of Table  6, we merged the explanations for Tables 6(b) and 6(c) under caption (b), which may have caused confusion. We will revise the caption to \"\n...(b) Analysis of model depth in the coarse module;\n(c) Analysis of model depth in the refinement module;\n(d) Evaluation of source and conservation loss terms; \n(e) Quantification of the impact of auxiliary input variables; \n(f) Assessment of sensitivity to temporal interval length... \n\" in the final version, avoiding potential confusion.\n\n\n> **Q12:** Table 6 (f): what are the units of the interval length? And is this the length of the prediction or the length of the context? Also, why are the short intervals worse than the intermediate ones? This feels counterintuitive. \n\n**Response:** Thank you for your thoughtful question. The interval length is measured in days for the Daily dataset and in months for the Monthly dataset. This interval length represents the length of the context used by the model. We observe performance peaks at a moderate interval length. Short contexts provide insufficient temporal information for the initializer to capture temporal dependencies, while long contexts introduce noise and uncorrelated information which may hinder model performance.\n\n> **Q13:** Line 260: change \"obatained\" to \"obtained\"\n\n**Response:** We appreciate your careful review, and we will revise the grammar in the final version.\n\n___\n\nThank you once again for your valuable feedback. We hope our responses address your concerns, and we would be happy to clarify further if needed. We look forward to your thoughts."}, {"review_id": "smDC6QhedV", "reviewer": "Reviewer_TCj8", "summary": "This paper tackles the problem of reconstructing ocean surface nutrient fields from extremely sparse, irregular observations, where standard inpainting or single-scale methods break down due to multiscale advection-diffusion dynamics and unstable eddy fluctuations. The authors propose NUTS, a two-scale framework that uses a homogenized PDE solver for large-scale flow to propagate coarse patterns stably and applies a conditional refinement module to reintroduce mesoscale variability based on the residual eddy field. Supported by theoretical stability guarantees from homogenization theory, NUTS provides state-of-the-art performance, and comprehensive ablation studies validate the value of each component.", "questions": "- In Figure 2, the boxes are colored in yellow, red, and green. Please explain what each color represents and why those specific colors were chosen, if it is intentional.\n\n- Have the authors employed higher‑order finite‑difference methods in their computations, and how does the approximation error vary with the order of the numerical methods?", "limitations": "See weakness and questions.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Strength**\n\n- The NUTS model makes a clear technical contribution by combining a PDE-based large-scale mean-flow solver with a refinement module for mesoscale eddy corrections. \n\n- The paper provides both theoretical justification and empirical evidence demonstrating the effectiveness of the proposed method.\n\n- The experiments include extensive ablation studies, demonstrating the robustness of the two-scale reconstruction framework.  \n\n**Weaknesses**\n\n- **No uncertainty quantification.** Data-assimilation and inpainting tasks are inherently ill-posed, so it is common to handle quantity uncertainty. However, NUTS produces only deterministic outputs. Although the authors acknowledge the inherent uncertainty and perturbations in the eddy and address them via the decomposed refinement module, they do not implement any mechanism for handling uncertainty. \n\n- **Insufficient qualitative results.** Despite strong quantitative results, the paper offers few visualizations of reconstructed fields, making it hard to judge prediction quality. The single output example in Figure 2 looks overly smooth and does not seem challenging. Also, in Figure 3, coefficient values span 10^-3 to 10^0, I am concerned that the model performance is biased toward larger values, resulting in a loss of fine-scale structure despite yielding small field errors. Since the benchmarks are not widely known, additional visual comparisons would greatly strengthen the evaluation. \n\n- **High model complexity.** The proposed NUTS model appears to have substantially more parameters than the baselines.", "was_revised": true, "final_justification": "I'll keep my original score.", "rebuttal": "We sincerely thank the reviewer for the professional and valuable feedback. Please find our detailed responses below.\n\n# Weakness\n\n> **W1:** No uncertainty quantification.\n\n**Response:** Thank you for raising this point. While NUTS does not explicitly model uncertainty, it is designed to handle key sources of variability—such as eddy-induced perturbations—through architectural decomposition and robust refinement. This makes it practically valuable in scenarios where accurate, physically consistent reconstruction is prioritized over full posterior characterization. That said, we agree that uncertainty quantification is an important next step, and NUTS provides a natural foundation for future extensions via ensemble or variational methods. We will clarify this perspective in the revised manuscript.\n\nDuring the rebuttal phase, we conducted a preliminary uncertainty analysis using a simple ensemble approach. Specifically, we perturbed the initializer output with Gaussian noise—analogous to injecting variability into the initial condition—where the noise at each grid point was scaled to 20\\% of the local temporal standard deviation of ground-truth nitrate concentrations from the {Monthly} dataset. We generated 10 ensemble members and computed the relative error in total nutrient mass between NUTS and the ground truth across four representative years. The \"$\\pm$\" values in **Table T1** denote the one-sigma spread across ensemble samples.\n\nWhile intentionally simple, this time-based perturbation reveals that NUTS can model meaningful initialization-driven uncertainties. As shown in **Table T1**, the relative errors remain low across years and months, while the ensemble spread is moderate—capturing plausible variability without overestimating uncertainty. This suggests that NUTS offers a credible starting point for uncertainty modeling. Future work will incorporate more structured perturbations—such as spatially coherent modes via EOFs—to better reflect dynamically consistent variability.\n\n**Table T1:** Relative error (\\%) between total nutrient mass from NUTS and ground truth across four years. The $\\pm$ indicates the one-sigma confidence interval.\n\n| Year | Jan.           | Mar.           | May           | Jul.           | Sep.           | Nov.           |\n| ---- | ------------- | ------------- | ------------- | ------------- | ------------- | ------------- |\n| 2011 | 1.08% $\\pm$ 0.27% | 0.92% $\\pm$ 0.26% | 1.49% $\\pm$ 0.31% | 3.42% $\\pm$ 0.25% | 1.28% $\\pm$ 0.22% | 1.64% $\\pm$ 0.19% |\n| 2014 | 1.65% $\\pm$ 0.21% | 2.62% $\\pm$ 0.26% | 2.81% $\\pm$ 0.27% | 2.61% $\\pm$ 0.27% | 0.96% $\\pm$ 0.24% | 2.33% $\\pm$ 0.26% |\n| 2017 | 2.38% $\\pm$ 0.28% | 0.83% $\\pm$ 0.28% | 3.08% $\\pm$ 0.34% | 3.98% $\\pm$ 0.25% | 3.07% $\\pm$ 0.24% | 3.79% $\\pm$ 0.30% |\n| 2020 | 5.33% $\\pm$ 0.34% | 3.17% $\\pm$ 0.30% | 3.76% $\\pm$ 0.32% | 3.76% $\\pm$ 0.24% | 3.94% $\\pm$ 0.16% | 4.17% $\\pm$ 0.28% |\n\n\n> **W2:** Insufficient qualitative results. The single output example in Figure 2 looks overly smooth and does not seem challenging. In Figure 3, I am concerned that the model performance is biased toward larger values, resulting in a loss of fine-scale structure despite yielding small field errors. \n\n**Response:** Thank you for raising this point. We agree that qualitative evaluation adds valuable context beyond metrics. In the final version, we will include additional qualitative analysis in the supplementary material, including: (1) NRMSE maps for NUTS and baselines to show spatial reconstruction accuracy, and (2) regional time series in dynamically complex zones such as the North Pacific and Southern Ocean.\n\nFigure 2 appears smooth due to the coarse color scale and the use of a simple example meant to illustrate model structure. We will clarify this in the caption to avoid confusion.\n\nFor Figure 3, we appreciate the concern. The figure is meant to illustrate spatial variability in nutrient dynamics—not reconstruction quality—and helps explain why phosphate is more challenging to recover. While we agree that visual support would help, we are unable to add new figures due to this year’s rebuttal policy. However, we will include additional qualitative discussion (e.g., error trends in high-variability regions) to demonstrate that NUTS preserves fine-scale structure even when coefficient values span several orders of magnitude.\n\n> **W3:** High model complexity vs. baselines.\n\n**Response:** We thank the reviewer for highlighting this concern. We acknowledge the model size: NUTS has 125.6M parameters versus 0.3M in 4DVarNet. However, this does not translate to significantly higher cost. On matched hardware (two RTX 6000 Ada GPUs, batch size 8), NUTS trains in 92s/epoch and infers in 58s—comparable to 4DVarNet’s 94s and 46s, respectively (Appendix D.6, Table 15). The reason is that 4DVarNet relies on iterative variational optimization, which is difficult to parallelize. NUTS uses batched, multi-step rollouts with efficient GPU utilization. In addition, NUTS delivers **lower error (NRMSE: 0.035 $\\pm$ 0.002)** and **comparable runtime**, despite having 400 $\\times$ more parameters. For context, NUTS is also 10--20 $\\times$ smaller than foundation-scale models like Prithvi (2.3B) and AtmoRep (0.7B), while being both faster and more accurate. We believe this reflects a favorable tradeoff between expressiveness and efficiency.\n\n# Questions\n\n> **Q1:** In Figure 2, the boxes are colored in yellow, red, and green. Please explain what each color represents and why those specific colors were chosen, if it is intentional. \n\n**Response:** Thank you for pointing this out. The colors in Figure 2 are used to group components by function: yellow indicates physics-based operations within the coarse module, red denotes intermediate variables passed between modules, and green highlights trainable components in the refinement stage. The gray initializer is also trainable and grouped under the coarse module. The color scheme was chosen for clarity rather than strict semantic coding. We will add a legend and clarify the color conventions in the final version.\n\n> **Q2:** Have the authors employed higher‑order finite‑difference methods in their computations, and how does the approximation error vary with the order of the numerical methods? \n\n**Response:** Thank you for the suggestion. During the rebuttal phase, we experimented with fourth-order central and second-order upwind differences for discretizing the advection–diffusion equation. On the Monthly nitrate dataset, this led to degraded performance: NRMSE increased from 0.151 to 0.166. We suspect this is due to numerical instability introduced by higher-order schemes, which may amplify noise in sparse observational settings.\n\n___\nThank you once again for your valuable feedback. We hope our responses have addressed your concerns. Please feel free to reach out if any further clarification is needed. We look forward to your continued thoughts."}, {"review_id": "N28CGI3YZd", "reviewer": "Reviewer_hsCa", "summary": "This paper addresses the important problem of reconstructing the surface ocean nutrient fields from sparse observations, leveraging the insight that ocean nutrient transport is governed by multiscale advection-diffusion dynamics. Here, unresolved mesoscale eddies leads to degraded reconstructions. The authors propose NUTS (Nutrient Two-Scale modeling), which uses deep learning to separate the dynamics across scales. NUTS models large-scale transport by first filtering the velocity field, and then using a homogenized PDE solver, where unresolved eddies are replaced with a learned effective diffusion term. The mesoscale details are further refined using a ViT (vision transformer) conditioned on residual eddy fields. The paper also introduces a new simulation dataset and evaluates NUTS on both simulated and real-world datasets, showing that NUTS outperforms several standard baseline methods.", "questions": "1. Why is it that mesoscale eddies carry more energy than the mean flow? In typical eddy simulation models, or reduced order models, the large-scale flow carries the majority of the energy. Is this affected by the choice of the threshold to separate scales that is typically used in multiscale ocean flow?\n1. As I understood, the learned diffusion term, which models the effect of the eddy components, is not coupled with the refinement module. Wouldn't the output of the refinement module directly affect the diffusion term due to the eddy components? Did you consider adding a constraint or loss term in the architecture? For instance, it seems plausible that over a long time rollout the diffusion term may drift out of sync with the refined details.\n1. In Table 3, since NUTS has 125M params v/s the closest baseline, the U-Net which has 31M params, why not make the U-Net larger to make the number of params in both methods similar?\n1. Could you report for training and inference times for the baseline methods, at least the U-Net and the 4D-VarNet?", "limitations": "yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "*Strengths* \n\n1. The paper tackles a significant problem in prediction of long-term biogeochemical cycles. It clearly presents the background and challenge of eddy instability. The work proposes NUTS, a two-scale method which ensures stability of the dynamics while not sacrificing small scale methods. This approach uses filtering, homogenized PDEs, and refinement; it is novel and physically motivated.\n\n1. The work provides extensive experimental validation. It creates a new MOM6 dataset and uses WOD data. It compares NUTS against many baselines (Table 3). It shows significant performance gains compared to the baseline methods, particularly FNO, U-Net and 4D-Var which are standard strong baselines.\n\n*Weaknesses*\n\n1. Potential Time Drift: The design isolates the refinement module. It acts as a spatial correction but as a post-processing step only It does not appear to feed its corrected field back into the PDE solver. This creates a risk of drifts over time, leading to poor detail in the long term rollouts. It is not clear if the learned diffusion model is consistent with the refinement module.\n\n1. Not Scalable for Long-Term Forecasting: The model excels at reconstruction in short term intervals, but it is not shown to work for long forecasting rollouts. Since spatio-temporal transformer is used for long range effects, it may not scale well to increased time rollouts.", "was_revised": true, "final_justification": "The paper addresses a significant problem, with a plausible approach with solid experimental results. The rebuttal addressed concerns with the baseline, as well as the concern with potential time drift.", "rebuttal": "We sincerely appreciate the time and effort you dedicated to reviewing our paper, as well as the thoughtful and constructive comments provided. Please find our responses to your questions below.\n\n# Weakness\n\n> **W1:** Potential Time Drift: The design isolates the refinement module. It acts as a spatial correction but as a post-processing step only. It does not appear to feed its corrected field back into the PDE solver. This creates a risk of drifts over time, leading to poor detail in the long term rollouts. It is not clear if the learned diffusion model is consistent with the refinement module.\n\n**Response:** Thank you for highlighting this concern. While the refinement module operates independently at each time step and does not explicitly feed back into the PDE solver, we address potential temporal drift via a mass conservation loss (Eqs.162-163) applied during training. This loss penalizes tracer mass discrepancies between the coarse and refined outputs, coupling the refinement and diffusion modules and enforcing physical consistency over time.\n\nTo empirically validate this design, we conducted an additional experiment during the rebuttal phase. We computed the monthly variation rate of total nutrient mass—relative to January—for two representative years (2014 and 2020). As shown in **Table T1**, the refinement module maintains variation within $\\pm$ 0.04\\% throughout the year, effectively flat. In contrast, NUTS—produced by summing the refined state and predicted source—exhibits substantial drift, with deviations up to 10.6\\% in 2014 and 16.1\\% in 2020, tracking ground-truth dynamics. This confirms that our refinement strategy avoids cumulative error and preserves long-term stability despite its non-recurrent design. We will clarify this mechanism and its empirical support in the final version.\n\n\n**Table T1:** Monthly variation rates (\\%) of total nutrient mass from Refinement Module, NUTS and ground-truth (G-T) across two years. The variation rate is defined as the ratio of each month’s total nutrient mass to that of January within the same year. We report the NRMSE between the variation rates over the interval.\n\n| Year | Data       | Feb   | Mar   | Apr   | May   | Jun   | Jul    | Aug    | Sep    | Oct    | Nov    | Dec   |\n| ---- | ---------- | ----- | ----- | ----- | ----- | ----- | ------ | ------ | ------ | ------ | ------ | ----- |\n| 2014 | Refinement | 0.00% | 0.01% | 0.03% | 0.03% | 0.04% | 0.03%  | 0.02%  | 0.00%  | 0.00%  | 0.00%  | 0.01% |\n|      | NUTS       | 0.17% | 2.14% | 5.40% | 3.10% | 3.54% | 6.78%  | 8.59%  | 10.60% | 10.04% | 6.97%  | 2.94% |\n|      | G-T        | 0.11% | 1.38% | 2.61% | 3.31% | 4.55% | 6.60%  | 8.08%  | 7.45%  | 7.28%  | 5.09%  | 0.43% |\n| 2020 | Refinement | 0.02% | 0.02% | 0.03% | 0.03% | 0.04% | 0.04%  | 0.04%  | 0.04%  | 0.03%  | 0.04%  | 0.04% |\n|      | NUTS       | 3.59% | 5.13% | 6.03% | 8.09% | 8.09% | 12.07% | 14.38% | 15.25% | 16.11% | 12.35% | 7.84% |\n|      | G-T        | 1.00% | 2.99% | 4.99% | 6.57% | 7.94% | 10.67% | 13.07% | 14.08% | 13.36% | 11.22% | 7.69% |\n\n\n> **W2:** Not Scalable for Long-Term Forecasting: The model excels at reconstruction in short term intervals, but it is not shown to work for long forecasting rollouts. Since spatio-temporal transformer is used for long range effects, it may not scale well to increased time rollouts.\n\n**Response:** Thank you for this thoughtful comment. We would like to clarify that NUTS is designed for **spatio-temporal reconstruction**, not long-horizon forecasting. Forecasting aims to extrapolate future states—often benefiting from longer rollouts—whereas reconstruction targets missing data within a fixed historical window. In this setting, the interval length is a tunable design choice. As shown in Figure 6(f), performance degrades as the interval grows too long, reflecting a tradeoff in reconstruction fidelity—not a failure to generalize in time.\n\nWe agree that long-term forecasting is a challenging and valuable direction, but it differs fundamentally in both objective and constraints. We will clarify this distinction in the final version to avoid confusion, and we consider extending NUTS to forecasting as an exciting direction for future work.\n\n# Question\n\n> **Q1:** Why is it that mesoscale eddies carry more energy than the mean flow? In typical eddy simulation models, or reduced order models, the large-scale flow carries the majority of the energy. Is this affected by the choice of the threshold to separate scales that is typically used in multiscale ocean flow?\n\n**Response:** Thank you for this thoughtful observation. We agree that at the global scale, the mean flow typically carries more energy. Our intention was to emphasize that in localized regions—such as western boundary currents and eddy-active zones—mesoscale eddies often exhibit higher local speed than the mean flow, which strongly influences nutrient transport.\n\nTo clarify this, we will revise Lines 125--126 to: \"*Mesoscale eddies are small-scale (10--100 km), high-energy structures that can dominate local transport processes.*\" and Lines 130--131 to: \"*...which can be large, as mesoscale eddies often exhibit higher local speed than the mean flow (Figure 2).*\"\n\n> **Q2:** As I understood, the learned diffusion term, which models the effect of the eddy components, is not coupled with the refinement module. Wouldn't the output of the refinement module directly affect the diffusion term due to the eddy components? Did you consider adding a constraint or loss term in the architecture? For instance, it seems plausible that over a long time rollout the diffusion term may drift out of sync with the refined details.\n\n**Response:** Thank you for the follow-up question. We address this concern in our response to W1: while the refinement module does not explicitly feed back into the diffusion term, we impose a mass conservation loss (Eq. 162–163) that couples the coarse and refined fields during training. This constraint ensures consistency between the diffusion dynamics and the refined output, effectively mitigating drift and preserving physical coherence over time. We will clarify this interaction in the final version.\n\n> **Q3:** In Table 3, since NUTS has 125M params v/s the closest baseline, the U-Net which has 31M params, why not make the U-Net larger to make the number of params in both methods similar?\n\n**Response:** Thank you for raising this point. During the rebuttal phase, we try scaling U-Net to 150M parameters and conduct experiments on the Monthly dataset, with nitrate being the target nutrient. We find that the reconstruction NRMSE drops from 0.187 to 0.171, achieving comparable results with NUTS, which is 0.151.\n\n> **Q4:** Could you report for training and inference times for the baseline methods, at least the U-Net and the 4D-VarNet?\n\n**Response:** Thank you for the suggestion. We report training/inference time and VRAM usage for NUTS and all baselines—including U-Net and 4D-VarNet—in **Table T2**. To ensure fair comparison, all models were benchmarked under the same hardware: two RTX 6000 Ada GPUs with identical batch size (8), as detailed in Appendix D.6 (Table 15).\n\nDespite having more parameters, NUTS achieves the lowest NRMSE while maintaining competitive runtime and memory usage. Notably, 4D-VarNet is over 400 $\\times$ smaller than NUTS in parameter count, yet its training (94s/epoch) and inference (46s) times are comparable. This is due to its iterative variational optimization, which requires repeated PDE solves and limits parallelism. In contrast, NUTS uses batched multi-step rollouts with data-driven initialization, making it more GPU-efficient despite its larger size.\n\n**Table T2:** Parameter, efficiency and Phosphate (sampling ratio=0.1\\%) reconstruction performance comparison of different models on the WOD (Daily) dataset.\n\n| Model     | Params     | Training Time (s) | Training VRAM (GB) | Inference Time (s) | Inference VRAM (GB) | NRMSE       |\n| --------- | ---------- | ------------------------ | ------------------------- | ------------------------ | ------------------------- | ----------------- |\n| 4D-VarNet | 0.3M       | 94                       | 6.8                       | 46                       | 3.0                       | 0.187 $\\pm$ 0.008     |\n| Marble    | 0.6M       | 23                       | 2.8                       | 7                        | 0.7                       | 0.363 $\\pm$ 0.058     |\n| FNO       | 4.8M       | 26                       | 14.2                      | 10                       | 4.5                       | 0.244 $\\pm$ 0.015     |\n| U-Net     | 31.0M      | 16                       | 6.4                       | 7                        | 2.9                       | 0.174 $\\pm$ 0.012     |\n| ViT       | 77.7M      | 62                       | 21.7                      | 20                       | 4.0                       | 0.263 $\\pm$ 0.034     |\n| AtmoRep   | 0.7B       | 87                       | 15.4                      | 58                       | 10.9                      | 0.206 $\\pm$ 0.013     |\n| Prithvi   | 2.3B       | 950                      | 26.1                      | 112                      | 18.4                      | 0.222 $\\pm$ 0.042     |\n| **NUTS**  | **125.6M** | **92**                   | **31.0**                  | **58**                   | **8.7**                   | **0.035 $\\pm$ 0.002** |\n\n___\nThank you again for your insightful feedback. We hope our responses have addressed your concerns. Please do not hesitate to contact us if further clarification is needed. We look forward to your continued input."}]

Author rebuttal:

Meta-review: This paper proposes a novel two-scale model (NUTS) for reconstructing ocean surface nutrients, effectively handling mesoscale eddy variability and outperforming baselines on key metrics. Results were validated on both simulated and real-world data. Reviewers agreed that not only were the theoretical principles well justified, the empirical results were also strong with good baselines and ablations. There were a few concerns w.r.t. model complexity and uncertainty quantification, which were mostly resolved during the discussion period.
Decision: Accept (poster)