REACT: Physically and Chemically Consistent Reconstruction of Marine Active Tracers

• Valid pH reconstruction must preserve carbonate consistency and source-free carbon conservation, because pH is the target while DIC is the conserved inventory.Source: §1 · p. 2
• The central modeling challenge is to separate conservative carbon transport, non-conservative carbon-cycle changes, and carbonate-equilibrium decoding.Source: §1 · p. 2
• REACT transports a latent carbonate state, applies active source correction, and decodes pH under carbonate constraints rather than conserving pH itself.Source: §1 · p. 2
• On simulation data, REACT reduces pH NRMSE by 14.7% and chemical consistency error by 24.0% over the best baseline.Source: §1 · p. 2
• Cross-temporal evaluations and ablations support robustness to coarse-to-fine error accumulation and the effectiveness of the framework's components.Source: §1 · p. 2

• Assimilation and inverse models combine sparse observations with background dynamics but require iterative optimization and repeated numerical model runs, making high-resolution long-term reconstruction costly.Source: §2 · p. 2
• Black-box methods learn completion from data but do not guarantee transport, conservation, or carbonate closure for ocean reconstruction.Source: §2 · pp. 2–3
• Figure 1 contrasts passive tracers, whose transported fields are conserved inventories, with pH, whose carbonate-system coupling means pH transport need not conserve carbon.Source: Figure 1 · p. 3
• Physics-constrained methods commonly place dynamics on the reconstructed variable, which suits passive tracers but not pH when carbon inventory is the physical state.Source: §2 · p. 3

• Chemical consistency checks whether reconstructed pH implies the corresponding DIC and pCO2 under carbonate equilibrium; low pH RMSE alone does not guarantee either.Source: §3 · p. 3
• In source-free transport with no-flux boundaries, the conserved quantity is DIC mass, not pH or H+ mass.Source: §3 · p. 3
• The reconstruction uses sparse pH observations, velocity, reference pCO2, and reference DIC to estimate pH under data, chemical, and physical losses.Source: §3 · p. 3
• The formulation therefore constrains the carbon state implied by pH and pCO2 to remain chemically and physically valid.Source: §3 · p. 4

4.1 A Naive Passive-Tracer Baseline for pH Reconstruction

• The passive baseline evolves pH with an advection–diffusion equation and trains on scalar pH error, so its transported variable is not the conserved carbon inventory.Source: §4.1 · p. 4
• Even in a source-free, no-flux setting, passive pH transport conserves pH mass but generically does not conserve the DIC implied by nonlinear carbonate chemistry.Source: §4.1 · p. 4

4.2 REACT: REconstructing Active Carbonate Tracers

• REACT initializes a latent carbon-related state, propagates it conservatively, adds an active source correction, and decodes the corrected state into pH.Source: §4.2 · pp. 4–5
• The source-free branch is constrained to conserve represented DIC before source correction, while the corrected state may change mass because it includes active sources.Source: §4.2 · p. 5
• Training combines pH supervision, carbonate closure, and source-free DIC conservation; small losses correspond to pH accuracy with chemical and physical consistency.Source: §4.2 · p. 5

4.3 Theoretical Analysis: Why REACT Uses a Carbon State

• If chemical and physical losses are zero under stated identifiability and realizability conditions, REACT closes carbonate chemistry and conserves represented DIC mass in the source-free branch.Source: §4.3 · p. 5
• Because carbonate mapping is nonlinear, conserving pH does not generally conserve DIC; REACT instead transports a carbon state and decodes pH through carbonate closure.Source: §4.3 · pp. 5–6

5.1 Experimental Setup

• The experiments use MOM6–COBALTv2 simulations for Month→Week and Week→Day tasks, plus WOD pH and SOCAT pCO2 observations; metrics include pH NRMSE, Chem., and local and global DIC drift.Simulation pH labels are sampled at 0.11%, 1%, and 10%; the observational pH records cover 0.11% of the grid.Source: §5.1 · pp. 6–7

5.2 Main Results (RQ1)

• REACT reduces pH NRMSE by 3.00%–14.7% on Month→Week and 5.90%–10.1% on Week→Day, while improving Chem. by 18.5%–24.0% and 18.7%–25.4%, respectively.Figure 3 also shows smaller spatial errors than NUTS, especially in the Arctic and Pacific regions.Source: §5.2 · p. 7
• On sparse real observations, REACT achieves pH NRMSE 5.25×10−3 versus 5.68 × 10−3 for 4DVarNet, and Chem. 0.102 versus 0.113.The evaluation trains on 75% of observed sites and tests on the remaining 25%.Source: §5.2 · pp. 7–8, Figure 4 · p. 7

5.3 Component Analysis (RQ2)

• Adding source correction lowers Month→Week pH error at 0.11% sampling from 4.08 × 10−3 to 5.79 × 10−4.Physical propagation alone misses non-conservative effects such as air–sea exchange, biology, and local model error.Source: §5.3 · p. 8, Table 3 · p. 8
• The DIC mass constraint reduces local drift from 8.84‰ to 2.26‰ and changes global drift from divergence to 9.73‰, while pH NRMSE slightly decreases.The same constraint reduces local and global drift in the U-Net ablation as well.Source: §5.3 · p. 8, Figure 5 · p. 8, Table 3 · p. 8
• The chemical constraint improves Month→Week Chem. from 1.94×10−2 to 1.59 × 10−2 while pH NRMSE changes from 5.68 × 10−4 to 5.84 × 10−4.The U-Net ablation shows the same trade-off: improved Chem. with a small increase in pH NRMSE.Source: §5.3 · p. 8, Table 3 · p. 8
• REACT’s uncertainty head yields two-standard-deviation coverage of 0.963, 0.971, and 0.958 for pH, pCO2, and DIC, respectively.The head estimates predictive standard deviation while keeping the deterministic predictive mean fixed.Source: Figure 6 · p. 8, §5.3 · p. 9

5.4 Ablation Study on Design Choices (RQ3)

• Design ablations favor moderate model depth, explicit spatiotemporal modeling, and pCO2 as the most important auxiliary variable.A depth-12 encoder achieves pH error 5.84 × 10−4, while depth 24 worsens drift; a depth-4 source/chemistry module reduces drift compared with depth 2. 2D CNN and ViT show very large global drift, 3D CNN fails, and removing pCO2 increases pH error from 5.84 × 10−4 to 2.17 × 10−3. Reconstruction horizon also trades short-term drift against long-range stability and error accumulation.Source: §5.4 · p. 9, Table 4 · p. 9

6 Conclusion

• The framework models pH through an underlying carbonate-system state, combining conservative transport, source correction, and carbonate-equilibrium constraints.Source: §6 · pp. 9–10
• REACT achieves the best pH NRMSE and chemical consistency on multiple reconstruction tasks.Source: §6 · p. 10
• Extension to other marine active tracers remains future work because of data and computational limitations.Source: §6 · p. 10

A.1 Oceanographic Data

• Observations provide physically grounded but sparse, irregular supervision, whereas simulations provide continuous gridded fields but are not direct observations and inherit model uncertainties.Source: §A.1 · p. 14

A.2 Assimilation and Inversion Methods

• Assimilation incorporates observations into model states, while inversion optimizes states against observations and constraints; both face limitations including computational cost or dependence on assumptions.Source: §A.2 · pp. 14–15

A.3 Data-Driven Reconstruction Methods

• Data-driven reconstruction uses proxy variables or learned field representations, but statistical correlation-based methods may lack explicit physical or chemical consistency.Source: §A.3 · p. 15

A.4 Physics-Informed Models

• Physics-constrained and hybrid models incorporate equations, numerical schemes, or domain knowledge, but many focus on transport-dominated variables rather than full biochemical consistency.Source: §A.4 · p. 16

B Theory for Carbon-State Reconstruction

• The theory distinguishes REACT's carbon-state certificate from passive pH transport: conserving pH mass does not generally conserve the DIC mass implied by carbonate chemistry.Source: B Theory for Carbon-State Reconstruction · p. 16

B.1 Notation and setup

• REACT transports a latent state without sources, adds an active correction, and decodes the corrected state into pH.Source: §B.1 · p. 17

B.2 Assumptions

• The theorem's assumptions include carbonate-map invertibility, observation identifiability, and a realizable decomposition of true DIC into a source-free branch and active residual.Source: §B.2 · p. 17

B.3 Positive result: carbon-state certificate

• With the stated assumptions and positive chemical-loss weight, REACT's zero chemical and physical losses certify carbonate closure and source-free represented-DIC conservation.Source: §B.3 · p. 18
• If chemical loss is zero, reconstructed pH closes carbonate chemistry on the target domain identified by the observations.Source: §B.3 · p. 18

2. Source-free represented-DIC conservation. Define

• If physical loss is zero, the source-free branch conserves represented DIC mass; continuity of the mass residual extends equality to every time in the interval.Source: §B.3 · p. 18, §2 · p. 18

B.3 Positive result: carbon-state certificate

• Under the realizability condition and chemically consistent true fields, a REACT realization can recover the true pH with zero chemical and physical losses.Source: §B.3 · p. 18

2. Source-free represented-DIC conservation. Define

• In the DIC-coordinate realization, the source-free branch represents the transported DIC field, whose integrated mass is conserved under the no-flux boundary condition.Source: §2 · p. 19

B.4 Negative result: passive pH transport is not a DIC-conservation certificate

• Equal pH mass does not determine equal implied DIC mass when the pH-to-DIC map is non-affine.The algebraic result establishes that two bounded pH fields can have equal pH mass but unequal mapped DIC mass.Source: §B.4 · pp. 19–20
• Passive pH advection–diffusion can conserve pH mass while changing implied DIC mass for some smooth initial fields.The result holds with divergence-free flow, no-flux boundaries, and diffusion positive definite on an open set.Source: §B.4 · p. 20
• Without diffusion, purely incompressible advection transports the full pH distribution, so integrals of fixed nonlinear functions of pH are also conserved.This special case does not certify passive pH reconstruction under advection–diffusion; with mixing, nonlinear carbonate chemistry makes pH-mass conservation insufficient for DIC-mass conservation.Source: §B.4 · p. 21

B.5 Consequence for REACT versus passive pH transport

• REACT places the source-free conservation constraint on a carbon coordinate, so zero physical loss certifies conservation of represented DIC.It decodes pH after conservative carbon transport and active carbon-cycle correction.Source: §B.5 · p. 22

C.1 Architectural Details

• A ViViT-style spatiotemporal transformer estimates the initial carbon-related transport state from sparse pH and pCO2 observations plus nine MOM6-derived variables.The input sequence is embedded as non-overlapping spatiotemporal tubelets; the output tokens are reshaped to the global latitude–longitude grid.Source: §C.1 · p. 22
• The conservative transport module propagates the carbon-related state using advection, with diffusion removed in the implementation.The state evolves under the physical transport operator from the reconstructed initial state.Source: §C.1 · p. 22
• A source module predicts residual corrections for unresolved non-conservative effects, including air–sea CO2 exchange and biological activity.The corrected state is obtained by adding the predicted source to the transported state.Source: §C.1 · pp. 22–23
• The pH decoder maps the corrected carbon-related state to pH conditioned on carbonate equilibrium constants K1 and K2.The chemical module uses carbonate-equilibrium formulas to calculate implied DIC from pH and pCO2, and implied pCO2 from pH and DIC, for consistency evaluation and constraints.Source: §C.1 · p. 23
• The formula-based DIC calculation differs from PyCO2SYS by a mean relative difference of 3.736 × 10−3 on MOM6 pH and surface CO2 inputs.The authors report that the formula implementation is an appropriate PyCO2SYS substitute while remaining compatible with neural-network training.Source: §C.1 · p. 23

C.2 Input and Output Details

• REACT outputs sea-surface pH with shape B × 1 × Tout × H × W; monthly-to-weekly reconstruction uses Tout = 4Tin and weekly-to-daily uses Tout = 7Tin.The chemical module is used when carbonate variables are needed for constraints or evaluation; the primary output remains pH.Source: §C.2 · p. 24

D.1 Datasets

• WOD surface pH and SOCAT surface CO2 observations from 1959–2022 are paired with MOM6-COBALTv2 reference fields on a global 1° × 1° grid.The MOM6 daily fields support both monthly-to-weekly and weekly-to-daily tasks; the real-observation experiment derives DIC with PyCO2SYS from pH, pCO2, salinity, and temperature.Source: §D.1 · p. 24, §D.4 · p. 26

D.2 Metrics

• The evaluation reports latitude-weighted, ground-truth-normalized NRMSE for pH, chemical consistency error from implied carbonate variables, and source-free DIC drift over clips and longer segments.The local drift is averaged across clips, while global drift is evaluated over longer segments.Source: §D.2 · p. 25

D.3 Implementation Details

• Training uses 250 epochs with AdamW, cosine annealing, and task-specific learning rates and mass-conservation loss weights.GradNorm adaptively balances active loss terms; experiments use eight NVIDIA 6000D GPUs with total batch size 8.Source: §D.3 · p. 26

D.4 Sampling Strategies

• Simulation observations are sampled from historical observation footprints at pH ratios of 0.11%, 1%, and 10%, with pCO2 fixed at 1.67%.For real observations, 75% of locations with pH, pCO2, and computed DIC are inputs and the remaining 25% are held out for evaluation.Source: §D.4 · p. 26, Figure 7 · p. 26

D.5 Hyperparameter setting.

• The appendix provides separate hyperparameter settings for REACT and its baseline models.The listed baselines include NUTS, 4D-VarNet, Marble, FNO, U-Net, ViT, AtmoRep, and Prithvi.Source: §D.5 · p. 27

D.6 Model efficiency

• REACT achieves the best pH NRMSE and Chem. while costing less computation than the Prithvi foundation-model baseline.The comparison is for the MOM6 monthly-to-weekly task at the 0.11% sampling ratio; REACT costs more than lightweight baselines.Source: §D.6 · p. 29, Table 7 · p. 29

E.1 Decomposed Chemical Errors for Table 2

• Across both reconstruction tasks and all sampling ratios, REACT has the lowest decomposed chemical errors for pCO2 and DIC.The reported pattern shows reductions in both carbonate-system components, not just one dominant term.Source: §E.1 · p. 29, Table 8 · p. 29

E.2 Additional Qualitative Results

• REACT better preserves spatial structures and produces smaller residual errors than baselines in the qualitative MOM6 comparisons.The figures compare ground truth and reconstructions, with errors computed as prediction minus ground truth.Source: §E.2 · pp. 29–30, Figure 8 · p. 30

E.3 Additional Real-Observation Results

• On WOD real-observation results, REACT has the lowest NRMSE for both pCO2 and DIC.Both metrics are reported in units of 10−1; the result is consistent with improvements in pH and chemical consistency.Source: §E.3 · p. 30, Figure 10 · p. 30

E.4 Decomposed Chemical Errors for Table 3

• Adding the chemical constraint reduces both pCO2 and DIC errors in the monthly-to-weekly REACT ablation.With Lchem, pCO2 error falls from 2.06 × 10−2 to 1.73 × 10−2 and DIC error from 1.82 × 10−2 to 1.45 × 10−2; the same pattern is reported for weekly-to-daily reconstruction and the U-Net ablation.Source: §E.4 · p. 31

E.5 Decomposed Chemical Errors for Monthly-to-Weekly Design Ablations

• Removing sea surface pCO2 substantially increases both decomposed errors in the monthly-to-weekly ablations.Table 10 reports the decomposed pCO2 and DIC errors for the design ablations.Source: §E.5 · p. 31, Table 10 · p. 32
• The 3D CNN initializer becomes unstable, consistent with its divergent behavior in the main ablation table.Source: §E.5 · p. 32

E.6 Weekly-to-Daily Design Ablations

• Removing pCO2 causes the largest degradation in both decomposed carbonate errors across weekly-to-daily auxiliary-variable ablations.The default configuration provides the best overall trade-off; a depth-24 initializer performs better overall but is retained at depth 12 for computational-cost reasons.Source: §E.6 · p. 32

E.7 Uncertainty Estimation

• Two-standard-deviation coverage is 0.963 for pH, 0.971 for pCO2, and 0.958 for DIC, close to the nominal Gaussian coverage of 0.95.Source: §E.7 · p. 32

F Limitations

• The learned source correction is not decomposed into individual physical or biogeochemical mechanisms, limiting attribution to underlying processes.Examples include air–sea exchange, biological activity, and unresolved mixing.Source: F Limitations · p. 33
• REACT targets sea surface pH and does not explicitly model vertical carbon transport or subsurface carbonate dynamics, limiting its characterization of three-dimensional carbonate chemistry.Source: F Limitations · p. 33

NeurIPS Paper Checklist

• The authors state that the paper’s claims reflect its scope and that limitations are discussed in Appendix F.Source: NeurIPS Paper Checklist · p. 34, §2 · p. 34

2. Limitations

• The authors report providing code and data through an anonymous repository and documenting the data-split protocol and training details.Source: §2 · p. 36