We utilize the hydrological cycle model component of H2MV within our model. This hydrological model estimates three primary water storages: snow, soil moisture, and groundwater. It estimates key water fluxes, including snowfall and snowmelt, soil and groundwater recharge, evapotranspiration (divided into transpiration, soil evaporation, and interception evaporation), and runoff (divided into fast and slow runoff) to update these water storages.

In the equations below, parameters with the superscript 〈s,t〉 indicate variables that vary both spatially (s) and temporally (t), while those with the superscript “s” refer solely to spatial variation. Parameters without any superscript represent globally constant parameters (denoted by the Greek letter β or Q10) that do not vary either in space or time and are learned by the neural network.

The main coupling mechanism between water and carbon cycles in H2CM is through linking transpiration and GPP by modelling WUE.

Transpiration (T) is modeled as described by : 1T〈s,t〉=fAPAR〈s,t〉⋅ETpot〈s,t〉⋅αT〈s,t〉inmmd-1, where transpiration is a function of the predicted vegetation state (fAPAR), potential evapotranspiration, and a neural network-learned parameter αT, which represents effective conductance or stress. The predictions of αT depend on relative soil moisture, vapor pressure deficit, net radiation, and static variables (Table ).

Parameters such as αT are explicit NN outputs. During training, the NN predicts αT from its inputs, and this value – together with variables like fAPAR – is passed to the process-based component to estimate transpiration, which contributes to total ET. Because ET is constrained by observations, training minimizes the loss between predicted and observed ET, thereby adjusting αT, among other parameters, iteratively. Moreover, αT is also influenced indirectly through other observational constraints (e.g., TWS, runoff, GPP), so multiple datasets jointly influence its learning. Further details on the hydrological model are available in .

We employ a simple, conceptual carbon cycle model to estimate carbon cycle fluxes (Fig. ).

GPP is the amount of carbon dioxide absorbed by vegetation to produce organic matter through photosynthesis. We model GPP as: 2GPP〈s,t〉=T〈s,t〉⋅αWUE〈s,t〉⋅CO2〈s,t〉⋅βCO2ingCm-2d-1 where GPP is a function of transpiration T, atmospheric CO₂ concentration, the global constant βCO2 describing the CO₂ fertilization effect and NN learned αWUE describing WUE without the CO₂ fertilisation effect. The βCO2 term is a globally constant, trainable parameter that regulates the strength of the CO₂ fertilization effect on GPP. Although βCO2 does not explicitly represent a dynamic CO₂ fertilization effect, the model's linear dependence on atmospheric CO₂ interacts with αWUE which is learned by the neural network as a nonlinear function of vapor pressure deficit, relative soil moisture, net radiation, and static variables (Table ).

NPP is the net carbon available after autotrophic plant respiration. NPP is modeled as: 3NPP〈s,t〉=GPP〈s,t〉⋅CUE〈s,t〉ingCm-2d-1 where NPP is a function of GPP and CUE. CUE is directly learned by the NN which receives air temperature, vapor pressure deficit, net radiation, atmospheric CO₂ concentration, relative soil moisture, fAPAR and NPP at previous time step, and static variables (Table ). Autotrophic respiration is simply calculated as the difference between GPP and NPP.

Heterotrophic respiration, Rh, refers to the process by which non-photosynthetic organisms (e.g., microorganisms) decompose organic matter, releasing carbon dioxide back to the atmosphere. We model Rh using a traditional Q10 function: 4Rh〈s,t〉=Q10Tair〈s,t〉-Tref10⋅Rb〈s,t〉ingCm-2d-1 where Q10, a global constant learned by the NN, describes the temperature sensitivity factor, indicating by which factor respiration increases with a 10 °C rise in air temperature. Tair is the current air temperature at time t, Tref is the reference temperature set to 15 °C), and Rb is the basal respiration rate essentially representing the availability of carbon for Rh and it is learned by a recurrent NN as a function of precipitation, net radiation, fAPAR and NPP at previous time step, and static variables (Table ). TER is the sum between autotrophic (Ra) and heteorotrophic (Rh) respiration representing the total amount of carbon dioxide being respired to the atmosphere. NEE is calculated as the difference between TER and GPP, where negative values indicate a land carbon sink.

Figure illustrates the overall architecture of H2CM:

A.

Static inputs and compression. Panel A1 represents the static environmental inputs, including land cover, soil properties, wetland extent, and digital elevation. These features are passed through a fully connected neural network (FC-NN 1; A2) that compresses spatially varying but temporally invariant information into a latent vector. FC-NN 1 has two hidden layers with 150 and 12 units, respectively. The 12-unit latent vector is used as compressed static features and serves as a shared input to all dynamic sequence layers. The outputs of this static network (A3) include spatially varying parameters such as maximum soil moisture capacity (SM_max) and αEi controlling interception evaporation.

When we integrate the carbon cycle with the hydrological cycle from H2MV, the number of NN predicted parameters and inputs to the model increases. This expansion raises the risk of the NN component making estimations based on unreasonable relationships, known as short-cut learning . This issue arises due to potential covariations among the inputs, where different inputs or their combinations might be used to estimate certain parameters, even if such associations are implausible based on established process knowledge.

To address this, we `guide' NNs by ensuring they learn from plausible inputs when estimating specific parameters. Instead of employing a single large NN to predict all parameters simultaneously, which is a common approach, we divide the neural networks into groups. Each group is restricted to learning from plausible relationships, enhancing the reliability of the model's predictions (Table  and Fig. ).

To evaluate the generalizability of H2CM, we use a 10-fold cross-validation (CV) setup. For this purpose, the spatial domain is divided into 10 folds composed of different grid cells. We then train 10 separate models, each time leaving out one fold of grid cells as the validation set and using the remaining folds for training. The training set is used to optimize the model, while the validation fold is exclusively for hyperparameter tuning, such as selecting the learning rate, and for early stopping. Furthermore, we keep a separate set of grid cells as a testing set, which is used only after training to assess the model's performance and generalizability. Importantly, the testing set is never exposed to the models during the training phase (Fig. ).

We divide the global data into training, validation, and testing sets using a spatial-only splitting method (Fig. ). This approach ensures that different grid cells are included in each of the training, validation, and testing sets. To address spatial autocorrelation among grid cells, we implement a spatial blocking technique as recommended by . In this approach, “blocks” refer to spatially contiguous groups of 5°×5° grid cells (25 grid cells in total) that are treated as single sampling units. These blocks are randomly selected from the global data, ensuring that the majority of the data (approximately 80 % of the grid cells) is allocated to the training set. The remaining ∼20 % of the data is reserved for validation and testing. In practice, each of the 10 cross-validation folds contains roughly 1/11 of the total data for validation, while ∼1/11 of the data is held out as a fixed testing set for final evaluation. The random selection of blocks also ensures that the training, validation, and testing sets approximately represent grid cells from all continents.

This split is crucial for accurately testing the model's generalizability on unseen testing grid cells and helps in understanding potential overfitting to the training data. Ideally, testing on both spatial and temporal dimensions, known as spatio-temporal splitting, would provide a more rigorous assessment. However, with the current set-up implementing proper time splitting for cross-validation is not feasible due to inconsistent and limited temporal coverage across different data constraints (Table ). To assess the model's generalizability across both space and time, we conducted an additional experiment in which the model was trained using data from 2001 to 2017 and evaluated on the subsequent two years (2018–2019; Appendix C). The results show qualitative agreement with the spatial CV setup, thereby justifying its validity in this study.

We use the mean squared error (MSE) as the loss function, following the approach in . The loss function MSE 5L(X,Θ,β)=1Nc∑c=1C∑i=1Ncyc,i-y^c,i2 evaluates the model's performance based on the inputs X, the neural network weights Θ, and the global constants β. In this context, C denotes the number of data constraints, Nc is the number of data points for each constraint c, and yc,i and y^c,i represent the observed and predicted values for the constraint c, respectively. Throughout the training process, Θ and β are adjusted to minimize the overall loss L. Note that we apply a Z-transformation to predictions and observations before computing the loss. This standardizes each variable by subtracting its mean and dividing by its standard deviation, removing the effect of different units and balancing the contributions of each data constraint to the total loss.

The loss function combines all data constraints into a single objective, where MSE is computed by concatenating the data across grid cells and monthly data for each variable. The only exception is for the data constraint on long-term NEE from CarboScope. In situ–based inversions such as CarboScope are generally more robust when averaged globally but become increasingly uncertain at the grid-cell level. Therefore, we first compute the spatial mean of both CarboScope and H2CM NEE monthly anomalies during training within each batch (subset of training samples processed together in one optimization step). This yields time series of spatial averages for which we calculate the loss.

We use the strategy outlined by to optimize H2CM for each CV fold separately, which includes utilizing the Adam optimizer . We set the initial learning rate to 0.005 and used a step-wise learning-rate scheduler (StepLR; ) that decays the rate by a fixed factor at fixed epoch intervals. Hyperparameters were tuned by training multiple model variants and selecting the configuration that achieved strong validation performance with stable training. During training, all inputs to the neural networks are standardized using Z-transformation (so deviations from the mean are preserved in standardized units). We also implement early stopping, a technique that monitors both validation and training losses. This approach halts further training if the model's performance on the validation set either declines or remains unchanged, thereby preventing overfitting to the training data. Throughout training, we keep track of the validation loss and select the final model based on the smallest total loss observed on the validation set. This selected model is then used for making final predictions, such as on the testing set.

In this section, we evaluate the performance of H2CM in reproducing the monthly, MSC, and monthly anomalies of GPP and NEE across the testing set that was not exposed to the model during training (Fig. ).

H2CM nearly perfectly reproduces the monthly and seasonal patterns of both GPP and NEE, with a Pearson's correlation coefficient (r) close to 1 (Figs.  and –). The monthly anomaly of GPP and NEE (based on OCO-2 data) has a Pearson's r of approximately 0.7 (mean across members), while the monthly anomaly of NEE based on CarboScope data has a Pearson's r of approximately 0.5. Uncertainties in the inversions of NEE monthly anomaly when locating in space and time also contribute to lowered performance metrics, as some of the variance in the inversions is noise, not signal.

The SDR for the monthly and seasonal patterns of GPP and NEE is close to 1, indicating the model's strong capability in accurately reproducing these patterns. The SDR for GPP monthly anomaly is greater than one, suggesting an overestimation of GPP monthly anomaly compared to FLUXCOM-X-BASE. This outcome is expected, as we do not explicitly constrain GPP monthly anomaly, and FLUXCOM-X-BASE is known to underestimate the GPP interannual variance . Conversely, H2CM underestimates NEE monthly anomaly in both short-term (OCO-2) and long-term (CarboScope) contexts, as indicated by a low SDR value.

In terms of RMSE, H2CM tends to exhibit higher errors for monthly and seasonal data compared to monthly anomalies. This elevated RMSE primarily reflects the large magnitude of the seasonal cycle, such that even relatively small absolute deviations lead to substantial RMSE values. Despite this, the near-unity Pearson's r and SDR indicate that the phase and relative amplitude of the seasonal cycle are well captured. In contrast, the monthly anomalies exclude the dominant seasonal component, resulting in smaller RMSE due to their reduced variance, but typically lower correlations because only irregular year-to-year variations remain.

Globally, H2CM reproduces GPP patterns from FLUXCOM-X-BASE with RMSEs of 0.1, 0.09, and 0.05 gC m⁻² d⁻¹ for monthly data, mean seasonal cycles (MSC), and monthly anomalies, respectively. For NEE from OCO-2 satellite inversions, RMSEs are 0.04, 0.03, and 0.02 gC m⁻² d⁻¹ for the same categories. Compared to CarboScope in-situ inversions, H2CM yields RMSEs of 0.07, 0.06, and 0.03 gC m⁻² d⁻¹. Bias is negligible (close to 0) for GPP and NEE (OCO-2), and for NEE (CarboScope) is 0.01 gC m⁻² d⁻¹ for monthly and MSC data, and zero for monthly anomalies (Table ).

Note that, although the results discussed here are based on the spatial-only cross-validation setup, Appendix C demonstrates that the model also generalizes well to unseen years, at least over short-term future periods. For example, when testing our model on unseen years for monthly GPP and NEE values, Pearson's r is close to 1 for GPP and 0.96 for NEE. These results hold for both the spatial split (original model) and spatio-temporal split experiments, with very similar ranges across cross-validation folds (Fig. ).

Model performance on water cycle-related data constraints is presented in Appendix B. We do not delve into the details here, as the performance is qualitatively similar to that of H2MV, which has been thoroughly discussed in .

In this section, we evaluate H2CM's performance in terms of learned global mean spatial patterns of GPP and NEE. We compare H2CM's estimations to the data constraints provided by FLUXCOM-X-BASE and OCO-2 satellite inversions, as well as to the estimations from TRENDY.

FLUXCOM-X-BASE, H2CM, and TRENDY show very similar spatial patterns of mean GPP with largest values in wet tropical regions such as South America, Central Africa, and Southeast Asia (Fig. a). While mean annual GPP from H2CM and FLUXCOM-X-BASE are nearly identical, TRENDY tends to show somewhat larger GPP in central South America, central Africa, and Southeast Asia, while lower GPP across much of the Northern Hemisphere (Fig. a). H2CM reproduces the spatial patterns of mean NEE from the OCO-2 MIP, whereas TRENDY's estimations show little spatial gradients (Figs. b and b). Specifically, both H2CM and OCO-2 inversions suggest the largest carbon sources in northeastern South America, tropical Asia, and in the Savannah belt south of the Sahara in Africa. In contrast, the eastern parts of North America and Europe, the southwest of South America, areas south of the Equator in Africa, and the eastern parts of China appear as carbon sinks in the period 2015–2019.

In this section, we qualitatively evaluate emerging global patterns in H2CM, focusing on key indicators such as precipitation, water, carbon, and light-use efficiency. These ratios, calculated based on mean annual values, help to diagnose and understand water–carbon cycle dynamics and coupling. Thus, this serves as credibility checks and a qualitative functional evaluation against patterns from existing studies in the literature.

H2CM estimates largest PUE values for high latitudes, particularly in regions such as western Canada, parts of Europe, and the Eurasian boreal regions (Fig. a). These findings strongly align with the results reported by that estimated rain use efficiency based on a light use efficiency model forced by satellite data, both in terms of spatial patterns and the magnitude of the estimations. Additionally, another study by employed a regression model based on meteorology and remote sensing, to estimate rain use efficiency, specifically focusing on Australia. Although they used GPP to estimate rain use efficiency instead of NPP as we did, their findings for Australia are also very consistent with our model's estimates for the region such as reduced PUE in Australia's interior. Overall, H2CM estimates low PUE in very wet and very dry regions, indicating that PUE peaks at intermediate moisture conditions. In dry areas, the low PUE pattern can be attributed to lower vegetation growth rates , enhanced losses by evaporation (lower T/ET, ), or water losses through runoff after rare but intense rainfall. In wet regions, a large fraction of precipitation is lost to runoff, and other factors such as light and nutrients may limit photosynthesis, which also can also cause a decline of PUE . We expect that PUE is very well constrained in H2CM because precipitation is a model input, and NPP is well constrained by data constraints on GPP and CUE.

H2CM estimates high WUE in regions such as northern North America (specifically Alaska and Canada), Northern Europe, the Eurasian boreal regions, certain areas in South America, Central Africa, the southernmost parts of Australia, and New Zealand (Fig. b). In contrast, H2CM predicts lower WUE for most arid regions. Low WUE estimations for arid regions is consistent with theory and observations predicting declining WUE with increasing VPD . The attenuated WUE in tropical forests might be related to larger costs of transporting water through tall trees . Our study defines WUE with respect to transpiration, while empirical studies often define it as a function of total evapotranspiration. This can lead to significant differences due to spatial variations of transpiration versus evapotranspiration . However, provides global WUE estimations as a function of GPP and transpiration based on process-based model simulations, which align well with H2CM's WUE estimations in terms of both spatial patterns and magnitudes. H2CM's WUE is constrained by GPP and ET data, while mainly fAPAR (also constrained by observations) and the respective process formulations constrain the partitioning of ET into transpiration and evaporation components.

H2CM generally estimates higher CUE values in higher latitudes and lower CUE values in most of the Southern Hemisphere (Fig. c). The range of these estimations is relatively narrow, approximately between 0.45 and 0.6, indicating that roughly 45 %–60 % of GPP is converted into net biomass. The patterns reflect the general expectation that the fraction of autotrophic respiration increases with temperature and biomass. This is expected because we have constrained the spatial pattern of mean annual CUE to align with predictions from the TRENDY ensemble of process models. Other studies have consistently reported similar findings in terms of magnitude and spatial patterns .

H2CM estimates high effective LUE for regions such as the northwest parts of North America, the eastern states of Canada and the United States, Europe, the Eurasian boreal regions, South America, Central Africa, and Southeast Asia (Fig. d). Low LUE for dry regions are expected due to increased water limitations. A study by , using FLUXNET site data and satellite-derived proxies, found results closely matching ours, particularly in spatial patterns. Similarly, used random forest to estimate LUE based on flux tower data, aligning mostly with our estimations, though they report lower LUE values for the northwest part of North America and the Eurasian boreal regions.

H2CM is a process-aware hybrid framework that integrates machine learning with simplified process-based formulations, and can be positioned within the broader family of model–data integration strategies that aim to combine physics with machine learning. This family includes approaches such as physics-informed neural networks , physics-guided machine learning (pgml, ), and differentiable modeling . Rather than reproducing the detailed parameterizations used in comprehensive land-surface models, H2CM employs conceptual process formulations to maintain interpretability and ensure parameter identifiability under the available observational constraints.

The primary objective of H2CM is to deliver a consistent, observation-constrained “reanalysis” of coupled carbon–water states and fluxes over recent decades by fusing diverse data streams within a process-informed ML architecture. H2CM is not intended to replace land surface models coupled into Earth system models, nor to provide long-term projections; such applications would require more complete representations of processes and feedback. Instead, H2CM can be perceived as a bridge between empirical and process-based modeling which enables a coherent interpretation of only partially observed coupled carbon–water cycle variability and interactions. H2CM is a global model in the sense that it is applied consistently across grid cells covering the global land surface, while it remains local as it does not simulate lateral processes among grid cells. Although the domain and grid are, in principle, arbitrary, the present design is tailored to comparatively coarse grid cells for improved consistency with spatially coarse observational constraints like TWS from GRACE or NEE inversions.

H2CM advances the field by combining previous approaches – such as fully data-driven models like FLUXCOM(-X-BASE) , and fully process-based models like TRENDY – by aiming at combining their strengths: it learns directly from global observations through its machine learning component, while respecting conceptual process understanding and mass balance. Relative to process-based TRENDY models, which rely on numerous parameterizations yet struggle with regional and climate-specific patterns in observation-based carbon fluxes, H2CM better captures key spatial–temporal features in observations (Fig. ). Likewise, compared with fully data-driven FLUXCOM approaches that depend primarily on site-level eddy covariance data and can underperform in tropical and subtropical regions with sparse coverage, H2CM leverages complementary data streams and process-aware structure to improve generalization (Fig. ).

Both, the rich information provided by complementary Earth Observations to H2CM, and the flexibility of machine learning to exploit these, contribute to the good performances of the model. Clearly, a key factor for the improved performance of H2CM compared to FLUXCOM in capturing NEE variations inferred from atmospheric inversions is the integration of this data constraint in the hybrid modeling approach. While this might also explain H2CM's improved performance compared to TRENDY, the machine learning component used to parameterize the conceptual process equations seems to contribute to the success significantly. A useful point of reference is the model-data-integration approach by , which shares a similar model structure of conceptual process equations of carbon and water cycles, and nearly identical observational constraints. The key difference is the representation of parameters in process equations: the study by assumes globally constant parameters, whereas H2CM uses neural networks to predict parameters that vary spatially and/or temporally. Overall, both approaches, being strongly data-constrained, capture large-scale patterns well. However, H2CM achieves higher performance, in particular in dry and wet tropical regions with partly complex linkages of carbon and water cycles. This illustrates the added value of the adaptive machine learning module in H2CM for better model performance. Taken together, H2CM's skill emerges from the synergy of conceptual process formulations, diverse observational constraints, and adaptive parameterization through machine learning.

H2CM heavily depends on data-driven learning, incorporating both meteorological forcings and data constraints. The primary constraints on the carbon cycle include GPP seasonality from FLUXCOM-X-BASE, NEE from OCO-2, and global NEE monthly anomalies from CarboScope. Each of these data sources is subject to uncertainties, which could propagate to H2CM. Accounting for data uncertainties explicitly in H2CM, e.g. based on a Bayesian approach, would be desireable conceptually. However, the lack of adequate and comparable quantitative information on data uncerainties across different data streams limits this approach in practice.

We tried to mitigate known issues of the observational constraints by making the loss function sensitive to the more robust patterns in the data, e.g. by using only the mean seasonal cycles of GPP by FLUXCOM-X-BASE due to known limitations for interannaul variability . However, uncertainty of the GPP mean seasonal cycle of X-BASE remain for example in the region of tropical South America. Thus, potential biases in the seasonal cycle of FLUXCOM-X-BASE GPP likely propagate to H2CM, which could then also compromise simulations of respiration due to the concomitant data constraint on NEE.

To constrain H2CM's NEE estimations, we utilize an atmospheric inversion ensemble using in-situ and OCO-2 satellite-based column XCO₂ data. Despite significant efforts and advancements in XCO₂ retrievals from OCO-2, the preferential sampling of clear-sky conditions and remaining retrieval biases may affect inferred NEE variations . Additional uncertainties from various methodological choices, particularly in atmospheric transport modeling, result in poorly constrained spatial inversion results . We used 1° OCO-2 MIP results to constrain NEE, which, despite the added value of extensive XCO₂ data, may be overly optimistic. Therefore, some discrepancies between H2CM-simulated and OCO-2 MIP-based NEE at 1° resolution (Fig. b) are due to this data constraint uncertainty. Since the issue of loose spatial constraints is even more pronounced for atmospheric inversions using only the sparse in-situ CO₂ network, we used only the spatial batch average as a data constraint from CarboScope.

Exploiting H2CM's explicit process interpretability for process understanding requires sufficiently strong observational constraints to overcome equifinality – i.e., different processes or pathways yielding similar outcomes when data are not informative enough. Examining the spread among 10 CV models trained on different cross-validation folds and with different random initializations indicates good qualitative robustness under varying training data . Nevertheless, the globally constant temperature sensitivity of respiration (Q10) and the scalar for the relative CO₂ fertilization effect on GPP (βCO2) show signs of non-identifiability. Across CV folds, the median Q10 (unitless) is 1.25, with a range of 1.24–1.27, lower than literature values for heterotrophic/soil respiration (1.4–2; ). For βCO2 the median value learned across CV folds is 35.82 (% 100 ppm⁻¹). The range of values extends from 26.88 (% 100 ppm⁻¹) to 62.71 (% 100 ppm⁻¹), which exceeds observational estimates of roughly 14 % 100 ppm⁻¹–16% 100 ppm⁻¹ . To probe this, we imposed informative priors on these global constants (Appendix D). Under these priors, H2CM recovered βCO2=15 (% 100 ppm⁻¹) and Q10=1.5 – matching the priors exactly – while other model results and performance remained qualitatively unchanged. These findings indicate that with the current data constraints, Q10 and βCO2 are not fully identifiable, underscoring the value of stronger observational constraints and the incorporation of process knowledge (e.g., via priors or loss-function regularization) for decadal-scale assessments with H2CM.

Deviations of H2CM simulations from the data constraints such as for NEE in tropical South America (Fig. ) are useful indications for missing information or issues in the data. This is because H2CM cannot fit the data constraints if (1) the relevant signals are not present in the model forcing data; or (2) the model's process formulations do not permit it, e.g. due to a missing process representation that cannot be captured by the flexibility of the machine learning component; or (3) because there is a conflict among different observational constraints that cannot be satisfied simultaneously given the process formulations.

While incorporating process understanding in H2CM also provides theoretical constraints that can help to regularize the NNs , a trade-off between model complexity and parameter identifiability is obvious. Thus the simple nature of process representations in H2CM is justified by the limited information content of the used observations to disentangle subprocesses, and by minimum requirements on interpretability. For example, we use net radiation as a single forcing term rather than separately prescribing shortwave and longwave components for simplicity as shortwave and net radiation are strongly correlated at daily time scale . Currently, H2CM lacks several important components of the land system, which are relevant for the coupled water and carbon cycles. For example, it does not yet explicitly model vegetation and soil carbon pools, as the current focus has been on spatial variations of carbon fluxes from sub-seasonal to interannual time scales. Furthermore, H2CM does not incorporate the effects of disturbances such as fire, or other drivers including land-use change. Note that dynamic vegetation changes are not explicitly represented in H2CM; however, temporal variations in fAPAR provide an implicit representation of phenological changes. There are emerging opportunities from the observational side which could support the integration of these components in H2CM in the future.