Learning variables were carefully selected based on their potential impacts on the variations of the air–sea turbulent heat fluxes (Grist et al., 2016; Kudryavtsev et al., 2014; Myslenkov et al., 2021; Song, 2020, 2021; Yan et al., 2024) to conduct the feature selection (see Sect. 2.3.1). These variables include Ta, sea surface air specific humidity (Qa), Mean Sea Level Pressure (SLP), Downward Long Wave Radiation Flux (LW), Downward Short Wave Radiation Flux (SW), Ts, sea surface specific humidity (Qs), Absolute Dynamic Topography (ADT), Sea Level Anomaly (SLA), Sea Surface Salinity (SSS), Sea Surface Density (SSD), Ocean Mixed Layer Current Velocity (CS), WS, Significant Wave Height (SWH), Wave period (Tp), as well as gradient of temperature (diff_T) calculated using the Ts and Ta, and gradient of humidity (diff_Q) calculated using the Qs and Qa.

Datasets of these learning variables were collected from multiple publicly available sources, as summarized in Table 2 and were used as the input features for training the neural network. The Multi Observation Global Ocean Sea Surface Salinity and Sea Surface Density (MOGOSD) dataset and Global Ocean Waves (GOW) Reanalysis dataset were spatially resampled to a 0.25° resolution using mean aggregation, while temporal mean aggregation to daily values was applied to the GOW dataset (originally at 3 h resolution) and Cross-Calibrated Multi-Platform (CCMP) wind vector analysis dataset (6 h resolution). Additionally, a daily ERA5 sea-ice mask was applied to the datasets to mitigate the impact of sea ice.

Seven widely used air–sea turbulent heat flux products, involving remote sensing-based JOFURO3, IFREMER and SeaFlux, as well as reanalysis-based ERA5 and MERRA2, hybrid-based OAFlux and machine learning-based OHF products were selected for inter-comparison, as summarized in Table 3.

The remote sensing-based JOFURO3, IFREMER, and SeaFlux products were developed by the Japanese Ocean Flux Data Sets under the Remote Sensing Observations (J-OFURO) research project, the Institute Français de Recherche pour l'Exploitation de la Mer (IFREMER), and the NASA Global Hydrology Resource Center (GHRC) Distributed Active Archive Center (DAAC), respectively. The reanalysis-based ERA5 and MERRA2 products were developed by the ECMWF and NASA Global Modeling and Assimilation Office (GMAO), respectively. The hybrid-based OAFlux and machine learning-based OHF products were developed or published by the Woods Hole Oceanographic Institution (WHOI) and NOAA Ocean Surface Bundle (OSB) Climate Data Record (CDR), respectively. With the exception of the OHF product calculating SHF and LHF simultaneously using a NN model without a constraint, all other products employed bulk aerodynamic methods to estimate SHF and LHF. The JOFURO3, IFREMER, and OAFlux products used the COARE3.0 model, while the SeaFlux used the COARE3.5 model. Differently, the ERA5 adopted the bulk aerodynamic method used by the ECMWF, and the MERRA2 used the method by the GEOS. These products provide SHF and LHF estimates at a 0.25° spatial resolution, except for the MERRA2 (0.5° × 0.625°) and OAFlux (1°). Additionally, most products provide daily SHF and LHF estimates, while only the OHF product provides estimates at a 3 h interval. For further inter-comparison, the daily mean aggregation was applied to the OHF product. More details about the seven products can be found in the review of Tang et al. (2024).

The study employed a random forest (RF) model to evaluate the importance scores of 17 oceanic and atmospheric learning variables (with datasets collected in Sect. 2.2) for target variables (SHF and LHF), aiming to filter out less influential variables. As shown in Table S2, the variable importance assessment revealed that diff_T and diff_Q showed the highest importance scores (71.56 % and 49.93 %) for SHF and LHF modelling, respectively; additionally, WS exhibited the second highest importance for both SHF (10.19 %) and LHF (27.59 %) modelling. Building upon the importance evaluation and through careful screening of highly correlated variables, we ultimately selected 11 key environmental features for subsequent air–sea turbulent heat flux modelling including SLP, LW, SW, SSS, ADT, CS, WS, SWH, Tp, diff_Q, and diff_T.

We selected the NN technique to build the BrTHF model due to its strong ability to capture the complex nonlinear relationships between the multiple inputs and multiple-target variables with high accuracy (Zhou et al., 2024; Fu et al., 2023; Cummins et al., 2023, 2024). Additionally, the technique enables the seamless integration of physical constraints, improving the reasonableness of the results (Zhou et al., 2024; Zhao et al., 2019; Shang et al., 2023).

In order to estimate the SHF and LHF with high accuracy in a physics-consistency framework, the β(= SHF/LHF) physical constraint was incorporated into the NN model using the custom multiple-objects (SHF, LHF and β) loss function as follows: 1Loss=a×LossSHF+b×LossLHF+c×Lossβ Loss_SHF, Loss_LHF and Loss_β represent the Mean Squared Error (MSE) of SHF, LHF and β, respectively. They were weighted using the factors of a (SHF), b (LHF) and c (β) to balance the different magnitudes of loss during optimization. To prevent potential gradient explosion during model training, predicted β[SHF′/LHF′, calculated using the predicted SHF (SHF^′) and LHF (LHF^′)] values were clipped within the observed range of β (from -5 to 5) during training: 2CLIPSHF′LHF′=MinSHF′LHF′,5SHF′LHF′>0MaxSHF′LHF′,-5SHF′LHF′<03Lossβ=MSESHFLHF,CLIPSHF′LHF′ Finally, after optimization, the final weights (a, b and c) for SHF, LHF, and β were set to 5, 1, and 250, respectively. The model was constructed using the Python TensorFlow library and consisted of one input layer, three fully connected hidden layers, two Batch Normalization layers applied after the first and second hidden layers, and one output layer. The numbers of neurons in the three hidden layers were 32, 64, and 16, respectively and the activation function of Leaky Rectified linear unit (ReLU) was used throughout the model.

To illustrate the superiority of the BrTHF model in terms of accuracy and physical consistency, another physics-free NN models, trained without integrating the β constraint, were also constructed to predict SHF and LHF separately for further comparison, where β was calculated to be SHF/LHF.

Figures 3, 4 and 5 present the normalized scatter density plots of the estimated daily SHF, LHF and β from the BrTHF and physics-free NN models, as well as the seven air–sea turbulent heat flux products against the observations obtained from 197 globally distributed buoys by the spatial ten-fold cross-validation strategy.

Most models and products showed data distributions closely aligned with the observed SHF, with the majority of samples clustered around the 1:1 line. The BrTHF model slightly overestimated SHF with a BIAS of 0.09 W m⁻², whereas the physics-free NN models, and ERA5 and IFREMER products showed more pronounced overestimations (from 0.42 to 4.05 W m⁻²). In contrast, the remaining five products exhibited notable underestimations (from -3.44 to -0.41 W m⁻²). As illustrated in Fig. 6, the variability of estimated SHF from the BrTHF and the physics-free NN models and ERA5 product closely matched the observed SHF, all with a Standard Deviation (STD) of approximately 16 W m⁻². Notably, the BrTHF model achieved the lowest RMSE (6.05 W m⁻²), outperforming both the physics-free NN models (6.29 W m⁻²) and the seven air-sea turbulent heat flux products (ranging up to 12.34 W m⁻² for OHF). Additionally, the BrTHF model combined with the physics-free NN models yielded the highest values of r (0.93), surpassing all seven other products. In summary, the BrTHF model showed the best overall performance in estimating SHF among all the models and products.

For LHF, similar to the results for SHF, the BrTHF model demonstrated the best agreement with observations, achieving the lowest RMSE (23.67 W m⁻²) and the highest value of r (0.91). Compared to the physics-free NN models and seven products, the BrTHF model reduced RMSE by 2.05 W m⁻² (physics-free NN models) to 12.38 W m⁻² (OHF) and improved r by 0.01 (physics-free NN model) to 0.1 (OHF). Additionally, the BrTHF model showed a slight overestimation of LHF (BIAS = 0.14 W m⁻²), lower than that of the SeaFlux, MERRA2, and ERA5 products. In contrast, the remaining products (JOFURO3, IFREMER, OAFlux, and OHF), along with the physics-free NN models, underestimated LHF, with the BIAS values ranging from -10.19 W m⁻² (OHF) to -1.61 W m⁻² (JOFURO3).

The BrTHF model exhibited a significantly different distribution of β compared to the physics-free NN models and the seven products, as shown in Fig. 5. The β estimates from the BrTHF model consistently fell within the observed range of -5 to 5, while the physics-free NN model and the seven products occasionally produced estimates outside this range. Specifically, approximately 0.9 % of β estimates from both the physics-free NN model and the seven products were out of range. The extreme positive and negative β estimates were found in the OHF (β= 14 997) and physics-free NN models (β= -25 703) products, respectively. The abnormal β estimates significantly impacted the accuracy of the physics-free NN models and the seven products as Fig. 6 indicated. When excluding the abnormal β samples from the physics-free NN models and seven products, the RMSEs ranged from 0.17 (physics-free NN models and SeaFlux) to 0.26 (OHF), with values of r ranging from 0.13 (OHF) to 0.46 (IFREMER), as shown in Fig. 6 and Table S3. However, when all estimates were considered, the performances of these model and products deteriorated sharply, with RMSEs rising from 0.87 (SeaFlux) to 39.21 (physics-free NN models), and values of r dropping from 0.06 (SeaFlux) to 0 (JOFURO3, MERRA2 and OHF). In contrast, the BrTHF model maintained robust outperformance, with the lowest RMSEs of 0.22 and 0.15, and higher r values of 0.25 and 0.43, both before and after removing the abnormal β samples from the physics-free NN models and the seven products. Notably, the BIAS values remained stable (ranging from -0.04 to 0.04) for all models and products, regardless of whether the abnormal samples were excluded.

To better understand the accuracy of SHF, LHF and β estimates from the BrTHF and physics-free NN models, as well as the seven products in different oceans, we conducted an additional evaluation by categorizing the observations according to the belonging ocean basins, as shown in Fig. 7. The major ocean boundaries, obtained from Marine Regions (https://www.marineregions.org/, last access: 31 March 2026), were used to define the ocean basins, which include the Arctic Ocean, South Pacific Ocean, North Pacific Ocean, South Atlantic Ocean, North Atlantic Ocean, and Indian Ocean.

For SHF, the BrTHF model exhibited overestimations in the South Pacific Ocean, North Atlantic Ocean, and Indian Ocean, while it underestimated SHF in the remaining three ocean basins. The values of BIAS ranged from -4.57 W m⁻² in the Arctic Ocean to 0.49 W m⁻² in the North Atlantic Ocean. Furthermore, the BrTHF achieved the lowest RMSEs in most ocean basins, ranging from 3.84 W m⁻² in the South Atlantic Ocean to 7.72 W m⁻² in the North Atlantic Ocean, except in the Arctic Ocean where the RMSE of 13.59 W m⁻² was higher than those of the ERA5 (12.5 W m⁻²) and MERRA2 (13.46 W m⁻²) products, as shown in Fig. 7b. Correlation analysis also demonstrated the robust performance of the BrTHF model in estimating SHF, with values of r exceeding 0.89 in most ocean basins, except for ocean basins in the Southern Hemisphere (ranging from 0.71 to 0.79) where the values of r for all models and products were reduced.

For LHF, the values of BIAS of the BrTHF model ranged from -4.15 W m⁻² in the Arctic Ocean to 1.19 W m⁻² in the North Pacific Ocean. In comparison, the BrTHF model showed more pronounced underestimations in the Arctic Ocean and Indian Ocean. Additionally, the BrTHF model outperformed the physics-free NN models and the seven products across most ocean basins, achieving the lowest RMSEs (ranging from 17.06 W m⁻² in the Arctic Ocean to 28.20 W m⁻² in the North Atlantic Ocean) and the highest values of r (ranging from 0.83 in the Indian Ocean to 0.94 in the North Atlantic Ocean) except for the Arctic Ocean where the value of r was 0.01 less than the physics-free NN models and the RMSEs were 2.45, 3.25 and 1.84 W m⁻² higher than the ERA5 and MERRA2 products and the physics-free NN models, respectively. Combining the overall and regional evaluations, we find that, except for the OHF product, the relative performance of SHF and LHF among the seven products is generally consistent with Tang et al. (2024). The OHF product shows notable degradation, particularly in high-latitude oceans, which may be due to limitations in the training datasets (sparse and unevenly distributed in-situ observations) and model training strategy (randomly splitting all observations into training, validation, and test sets without accounting for spatial dependencies), leading to reduced generalizability.

The BrTHF model consistently performed better in estimating β across most ocean basins, both before and after removing the abnormal β samples that deviated from the observed range (-5 ≤β≤ 5). In contrast, the physics-free NN models and the seven products did not perform as well. Specifically, the BrTHF model exhibited the lowest RMSEs in almost all ocean basins except in the South Atlantic Ocean after removing β outliers. Moreover, in terms of correlation analysis, the BrTHF model achieved higher values of r in most ocean basins before and after the removal of abnormal β samples, among all models and products.

The primary advantage of the BrTHF model lies in its accurate estimation of β, which shows the most pronounced improvement among all flux components. As a key indicator of surface energy partitioning, β is widely used within the surface energy balance framework to ensure physically consistent and reliable estimates of SHF and LHF (Jo et al., 2004; Yang and Roderick, 2019; Yang et al., 2025). In addition, β serves as an effective diagnostic variable in the studies of large-scale climate variability (e.g., El Niño) (Jo, 2002; Weisberg and Wang, 1997) and in investigations of how surface energy constraints regulate the hydrological cycle (e.g., precipitation) (Cai and Lu, 2009; Wang et al., 2021). With its enhanced representation of β, the BrTHF product is expected to provide more reliable support for such applications.

To achieve these improvements in β and fluxes, the BrTHF model was designed differently from our previous study (Wang et al., 2024), which simultaneously predicted SHF, LHF and β in the constructed RF model. Specifically, this study employed an NN model constrained by the Bowen ratio to jointly estimate SHF and LHF. The new approach avoided the issue of selection of β derived from either the calculated β [βcal equals predicted SHF (SHF_pre) divided by predicted LHF (LHF_pre)] or the predicted β (βpre), as reported by Wang et al. (2024). Furthermore, the custom loss function in our new approach provides a flexible approach to adjust the weights of SHF, LHF, and β, allowing the model to balance attention among these variables. As a result, the accuracy of SHF, LHF, and β from our newly developed BrTHF model outperformed that of the mainstream air–sea turbulent heat flux products and the physics-free NN models on both global and regional scales. In contrast, the accuracy of SHF and LHF in the model constructed by Wang et al. (2024) was marginally lower than that of the physics-free RF model.

Based on Fig. 2 and Table S8, we observe that the spatial coverage of observations varies across different ocean regions: the Northern Hemisphere generally has higher coverage than the Southern Hemisphere, with the Northern Pacific Ocean exhibiting the highest coverage, while the Arctic Ocean shows the lowest. Comparing spatial coverage with accuracy metrics reveals a more complex relationship between model performance and data coverage. Specifically, the values of r tend to be lower in regions with lower coverage – a pattern consistent across SHF, LHF, and β. However, RMSE does not follow this trend. For SHF and β, RMSEs in the Northern Hemisphere are generally higher than those in the Southern Hemisphere. Similarly, for LHF, RMSEs are higher in the Northern Hemisphere except in the Indian Ocean, where the pattern differs.

We applied a spatial 10-fold cross-validation, which provides a more generalized assessment than traditional random cross-validation, to evaluate the BrTHF model. However, it is important to acknowledge that the spatial distribution of the training dataset is inherently imbalanced, with a heavy concentration of observations in the Tropics and the Northern Hemisphere. In contrast, the Southern Hemisphere – particularly the Southern Ocean – suffers from sparse or even missing observational coverage. Given that the environmental conditions in these underrepresented or data-sparse regions may differ significantly from those captured in the training dataset, the selected input variables for the observations may lead to large uncertainty in the model's performance in these areas. To further assess the model's ability to extrapolate to such regions, we conducted an additional targeted cross-validation. Specifically, we excluded stations from the Southern Ocean [i.e., Southern Ocean Flux Station (SOFS) and Global Southern Ocean Station (GSOS)] from the training dataset and used them solely for validation. Results presented in Tables S4 and S5 show that the BrTHF model achieved the best performance in terms of LHF and β at the SOFS with lower RMSE of 15.6 W m⁻² and 0.73 and higher values of r of 0.96 and 0.34, respectively, while its SHF was slightly outperformed by ERA5 and the physics-free NN model. At the GSOS, BrTHF yielded more accurate estimates for SHF and β with RMSEs of 6.38 W m⁻² and 0.74 and values of r of 0.95 and 0.16, respectively, compared to other products, while its LHF was marginally less accurate than that of SeaFlux and the physics-free NN model. Moreover, under both spatially-informed cross-validation and targeted cross-validation, the model demonstrates comparable accuracy at the two sites, as shown in Tables. S4–S7. These findings suggest that BrTHF retains competitive accuracy of SHF, LHF and β even in regions entirely excluded from training, reflecting promising generalization. While these results are encouraging, it is important to note that the validation remains limited to a small number of sites with available observations. Therefore, the reported r values and RMSE reflect model performance in these specific locations and do not necessarily guarantee similar accuracy in broader, unobserved ocean regions. Consequently, the BrTHF product should be viewed as being optimized within the geographical coverage of existing buoy networks. In remote regions far from the observation-rich regions, such as the high-latitude Southern Ocean, the model inputs may fall outside the range of values represented in the training data, leading to increased uncertainty. Users should therefore exercise caution when interpreting the global-scale performance, particularly in data-sparse basins where spatial sampling limitations are most pronounced.

The generalization capability of the model can also affect the accuracy of simulated long-term trends. In Fig. 13, we present the spatial distributions of long-term trends for SHF, LHF, and β simulated by the BrTHF product. Considering the scarcity of training data in high-latitude oceans, the simulated long-term trends in these regions may be associated with larger uncertainties. However, due to the lack of long-term observations in high-latitude oceans, we cannot validate the simulated trends using observational records as has been done in previous studies for mid- and low-latitude regions (Weller et al., 2022; Tang et al., 2024). To address this, we examined the spatial distribution of long-term trends from the other seven widely used products, as illustrated in Figs. S2–S4. Specifically, in these high-latitude regions, the trends simulated by the BrTHF are largely consistent with those of most other products – for example, SHF exhibits a pronounced increase in the Kara Sea, Gulf Stream, Baffin Bay, Brazil Current, Sea of Okhotsk, and Sea of Japan, with differences mainly in magnitude.

Given the key role of the β constraint in the BrTHF model, it is also important to assess the sensitivity of the estimated SHF and LHF to the imposed β range. A series of sensitivity experiments with progressively relaxed β constraints indicate that the BrTHF model exhibits weak sensitivity to the specific choice of the β range (see Table S9). The resulting variations in RMSE are small relative to their mean values, while values of r and BIAS remain largely unchanged across different β configurations. These results suggest that the improved performance of BrTHF is not driven by a particular predefined range of β, but instead reflects the robustness of the Bowen ratio–constrained machine learning framework.

Beyond the sensitivity to the imposed β range, we further examined the robustness of the BrTHF model to uncertainties in its environmental inputs, using one-at-a-time perturbation experiments (Table S10). The results indicate that the model is generally insensitive to perturbations (from ±5 to ±20 %) in most auxiliary inputs, with changes in SHF and LHF RMSE typically remaining within 1 %. Noticeable sensitivities are mainly associated with WS, diff_Q, and diff_T, which are physically expected given their direct roles in controlling air–sea turbulent heat exchanges. Even under extreme perturbations, the resulting variations in SHF and LHF remain within physically reasonable ranges, suggesting that the BrTHF model does not rely excessively on precise tuning of individual inputs. For β, most perturbations lead to minor changes; however, perturbations of diff_Q can induce large variability due to the ratio-based nature of β, further highlighting the necessity of the β constraint.

To further address potential methodological inconsistencies between buoy-derived fluxes and flux products, we conducted an additional baseline experiment. Specifically, the COARE3.5 model was forced with the same subset of daily meteorological variables used to drive the BrTHF model, thereby providing a reference under identical forcing conditions. The results (Table S12) show that the COARE3.5-driven estimates exhibit substantially larger errors for SHF and LHF, with RMSEs of approximately 10.3 and 34.4 W m⁻², respectively, accompanied by systematic underestimations relative to buoy observations. More importantly, physically unrealistic values of the β emerge in the COARE3.5 estimates, leading to a degradation in β accuracy (RMSE = 6.35). These findings suggest that, even when driven by the same meteorological inputs, traditional bulk formulations remain highly sensitive to forcing uncertainties. In contrast, the BrTHF model demonstrates improved robustness by explicitly incorporating physical constraints within the machine-learning framework.

Finally, we examined the potential impact of temporal aggregation, as the buoy-derived daily fluxes used for model training and evaluation were calculated from daily-averaged meteorological variables via COARE3.5, which may introduce biases due to the nonlinearity of bulk flux formulations. By recalculating fluxes from high-frequency buoy observations and aggregating to daily means, we found that although absolute errors differ to some extent, the relative performance among different flux products and models remains unchanged, with the BrTHF model still achieving the best overall accuracy (Table S11). This confirms that our conclusions are robust to the choice of daily flux calculation strategy.

Although the results demonstrated significant improvements in the accuracy and physical consistency of SHF, LHF, and β estimates from the BrTHF model compared to those from the physics-free NN models and the seven products, the BrTHF product also has some limitations. First, extreme values of β may arise either from physically plausible but rare ocean conditions or from numerical instability or poorly constrained estimates associated with uncertainties in model-derived fluxes and input variables. In practice, these two sources are difficult to distinguish. To ensure robust model training and stable performance across the vast majority of β conditions (approximately the 1 %–99 % of the β distribution), we applied a conservative β constraint (-5 to 5) during training. This compresses the standard deviation and constrains the extreme tails of the predicted β, but substantially improves model stability and accuracy for the majority of ocean conditions. Although this choice leads to a narrower dynamic range compared to some other products, it ensures that the 1 %–99 % of the β distribution is well-represented for most practical applications. It should be noted that this strategy represents an interim solution rather than a final one. With future improvements in the quality and spatiotemporal representativeness of observational datasets, physically plausible extremes (e.g., 0 %–1 % and 99 %–100 % of the β distribution) could be better constrained and incorporated into model training, allowing expansion of the dynamic range of β. Secondly, the estimated SHF and LHF values exhibited a narrower distribution compared to the observations. This issue possibly stems from the uncertainty of the BrTHF model that was constructed from the uneven distribution of SHF and LHF in the observation datasets, which contain a low proportion of extreme samples, especially negative LHF values. Moreover, due to the insufficient observations, validation and modelling in high-latitude oceans, especially in the Southern Ocean, were limited. To address these problems, more experiments are highly recommended to collect observations covering more regions of oceans with better spatial and temporal representativeness, which could further enhance the product.

The BrTHF model demonstrated the feasibility and potential of jointly estimating multiple interrelated air–sea variables through a machine learning model that incorporates appropriate physical constraints to account for their interrelations. In the future, the predicted variables in the BrTHF model could be expanded to include surface radiation, heat storage, and precipitation over the ocean, by integrating the physical mechanisms of energy and water exchange. This would allow for the collaborative optimization of estimates across all components of the air–sea energy and water budgets, potentially contributing to achieving global closure of the air-sea energy and water budgets.