Physics-informed deep-learning model for mitigating spatiotemporal imbalances in FLUXNET2015 global evapotranspiration data

Physics-informed deep-learning model for mitigating spatiotemporal imbalances in FLUXNET2015 global evapotranspiration data | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Physics-informed deep-learning model for mitigating spatiotemporal imbalances in FLUXNET2015 global evapotranspiration data Jiancheng Wang, Tongren Xu, Sayed M. Bateni, Shaomin Liu, Changhyun Jun, and 7 more This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-5150315/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract As a key component of the water cycle, evapotranspiration (ET) plays a critical role in agricultural management and climate prediction. While numerous long-term observation sites have been established in Europe and North America (data-rich regions), fewer short-term observation sites exist in South America and, particularly, in Africa (data-poor regions). Several machine learning approaches have been developed for ET estimations. However, most existing studies in this field have used training and testing data from the same region, potentially leading to poor extrapolation in unseen areas. This paper proposes a physics-informed deep-learning model that considers external environmental variables, enabling more accurate identification of different underlying surfaces. Our results demonstrate that the proposed model effectively transfers the knowledge acquired from its training on data-rich regions to data-poor regions, thereby mitigating spatiotemporal imbalances in global in-situ ET observations. Overall, this approach can support the sustainable development of data-deficient regions or countries. Hydrology Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Introduction Evapotranspiration (ET), a crucial process in the hydrological cycle, refers to the transfer of water vapor from the Earth’s surface to the atmosphere 1 . As global warming intensifies, accurate estimations of ET become increasingly vital for precise weather prediction, agricultural production, water resource planning, and human livelihood 2 – 5 . Current ET estimation approaches can be broadly categorized into physics- and data-driven models. Among these, physics-driven models explicitly compute ET using mathematical formulas that represent physical phenomena such as energy and water balances. However, empirically estimating certain parameters using a unified model becomes challenging owing to the variability of the underlying surfaces 6 . Meanwhile, data-driven methods rely on large-scale data to automatically update model parameters, offering greater precision in ground truth approximations. However, these models are constrained by the opacification of their "black-box" nature, as well as their limited extrapolation performance. The inherent complexities of ET further complicate accurate ET estimations, as they require extensive observational data for validation. Moreover, owing to the uneven distribution of site data, regions with dense observation networks effectively support model validation, while those with sparse data monitoring experience greater uncertainty. This disparity impacts disaster prediction accuracy and complicates efforts to realize the Sustainable Development Goals (SDGs). In this study, we develop a physics-informed ET estimation method using a deep-learning (DL) model as its core framework. The sensitivity of the DL to various external environmental factors is improved by incorporating an additional parameter layer. This enhancement allows the DL model to better extrapolate insights acquired from training data to other data-poor regions, such as Africa, South America, and Siberia. Two types of input variables are incorporated: static and calculation variables. Among these, calculation variables are used to determine ET values, while static variables are extracted by the parameter layer to gauge the external environment, allowing appropriate adjustments to the calculation coefficients of the calculation variables. This approach facilitates the development of a unified model that better accounts for the physical conditions of diverse underlying surfaces, leading to improved ET estimation accuracy. The extrapolation performance of the proposed model, termed self-attention influence (SAI), is compared with that of several commonly used basic models (random forest (RF) and self-attention (SA)) and a computer vision model (multi-task and multi-scenario (MTMS) 7 ) for underlying surface classification. Furthermore, the relationship between ET and input variables is visualized using explainable methods, unveiling the internal mechanisms of the black-box framework. Finally, the global-scale ET results from these models and renowned ET products are compared across some basins to demonstrate the advantages of the SAI framework. The next SDGs remain a daunting challenge 8 , 9 . In Africa, development vulnerability is particularly affected by water resources 10 , 11 , and drought conditions substantially exacerbate this vulnerability 12 , 13 . Furthermore, global land ET shows a significant positive linear increase trend, which leads to increased water loss on land 14 . This complex coupling process exacerbates the uncertainty in the components of the water cycle in some regions such as Africa and South America 15 . Furthermore, data availability in these regions is limited; hence, they are commonly referred to as data-poor regions, in contrast to data-rich regions where extensive site data are maintained. Consequently, numerous models are routinely validated using data from these data-rich regions, overlooking the requirements for cross-domain estimation. This data imbalance requires urgent attention. Although ET estimation in data-poor regions is a critical aspect of the SDGs, most current models struggle to address this issue effectively. Most physics-driven models designed for ET estimation primarily rely on micro-meteorological theories based on the Penman–Monteith or Priestley–Taylor formulas 16 , 17 , calculation formulas based on balance equations 18 – 22 , and land surface hydrological models encapsulated through physical modules 23 – 27 . Meanwhile, traditional data-driven methods include empirical or semi-empirical models 28 – 30 and machine learning frameworks 31 – 35 . With the accumulation of satellite-derived and in-situ observations 36 , 37 , DL methods have gradually emerged as the mainstream approaches in data-driven research 38 – 40 . Notably, DL methods enhance estimation accuracy through high-dimensional feature space mapping and nonlinear fitting. Both physics- and data-driven methods offer numerous advantages in ET estimation and have been further refined to address global hydrological challenges 41 , 42 . However, physics-driven models are currently incapable of comprehensively simulating real-world conditions through high-density formula superposition, which limits their accuracy improvements. Conversely, the performance of data-driven models depends on the richness of the training data. Owing to the limited understanding of physical ET processes and the “black-box” nature of the utilized models, considerable estimation errors can arise if the data distributions of the training and testing sets exhibit deviations, particularly in data-poor regions. Consequently, developing models that can leverage the insights acquired from data-rich regions to improve performance in data-poor regions is essential, contributing toward data transmission and boosting ET estimation accuracy in Africa and other data-poor regions. To address these problems, hybrid methods that integrate physics- and data-driven models have been proposed 43 . These models are typically combined in either series or parallel configurations 44 , 45 . However, the issue of cross-domain estimation remains unresolved, and comprehensive evaluations of the extrapolation performance of these models remain challenging. To address the “black-box” nature, explainable methods must be employed 46 . One promising approach in this context is to learn specific information about ET processes in data-rich regions and apply this knowledge to data-poor regions. Notably, the “physics-constrained” approach introduces strong constraints by incorporating explicit equations through strategies such as adding the energy balance equation to the loss function. Alternatively, the “physics-informed” approach applies weak constraints by enhancing the ability of the model to perceive certain situations, enabling it to learn common characteristics across different regions. Given the inherent differences in ET trends between data-rich and data-poor regions, perceiving such shared features can improve ET estimation in data-poor regions. The proposed model perceives external environmental factors through an appropriate structural design, alleviates the data imbalance issue, and improves ET estimation accuracy in data-poor regions. Results High accuracy of SAI model in data-poor regions We designed three experimental scenarios to evaluate the extrapolation performance of the models when using unbalanced data (See Method). Notably, the estimation accuracies of the models in the data-rich-region test differed from those in the data-poor-region test (Table. S1). Overall, the RF model demonstrated the highest accuracy in the data-rich-region test (with a root mean square error (RMSE) of 0.62 mm/d and a coefficient of determination ( R 2 ) of 0.77) owing to strong interpolation capability, which allowed it to accurately estimate values based on data from neighboring points in the data-rich regions. Among DL models, the SA framework demonstrated the lowest accuracy (RMSE of 0.69 mm/d and R 2 of 0.74). Meanwhile, the proposed SAI model demonstrated the highest estimation accuracy (RMSE of 0.64 mm/d and R 2 of 0.76), nearly matching the performance of the RF model. These results indicate that most methods perform well in data-rich regions (such as North America) owing to the abundance of available data, which facilitates data adaptation. In the data-poor-region test, test data from Africa, South America, and Siberia were excluded from the training process to evaluate the extrapolation performance (Fig. S1). Unlike the data-rich-region test, the RF model demonstrated the lowest estimation accuracy in the data-poor-region test (RMSE of 1.02 mm/d and R 2 of 0.49). Comparatively, the estimation accuracies of the DL models were higher, with the SAI achieving the highest accuracy (RMSE of 0.76 mm/d and R 2 of 0.68), outperforming both the MTMS (RMSE of 0.80 mm/d and R 2 of 0.67) and SA (RMSE of 0.80 mm/d and R 2 of 0.68). Notably, the inclusion of a physics-informed module substantially improved estimation accuracy by addressing data imbalance issues, particularly in Africa and South America. This module also facilitated the transfer of information gathered from the data-rich regions to the data-poor regions, which is challenging for other learning models. According to the underlying surface classification framework of the International Geosphere-Biosphere Programme (IGBP), we compared the bias distributions (model estimations minus in-situ observations) of the estimation results. Interestingly, the following conclusions were derived from the data-rich-region test (Fig. 1 ): The SA model demonstrated the worst performance and underestimated the ET trends of the evergreen broadleaf forest (EBF). Although the RF model achieved the highest accuracy, it considerably overestimated the grassland (GRA) ET. Finally, the estimation accuracy of the MTMS was consistent with that of the SAI, and the physics-informed architecture effectively balanced the estimation accuracy for various underlying surfaces. In the data-poor-region test, the performance of the four models varied substantially (Fig. 2 ). In particular, the RF model performed poorly in the savannas (SAV) owing to variations in the relationship between input variables and ET during extrapolation based on feature comparison. Meanwhile, both the RF and MTMS models tended to overestimate ET values in the mixed forest (MF). Although the MTMS model adopted a multi-scenario architecture, the environmental characteristics of the MF conflated with those of the other underlying surface types owing to the model’s limited perception of external factors. The SA model generally underestimated the ET trends of both the deciduous broadly forest (DBF) and open shrubland (OSH). This is because a uniform model often sacrifices accuracy on specific underlying surfaces to minimize the overall loss function. Conversely, the proposed SAI demonstrated high accuracy across various underlying surfaces. This is because the SAI model learns external information from data-rich regions and adapts it well to data-poor regions. To avoid the contingency of above test results, we conducted additional experiments to support the results (Table. S2) by excluding data from specific regions (i.e., Australia and Asia) as the test sets (see Method). The results of the Australia test were consistent with those derived from the data-poor-region test. Specifically, the SAI model demonstrated the best performance (RMSE of 0.85 mm/d and R 2 of 0.71), followed by the MTMS (RMSE of 0.88 mm/d and R 2 of 0.69), SA (RMSE of 0.99 mm/d and R 2 of 0.63), and RF (RMSE of 1.05 mm/d and R 2 of 0.58). In the Asia test, the SAI and RF models demonstrated the highest and lowest accuracies (RMSEs of 0.88 mm/d and 0.96 mm/d and R 2 values of 0.58 and 0.51), respectively. Interestingly, the SA (RMSE of 0.90 mm/d and R 2 of 0.56) model outperformed the MTMS model (RMSE of 0.91 mm/d and R 2 of 0.57). This is because the multi-scenario architecture alone is insufficient for the DL models. Overall, the SAI framework consistently achieved higher accuracy across most underlying surfaces compared to the other models. A comparative analysis of the estimation accuracies of the models across various underlying surfaces in the Australia and Asia tests (Fig. S2 and S3) revealed that the RF, SA, and MTMS frameworks exhibited substantial errors for certain underlying surfaces owing to inadequate perception. In contrast, the SAI model delivered reasonable estimations for all types of underlying surfaces. In summary, the SAI model transfers the common information learned from data-rich regions to data-poor regions, substantially improving estimation accuracy in several challenging unseen regions such as Africa and South America. ET prediction mechanism of machine learning approaches To gain insights into the black-box nature of the four methods and to comprehend the relationship between input variables and ET, we used SHapley Additive exPlanations (SHAP) values (see Method) to quantify the contributions of different input variables. The results revealed that the RF model was more sensitive to solar-induced fluorescence (SIF) data (Fig. 3 ). In contrast, the DL methods prioritized incoming shortwave radiation (SW_IN), with site variables being generally more important than remote sensing (RS) variables. This is because in-situ measurements within the same spatial domain were consistent with the data distribution, as confirmed by a partial dependence plot (PDP) analysis (Method, Fig. S4). Owing to differences in model architectures, static variables (such as digital elevation model (DEM) and clay content (CLAY)) were not directly involved in the ET calculations of the MTMS and SAI models. These variables were instead used to control the data processing weights of the expert and influence layers. Consequently, the average SHAP values of these static variables were low. However, the SHAP values of the static variables considered in the SAI were higher than those in the MTMS owing to the presence of the influence layer, which enhanced the model’s ability to perceive external environmental factors. Furthermore, the RF model exhibited extremely low sensitivity to relative humidity (RH) (Fig. S4). Meanwhile, the SA model displayed the least sensitivity to cumulative weekly precipitation (acc_rain7), failing to capture the time-lag characteristics of precipitation. All models failed to capture the seasonal variation in ET (Fig. S4) owing to the lack of consideration for time-series continuity and confusion caused by differences between the northern and southern hemispheres. We further analyzed the impact of interactions between two variables on ET, using soil moisture (SM) and leaf area index (LAI) as examples (Fig. 4 ) owing to their medium SHAP importance. The interactions were categorized into three types: In Fig. 4 , the red points correspond to conditions of high SM but low LAI, indicating a negative impact on ET in areas with sparse vegetation even with high SM. When LAI is below approximately 1.2 m²/m², the interaction between SM and LAI negatively impacts ET, with low SM levels exhibiting a smaller negative correlation. When LAI exceeds 1.2 m²/m², higher SM levels exhibit a stronger positive relationship with ET. These findings suggest that the interactions between input variables are more complex than individual variable correlations. The SAI model demonstrates the most pronounced clustering, indirectly reflecting the advantages of its physics-informed architecture. Uncertainty of the SAI model at basin scale Using remote sensing and reanalysis data, we generated global ET estimates using the four models (Fig. S5 shown result from SAI model). Next, these ET estimates were compared with the data of two ET products across 15 global basins (top 5 data-rich basins and top 10 data-poor basins 18 , Table. S3, Fig. S6). The selected products, FLUXCOM and GLEAM, are representatives of machine learning and remote sensing approaches, respectively. Given the lack of ground truth values at the basin scale, the three-cornered hat (TCH) method was employed to determine the relative uncertainties 47 . As illustrated in Fig. 5 , the uncertainties in the ET estimates of the models range from 1 to 26 mm/month. Overall, the SAI model demonstrates the lowest uncertainty (3.027 mm/month), followed by SA and RF (4.653 mm/month and 5.413 mm/month, respectively). The MTMS model, GLEAM, and FLUXCOM (10‒18 mm/month) exhibit the highest uncertainties. Notably, all models (products) demonstrate lower relative uncertainties in data-rich basins owing to their adaptation to large datasets. However, the SAI model maintains the lowest uncertainty in data-poor basins. This finding aligns with the in-situ results, demonstrating the ability of the SAI model to transfer knowledge from data-rich regions to data-poor regions. Generally, all models have less uncertainty in data-rich basins than in data-poor basins. For example, in the drought-prone African region and the cold Siberian region, the uncertainty of MTMS and FLUXCOM is very high, especially FLUXCOM shows overestimation in some basins (blue background), while the uncertainty of the SAI model is relatively lower. However, the uncertainties in the Amazon basin and the Congo basin are relatively low across the models, but the consistency is poor (see Fig. S7). Due to the few data available in South America and Africa, ET estimations fluctuate widely and are difficult to determine. Spatial patterns of ET trends The Mann-Kendall trend test was employed to examine the spatial patterns of ET trends from 2000 to 2021 (with most pixel points reaching a 95% level of significance, Fig. 6 ). Overall, the ET results demonstrated an increasing trend on the global scale; however, this trend varied across different regions. In Siberia and northwestern Canada, a weaker ET growth trend was observed, which is confirmed by greening of the Pan-Arctic and intensified permafrost thawing 48 – 50 . In contrast, the southern forests of Canada exhibited a weaker declining trend 51 . In the Middle East, a strong decline in ET was detected along the southeastern Caspian Sea, potentially attributed to decreasing precipitation 52 , 53 . Most European regions exhibited an increasing ET trend 54 . However, in the Amazon basin, the increase in ET was not evident, likely owing to human activities such as deforestation 55 , 56 , Meanwhile, in central Africa, a strong increasing ET trend was observed, although the uncertainty in ET values was high in these areas. In Australia, the eastern coast exhibited an increasing ET trend, while the southern and central regions experienced a strong decline in ET, possibly influenced by changes in the mean sea surface temperature 57 . Additionally, India exhibited a notable increase in ET. These trends are generally consistent with those reported in several previous global studies 1 . Discussion Accurate estimations of ET trends are crucial for understanding the water cycle, particularly in regions where unpredictable disasters can impede sustainable development, such as in parts of Africa. However, the uneven global distribution of ET observation sites introduces substantial uncertainties in ET estimates for data-poor regions. Therefore, transferring information from data-dense regions to data-poor areas, such as Africa and South America, is essential to improve ET estimation accuracy. To address this challenge, we developed a physics-informed DL model, called the SAI model, for ET estimation. Notably, the SAI framework exhibits high-precision extrapolation capabilities, making it well-suited for estimating ET in data-poor regions. By categorizing the input variables into static and calculation variables and incorporating the influence layer, the SAI model can simulate real-world ET conditions, thereby improving the estimation accuracy. Notably, the SAI model demonstrates improved ET estimation accuracy across various types of land covers, providing high-precision predictions in diverse scenarios. Furthermore, using explainable analysis methods, the black-box nature of the model is unveiled, demonstrating the rationality of its design and further enhancing trust in its results. Finally, based on reanalysis and remote sensing data, generating monthly global results at the 0.25°scale and integrating them to the basin scale. The SAI model has the lowest uncertainty proved by TCH method, especially in data-poor basins. Methods Study data Data were obtained from FLUXNET 2015 Tier One ( https://fluxnet.org/data/fluxnet2015-dataset/ ) and the National Tibetan Plateau Data Center ( https://data.tpdc.ac.cn/home ). Thirteen variables were selected as model inputs (Table S4): including site-specific observation variables such as wind speed (WS), air temperature (Ta), RH, air pressure (Press), acc_rain, SW_IN, month, and IGBP land cover type. Additional variables included LAI 58 , SIF 59 , DEM 60 , SM 61 , and soil texture (CLAY, SAND, or SILT) 62 . Given the lack of SM data for some sites, remote-sensing-derived SM values were used consistently. Furthermore, cumulative precipitation was selected as the input, instead of daily precipitation, to account for time lag, and based on preliminary experiments, we finally selected cumulative weekly precipitation (acc_rain7) as our input variable (Table. S5). The adopted data processing techniques included gap-filling approaches and energy closure corrections using the Bowen ratio method. These methods resulted in a high-quality dataset. Global ET pixel scale results were generated, and the relevant FLUXNET 2015 meteorological station data were replaced with ERA5 reanalysis data, and SW_IN was obtained from GLDAS (Table. S6). RF and SA RF, the fundamental model in our study, comprises an ensemble of decision trees that vote to make predictions. The RF model is widely used for interpolation and, in some cases, achieves greater accuracy than DL methods. Hence, we established an RF model for comparison. In our experiment, the hyperparameters of the RF model were set using the Python sklearn package. Particularly, the number of decision trees was set as 100, maximum depth was set to None (allowing the node of each tree to be always split), and random seed (random_state) was set to 100. In recent years, Transformers have emerged as the most prominent DL methods, with the SA mechanism at their core 63 . Compared to other DL methods such as the convolutional neural network 64 , which compensates for the locality problem of receptive fields through deeper hidden layers with convolutional kernels, or the long short-term memory frameworks 65 , 66 , which can easily lose information in long time-series data, the SA mechanism directly computes the correlations between among individual input variables to improve accuracy. Hence, an ET estimation model based on the SA mechanism (Fig. 7 a) was developed for comparison. The principle and experimental architecture of the SA model: First, the input data are transformed by the Linear layer operation: $$\:\begin{array}{c}{X}_{hidden}=W\times\:{X}_{input}+b \left(1\right)\end{array}$$ where \(\:{X}_{input}\) denotes the input data value, weight W and offset b represent the parameters to be learned, and \(\:{X}_{hidden}\) represents the output obtained after feature extraction. Second, the output feature vector, as the input, passes through 10 layers of blocks, each incorporating a multi-head attention mechanism, a feedforward operation, and residual connections: $$\:\begin{array}{c}{\stackrel{\sim}{X}}_{hidden}=FF\left(MH\left({X}_{input}\right)+{X}_{input}\right)+MH\left({X}_{input}\right)+{X}_{input} \left(2\right)\end{array}$$ where FF represents a feedforward operation designed for feature extraction, and MH denotes the multi-head attention mechanism. Notably, in the residual connection, the feature vector post-operation (FF(·) or MH(·)) is added to the original input vector. The FF structure involves a layer-by-layer accumulation of the linear layer (Eq. 1) and rectified linear unit \(\:\:\) (ReLU) function: $$\:\begin{array}{c}ReLU\left(X\right)=\text{max}\left(0,X\right) \left(3\right)\end{array}$$ The ReLU function introduces nonlinearity into the model through the \(\:\text{m}\text{a}\text{x}\) function. As the core of the Transformer architecture, the SA mechanism enables the model to learn the relationships between input data. The following calculations are performed: $$\:\begin{array}{c}\stackrel{\sim}{V}=Attention\left(Q,K,V\right)=Softmax\left(\frac{Q{K}^{T}}{\sqrt{{d}_{K}}}\right)*V \left(4\right)\end{array}$$ $$\:\begin{array}{c}Q={W}_{Q}*X+{b}_{Q} \left(5\right)\end{array}$$ $$\:\begin{array}{c}K={W}_{K}*X+{b}_{K} \left(6\right)\end{array}$$ $$\:\begin{array}{c}V={W}_{V}*X+{b}_{V} \left(7\right)\end{array}$$ $$\:\begin{array}{c}Softmax\left({z}_{i}\right)=\frac{{e}^{{z}_{i}}}{{\sum\:}_{i=1}^{n}{e}^{{z}_{i}}} \left(8\right)\end{array}$$ where \(\:{W}_{Q}\) , \(\:{W}_{K}\) , \(\:{W}_{V}\) and \(\:{b}_{Q}\) , \(\:{b}_{K}\) , \(\:{b}_{V}\) are similar to the parameters of the linear layer (Eq. 1). Q, K, and V represent the query, key, and value of the input data respectively. First, the weight coefficient \(\:\alpha\:=Q{K}^{T}\) is calculated according to the query and key to learn the correlation between inputs. Subsequently, the correlation is multiplied by the corresponding value to output the result. The Softmax function maps the original output to values (0,1), ensuring that the cumulative sum of these values is one (satisfying the nature of probability). Eq. 4 is divided by \(\:\sqrt{{d}_{K}}\) to prevent the input value to the Softmax function from being excessively large, which could cause the partial derivative to approach zero, and the result ofαsatisfies a distribution with an expectation of zero and a variance of one, similar to normalization. MH slices a vector into different dimensions to capture various patterns through multiple attention mechanisms. MTMS Studies on global ET trends are typically performed across various underlying surface conditions, similar to recommendation systems in computer vision. To accommodate this, we designed the MTMS model (Fig. 7 b). Building upon the SA model, we introduced an expert decision layer in each block to create a multi-scenario framework. The input variables were categorized into two groups: variables for calculating ET (calculation variables) and variables for distinguishing underlying surfaces (static variables). Among these, the static variables were controlled by a fully connected layer, called the parameter layer, which determined the weights for data processing in the expert decision layer. The principle of parameter layer is similar to that of FF, differing only in the number of layers and hidden neurons. The final output of the model included an ET estimate and sensible heat flux (H), ensuring that the data processed by the model shared common features of energy flux, thereby preventing overfitting. The loss function was designed as the sum of the mean-square errors (MSEs) of the two variables, with weights of 0.7 and 0.3: $$\:\begin{array}{c}Loss=0.7*\frac{1}{n}{\sum\:}_{i=1}^{n}{\left({ET}_{output}-{ET}_{truth}\right)}^{2}+0.3*\frac{1}{n}{\sum\:}_{i=1}^{n}{\left({H}_{output}-{H}_{truth}\right)}^{2} \left(9\right)\end{array}$$ SAI A physics-informed DL method, called the SAI method (Fig. 7 c), was proposed herein. Although the MTMS model theoretically enables the recognition of underlying surfaces during training, the expert decision layers may become overly focused on certain channels, potentially degrading the performance of the base SA model. To address this, we leveraged the advantages of the SA mechanism and introduced an influence layer derived from static variables, which directly affects the learning of correlations between input variables. The size of this layer is the same as that of the \(\:\alpha\:(=Q{K}^{T})\) matrix in the attention layer (Eq. 4). By multiplying the corresponding positions of the two matrices directly, external variables are incorporated into the correlation learning process of the attention mechanism: $$\:\begin{array}{c}{I}_{E}=ReSize\left(Influence\left({X}_{static}\right),Q{K}^{T}\right) \left(10\right)\end{array}$$ $$\:\begin{array}{c}\stackrel{\sim}{V}=Attention\left(Q,K,V\right)=Softmax\left(\frac{Q{K}^{T}\text{*}{I}_{E}}{\sqrt{{d}_{K}}}\right)*V \left(11\right)\end{array}$$ where \(\:{X}_{static}\) represents the value of the static variables, and Resize(A, B) reshapes the size of matrix A to that of matrix B. The benefits of the SAI framework are twofold: First, it ensures that the learning of correlations between input variables is influenced by static variables from the external environment, thus avoiding the opacity induced by the “black-box” nature of the model. Furthermore, the interactions between different variables may vary across different underlying surfaces. However, most models tend to maintain constant weights for the correlations of variables once training is completed, leading to poor accuracy. Hence, to improve estimation accuracy, we attempted to enhance the model’s awareness of its external environment. Hyperparameter settings of the DL model The number of hidden layer neurons in the DL structure was set to 48, with \(\:FF\) having two or three layers. The number of heads in the multi-head attention mechanisms was three, and the expert layer included six experts. A dropout rate of 0.1 was set to prevent overfitting. During training, Batch_size is set as 64. The warm up training method is used, with a learning rate of 0.1 and attenuation factor of 0.98. Experimental Setup Numerous models have been previously reported for diverse ET estimation tasks, where the training and testing sets are randomly or sequentially divided from the same dataset. The model proposed in this study was aimed at improving extrapolation ability and enhance ET estimation accuracy in data-poor regions. To assess this performance, we designed three experiments: The data were divided into a training set (including validation set) and a test set for the three experiments (Fig. S8). Experiment 1: The data-rich-region test set was randomly selected from North America, Europe, Asia, and Australia, with numbers consistent with those of poor regions. The data-poor-region test set corresponded to other regions with fewer sites (approximately 3% of the total data). To minimize the contingency of the experimental results, two additional experiments were designed. In Experiment 2, all sites from the Asian region were selected as the test set, while in Experiment 3, all sites from the Australian region were selected as the test set. The remaining data were used for training and validation to further verify the extrapolation of the performance model. SHAP value principle The SHAP approach, inspired by game theory, is designed to interpret the predictions of machine learning models 67 . SHAP generates a value for each input feature (named a SHAP value). This value indicates how the feature contributes to the prediction of the outcome (positively or negatively). The principle of SHAP can be summarized as calculating the marginal contribution of an input to the result for a given sample: $$\:\begin{array}{c}{\varPhi\:}_{j}={\sum\:}_{S\subseteq\:N\backslash\:\left\{j\right\}}\frac{\left|S\right|!\left(N-\left|S\right|-1\right)!}{N!}\left(v\left(S\cup\:\left\{j\right\}\right)-v\left(S\right)\right) \left(11\right)\end{array}$$ where j denotes the input variable, S represents a subset of the input space, \(\:(v\left(S\cup\:\left\{j\right\}\right)-v(S\left)\right)\) denotes the marginal contribution of j to S, and \(\:\frac{\left|S\right|!\left(N-\left|S\right|-1\right)!}{N!}\) signifies the weight. In machine learning, \(\:v\left(S\right)=v\left(S\cup\:\left\{{j}_{mean}\right\}\right)\) is usually set to the model result value with the mean value of j. Partial dependency plot (PDP) principle PDPs illustrate the dependencies between the objective function (our models) and a set of features while marginalizing other features. These plots represent functional relationships by applying a model to a dataset and varying the value of the feature of interest while maintaining the other feature values constant. This process helps analyze the model output to determine the effect of the feature variables on the model's prediction results, such as a near-linear, monotonic, or more complex relationship. Global pixel-scale outputs and basin scale validation Based on reanalysis and remote sensing data (Table. S6), we retrained four models and produced global pixel-scale results. Five data-rich basins and 10 data-poor basins were selected according to the site density in the basin area. The four models are analyzed with FLUXCOM and GLEAM using the TCH method to account for uncertainties, owing to the lack of relative truth values for pixel-scale data. Next, given the differences in the vacancy value positions of different products (or models), we only selected the common pixels in the basin as the test set (the TCH method does not represent the entire basin (Fig. S6). The results of the SAI model were output through partial interpolation and averaging (Fig. 6 , S6)). Declarations Acknowledgments Funding: this work was funded by the National Natural Science Foundation of China (42171315). Author contributions: Conceptualization: Jiancheng Wang, Tongren Xu Methodology: Jiancheng Wang, Tongren Xu, Sayed M. Bateni Investigation: Sayed M. Bateni, Shaomin Liu Visualization: Jiancheng Wang, Ziwei Xu Supervision: Changhyun Jun, Dongkyun Kim, Xiaoyan Li, Xin Li Writing—original draft: Jiancheng Wang, Tongren Xu Writing—review & editing: Xiaofan Yang, Gangqiang Zhang, Wenting Ming Competing interests: the authors declare that they have no competing interests. Data and materials availability: All data needed to replicate these analyses and the code used to make these main figures are available at https://github.com/mayismine/SAI.git. References Yang Y et al (2023) Evapotranspiration on a greening Earth. Nat Rev Earth Environ 4:626–641 Rasmussen R et al (2014) Climate Change Impacts on the Water Balance of the Colorado Headwaters: High-Resolution Regional Climate Model Simulations. J Hydrometeorol 15:1091–1116 Cai J, Liu Y, Lei T, Pereira LS (2007) Estimating reference evapotranspiration with the FAO Penman–Monteith equation using daily weather forecast messages. 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J Hydrol 624:129897 Yin Z, Wang H, Liu XA (2014) Comparative Study on Precipitation Climatology and Interannual Variability in the Lower Midlatitude East Asia and Central Asia. J Clim 27:7830–7848 Rezaei A, Karami K, Tilmes S, Moore JC (2024) Future water storage changes over the Mediterranean, Middle East, and North Africa in response to global warming and stratospheric aerosol intervention. Earth Syst Dynam 15:91–108 Nistor MM, Satyanaga A, Dezsi S, Haidu I (2022) European Grid Dataset of Actual Evapotranspiration, Water Availability and Effective Precipitation. Atmosphere 13 Baker JCA et al (2021) Evapotranspiration in the Amazon: spatial patterns, seasonality, and recent trends in observations, reanalysis, and climate models. Hydrol Earth Syst Sci 25:2279–2300 Lapola DM et al (2023) The drivers and impacts of Amazon forest degradation. Science 379:eabp8622 Liang S et al (2022) Interplay of greening and ENSO on biosphere–atmosphere processes in Australia. 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Proceedings of the. ; Long Beach, California, USA: Curran Associates Inc.; 2017. pp. 6000–6010 García-Pedrero A, Gonzalo-Martin C, Lillo M, Rodríguez-Esparragón D, Menasalvas E (2017) Convolutional neural networks for estimating spatially distributed evapotranspiration Feng D, Liu J, Lawson K, Shen C, Differentiable (2022) Learnable, Regionalized Process-Based Models With Multiphysical Outputs can Approach State‐Of‐The‐Art Hydrologic Prediction Accuracy. Water Resour Res 58 Lees T et al (2022) Hydrological concept formation inside long short-term memory (LSTM) networks. Hydrol Earth Syst Sci 26:3079–3101 Lundberg SM, Lee S (2017) A Unified Approach to Interpreting Model Predictions. Neural Information Processing Systems. ; 2017 Additional Declarations The authors declare no competing interests. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-5150315","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":358808143,"identity":"164aa8c0-dcf2-4cf0-aa53-7ee5c97184f5","order_by":0,"name":"Jiancheng Wang","email":"","orcid":"","institution":"State Key Laboratory of Earth Surface Processes and Resource Ecology, School of Natural Resources, Faculty of Geographical Science, Beijing Normal University","correspondingAuthor":false,"prefix":"","firstName":"Jiancheng","middleName":"","lastName":"Wang","suffix":""},{"id":358808144,"identity":"0ffa4a3b-5f38-4e4f-947a-e2983e4d8cf1","order_by":1,"name":"Tongren Xu","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAqElEQVRIiWNgGAWjYLCCjw1gyoBY9cwMjDNJ1sLMS5IWgxv5xz7b7riT2MDevE2CoeYOYS2SM5KZZ+eeeZbYwHOsTILh2DPCWvglkpmZc9sOJzZI5JhJMDYcJqyFDaTFEqRF/g2RWsC2MIJt4SFSi2TPY2PG3jOHjdt40ootEo4RocXgeOJjhp87Dsv2sx/eeONDDRFa4IANRCSQoGEUjIJRMApGAR4AAJ4PNRFPqt4LAAAAAElFTkSuQmCC","orcid":"","institution":"State Key Laboratory of Earth Surface Processes and Resource Ecology, School of Natural Resources, Faculty of Geographical Science, Beijing Normal University","correspondingAuthor":true,"prefix":"","firstName":"Tongren","middleName":"","lastName":"Xu","suffix":""},{"id":358808145,"identity":"e457a09f-0274-4bfa-9f59-9d1753071eca","order_by":2,"name":"Sayed M. Bateni","email":"","orcid":"","institution":"Department of Civil, Environmental and Construction Engineering \u0026 Water Resources Research Center, University of Hawaii at Manoa","correspondingAuthor":false,"prefix":"","firstName":"Sayed","middleName":"M.","lastName":"Bateni","suffix":""},{"id":358808146,"identity":"dc378e71-568a-4195-9e2a-dc6132a16cd5","order_by":3,"name":"Shaomin Liu","email":"","orcid":"","institution":"State Key Laboratory of Earth Surface Processes and Resource Ecology, School of Natural Resources, Faculty of Geographical Science, Beijing Normal University","correspondingAuthor":false,"prefix":"","firstName":"Shaomin","middleName":"","lastName":"Liu","suffix":""},{"id":358808147,"identity":"eb46d8c1-b695-49ad-a13b-12fe0b3b1f78","order_by":4,"name":"Changhyun Jun","email":"","orcid":"","institution":"School of Civil, Environmental and Architectural Engineering, College of Engineering, Korea University","correspondingAuthor":false,"prefix":"","firstName":"Changhyun","middleName":"","lastName":"Jun","suffix":""},{"id":358808148,"identity":"1e1717e4-dd47-4481-9e3e-f531a23546cc","order_by":5,"name":"Dongkyun Kim","email":"","orcid":"","institution":"Department of Civil Engineering, Hongik University","correspondingAuthor":false,"prefix":"","firstName":"Dongkyun","middleName":"","lastName":"Kim","suffix":""},{"id":358808149,"identity":"6122aacf-2bef-440f-9bd6-74c8b15db7e2","order_by":6,"name":"Xiaoyan Li","email":"","orcid":"","institution":"State Key Laboratory of Earth Surface Processes and Resource Ecology, School of Natural Resources, Faculty of Geographical Science, Beijing Normal University","correspondingAuthor":false,"prefix":"","firstName":"Xiaoyan","middleName":"","lastName":"Li","suffix":""},{"id":358808150,"identity":"86dcc99b-c9b2-43af-8b67-7d899b0ed5ac","order_by":7,"name":"Xin Li","email":"","orcid":"","institution":"National Tibetan Plateau Data Center, State Key Laboratory of Tibetan Plateau Earth System, Environment and Resources, Institute of Tibetan Plateau Research, Chinese Academy of Sciences","correspondingAuthor":false,"prefix":"","firstName":"Xin","middleName":"","lastName":"Li","suffix":""},{"id":358808151,"identity":"d0d3aa85-211e-4cf8-aa54-3be0ce1e0d54","order_by":8,"name":"Xiaofan Yang","email":"","orcid":"","institution":"State Key Laboratory of Earth Surface Processes and Resource Ecology, School of Natural Resources, Faculty of Geographical Science, Beijing Normal University","correspondingAuthor":false,"prefix":"","firstName":"Xiaofan","middleName":"","lastName":"Yang","suffix":""},{"id":358808152,"identity":"80daf490-bd08-4273-b470-d60f7b2acba8","order_by":9,"name":"Ziwei Xu","email":"","orcid":"","institution":"State Key Laboratory of Earth Surface Processes and Resource Ecology, School of Natural Resources, Faculty of Geographical Science, Beijing Normal University","correspondingAuthor":false,"prefix":"","firstName":"Ziwei","middleName":"","lastName":"Xu","suffix":""},{"id":358808153,"identity":"e11d66b5-9177-4e62-8ff1-08478405dd49","order_by":10,"name":"Gangqiang Zhang","email":"","orcid":"","institution":"State Key Laboratory of Earth Surface Processes and Resource Ecology, School of Natural Resources, Faculty of Geographical Science, Beijing Normal University","correspondingAuthor":false,"prefix":"","firstName":"Gangqiang","middleName":"","lastName":"Zhang","suffix":""},{"id":358808154,"identity":"13b29a0d-f3b1-49b1-92f9-aa4a5c8df665","order_by":11,"name":"Wenting Ming","email":"","orcid":"","institution":"State Key Laboratory of Earth Surface Processes and Resource Ecology, School of Natural Resources, Faculty of Geographical Science, Beijing Normal University","correspondingAuthor":false,"prefix":"","firstName":"Wenting","middleName":"","lastName":"Ming","suffix":""}],"badges":[],"createdAt":"2024-09-25 08:36:52","currentVersionCode":1,"declarations":{"humanSubjects":false,"vertebrateSubjects":false,"conflictsOfInterestStatement":false,"humanSubjectEthicalGuidelines":false,"humanSubjectConsent":false,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":false},"doi":"10.21203/rs.3.rs-5150315/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-5150315/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":65337618,"identity":"e118f909-6b4e-4fc9-a8c1-63e545cb3ec3","added_by":"auto","created_at":"2024-09-26 08:29:04","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":202643,"visible":true,"origin":"","legend":"\u003cp\u003eBias distribution and scatter density in the data-rich-region test. The top graph displays the bias results (model estimations minus \u003cem\u003ein-situ\u003c/em\u003eobservations) of the four models. Meanwhile, the bottom graphs present scatter density plots for various underlying surfaces (grassland: GRA; savannas: SAV; evergreen broadleaf forest: EBF; deciduous broadly forest: DBF; mixed forest: MF; evergreen needleleaf forest: ENF; and open shrubland: OSH) classified in the test set relative to model estimations (mm/d) and \u003cem\u003ein-situ\u003c/em\u003e observations (mm/d). Rows 1 to 4 present the ET estimations of the RF, SA, MTMS, and SAI models, respectively.\u003c/p\u003e","description":"","filename":"image1.png","url":"https://assets-eu.researchsquare.com/files/rs-5150315/v1/0a255487484784482cdd7bf9.png"},{"id":65336228,"identity":"fbd784f6-9e19-4733-b715-b16a918230bc","added_by":"auto","created_at":"2024-09-26 08:13:05","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":208652,"visible":true,"origin":"","legend":"\u003cp\u003eBias distribution and scatter density in the data-poor-region test. The top graph displays the bias results (model estimations minus \u003cem\u003ein-situ\u003c/em\u003e observations) of the four models. Meanwhile, the bottom graphs depict scatter density plots for various underlying surfaces (grassland: GRA; savannas: SAV; evergreen broadleaf forest: EBF; deciduous broadly forest: DBF; mixed forest: MF; evergreen needleleaf forest: ENF; and open shrubland: OSH) classified in the test set relative to model estimations (mm/d) and \u003cem\u003ein-situ\u003c/em\u003e observations (mm/d). Rows 1 to 4 present the ET estimations of the RF, SA, MTMS, and SAI models, respectively.\u003c/p\u003e","description":"","filename":"image2.png","url":"https://assets-eu.researchsquare.com/files/rs-5150315/v1/b968364e90c2f29a7ba72361.png"},{"id":65336224,"identity":"4d776b17-ebe1-4894-8dcc-b1d4935bd43b","added_by":"auto","created_at":"2024-09-26 08:13:04","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":121048,"visible":true,"origin":"","legend":"\u003cp\u003eAnalysis of the four models based on SHAP values. Bar plots indicate the order of importance of the four models (calculated based on the average |SHAP| value of all samples), while violin plots of the SHAP values of the samples are displayed in the middle.\u003c/p\u003e","description":"","filename":"image3.png","url":"https://assets-eu.researchsquare.com/files/rs-5150315/v1/d2febdadd85ac577bd79c724.png"},{"id":65336451,"identity":"4938a85a-5288-48d8-a0de-20aba548a309","added_by":"auto","created_at":"2024-09-26 08:21:04","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":200278,"visible":true,"origin":"","legend":"\u003cp\u003ePDP analysis results of LAI and SM interactions based on the four models. The \u003cem\u003ex\u003c/em\u003e axis represents LAI (m\u003csup\u003e2\u003c/sup\u003e/m\u003csup\u003e2\u003c/sup\u003e), \u003cem\u003ey\u003c/em\u003e axis represents SHAP (mm/d), and colored points denote SM (m\u003csup\u003e3\u003c/sup\u003e/m\u003csup\u003e3\u003c/sup\u003e).\u003c/p\u003e","description":"","filename":"image4.png","url":"https://assets-eu.researchsquare.com/files/rs-5150315/v1/dc1b0657a5ffa9f5dd685a70.png"},{"id":65336230,"identity":"665de7e3-c258-47ec-9097-67e6fa66dc5f","added_by":"auto","created_at":"2024-09-26 08:13:05","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":360340,"visible":true,"origin":"","legend":"\u003cp\u003eUncertainties in the ET estimates of models for 20 basins, analyzed using the TCH method.\u003c/p\u003e","description":"","filename":"image5.png","url":"https://assets-eu.researchsquare.com/files/rs-5150315/v1/b0830f2482c8b6d4f7a4ac7a.png"},{"id":65336229,"identity":"ed4c18e6-c2f2-4047-9bac-1ad9cdb59510","added_by":"auto","created_at":"2024-09-26 08:13:05","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":529665,"visible":true,"origin":"","legend":"\u003cp\u003eSpatial patterns of global ET trends.\u003c/p\u003e","description":"","filename":"image6.png","url":"https://assets-eu.researchsquare.com/files/rs-5150315/v1/b801764a66272e24ef957973.png"},{"id":65336225,"identity":"4c77c548-92ac-47a7-b8b8-4287bf069b5d","added_by":"auto","created_at":"2024-09-26 08:13:04","extension":"png","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":229152,"visible":true,"origin":"","legend":"\u003cp\u003eDL Models. (A) SA mechanism, based on the Transformer model architecture. The encoder has 10 layers. (B) MTMS model, developed with SA as its foundational framework, with an added expert layer (multi-scenario) and multi-task mechanisms. (C) SAI model, based on the MTMS model, with an additional layer (influence layer) incorporated into the variable correlation analysis.\u003c/p\u003e","description":"","filename":"image7.png","url":"https://assets-eu.researchsquare.com/files/rs-5150315/v1/f35d28a3b8ccb9cfd214374f.png"},{"id":65338057,"identity":"dbfef30d-2490-4a28-8a9c-65126237c675","added_by":"auto","created_at":"2024-09-26 08:37:06","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":2210145,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-5150315/v1/501f5315-632a-4985-ad35-639dacd1d5f0.pdf"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"\u003cp\u003ePhysics-informed deep-learning model for mitigating spatiotemporal imbalances in FLUXNET2015 global evapotranspiration data\u003c/p\u003e","fulltext":[{"header":"Introduction","content":"\u003cp\u003eEvapotranspiration (ET), a crucial process in the hydrological cycle, refers to the transfer of water vapor from the Earth\u0026rsquo;s surface to the atmosphere\u003csup\u003e\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e\u003c/sup\u003e. As global warming intensifies, accurate estimations of ET become increasingly vital for precise weather prediction, agricultural production, water resource planning, and human livelihood\u003csup\u003e\u003cspan additionalcitationids=\"CR3 CR4\" citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e\u003c/sup\u003e. Current ET estimation approaches can be broadly categorized into physics- and data-driven models. Among these, physics-driven models explicitly compute ET using mathematical formulas that represent physical phenomena such as energy and water balances. However, empirically estimating certain parameters using a unified model becomes challenging owing to the variability of the underlying surfaces \u003csup\u003e\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e\u003c/sup\u003e. Meanwhile, data-driven methods rely on large-scale data to automatically update model parameters, offering greater precision in ground truth approximations. However, these models are constrained by the opacification of their \"black-box\" nature, as well as their limited extrapolation performance. The inherent complexities of ET further complicate accurate ET estimations, as they require extensive observational data for validation. Moreover, owing to the uneven distribution of site data, regions with dense observation networks effectively support model validation, while those with sparse data monitoring experience greater uncertainty. This disparity impacts disaster prediction accuracy and complicates efforts to realize the Sustainable Development Goals (SDGs).\u003c/p\u003e \u003cp\u003eIn this study, we develop a physics-informed ET estimation method using a deep-learning (DL) model as its core framework. The sensitivity of the DL to various external environmental factors is improved by incorporating an additional parameter layer. This enhancement allows the DL model to better extrapolate insights acquired from training data to other data-poor regions, such as Africa, South America, and Siberia. Two types of input variables are incorporated: static and calculation variables. Among these, calculation variables are used to determine ET values, while static variables are extracted by the parameter layer to gauge the external environment, allowing appropriate adjustments to the calculation coefficients of the calculation variables. This approach facilitates the development of a unified model that better accounts for the physical conditions of diverse underlying surfaces, leading to improved ET estimation accuracy. The extrapolation performance of the proposed model, termed self-attention influence (SAI), is compared with that of several commonly used basic models (random forest (RF) and self-attention (SA)) and a computer vision model (multi-task and multi-scenario (MTMS)\u003csup\u003e\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e\u003c/sup\u003e) for underlying surface classification. Furthermore, the relationship between ET and input variables is visualized using explainable methods, unveiling the internal mechanisms of the black-box framework. Finally, the global-scale ET results from these models and renowned ET products are compared across some basins to demonstrate the advantages of the SAI framework.\u003c/p\u003e \u003cp\u003eThe next SDGs remain a daunting challenge\u003csup\u003e\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e,\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e\u003c/sup\u003e. In Africa, development vulnerability is particularly affected by water resources \u003csup\u003e\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e,\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e\u003c/sup\u003e, and drought conditions substantially exacerbate this vulnerability \u003csup\u003e\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e,\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e\u003c/sup\u003e. Furthermore, global land ET shows a significant positive linear increase trend, which leads to increased water loss on land\u003csup\u003e\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e\u003c/sup\u003e. This complex coupling process exacerbates the uncertainty in the components of the water cycle in some regions such as Africa and South America\u003csup\u003e\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e\u003c/sup\u003e. Furthermore, data availability in these regions is limited; hence, they are commonly referred to as data-poor regions, in contrast to data-rich regions where extensive site data are maintained. Consequently, numerous models are routinely validated using data from these data-rich regions, overlooking the requirements for cross-domain estimation. This data imbalance requires urgent attention. Although ET estimation in data-poor regions is a critical aspect of the SDGs, most current models struggle to address this issue effectively.\u003c/p\u003e \u003cp\u003eMost physics-driven models designed for ET estimation primarily rely on micro-meteorological theories based on the Penman\u0026ndash;Monteith or Priestley\u0026ndash;Taylor formulas\u003csup\u003e\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e,\u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e\u003c/sup\u003e, calculation formulas based on balance equations\u003csup\u003e\u003cspan additionalcitationids=\"CR19 CR20 CR21\" citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e22\u003c/span\u003e\u003c/sup\u003e, and land surface hydrological models encapsulated through physical modules\u003csup\u003e\u003cspan additionalcitationids=\"CR24 CR25 CR26\" citationid=\"CR23\" class=\"CitationRef\"\u003e23\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR27\" class=\"CitationRef\"\u003e27\u003c/span\u003e\u003c/sup\u003e. Meanwhile, traditional data-driven methods include empirical or semi-empirical models\u003csup\u003e\u003cspan additionalcitationids=\"CR29\" citationid=\"CR28\" class=\"CitationRef\"\u003e28\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e30\u003c/span\u003e\u003c/sup\u003e and machine learning frameworks\u003csup\u003e\u003cspan additionalcitationids=\"CR32 CR33 CR34\" citationid=\"CR31\" class=\"CitationRef\"\u003e31\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR35\" class=\"CitationRef\"\u003e35\u003c/span\u003e\u003c/sup\u003e. With the accumulation of satellite-derived and \u003cem\u003ein-situ\u003c/em\u003e observations\u003csup\u003e\u003cspan citationid=\"CR36\" class=\"CitationRef\"\u003e36\u003c/span\u003e,\u003cspan citationid=\"CR37\" class=\"CitationRef\"\u003e37\u003c/span\u003e\u003c/sup\u003e, DL methods have gradually emerged as the mainstream approaches in data-driven research\u003csup\u003e\u003cspan additionalcitationids=\"CR39\" citationid=\"CR38\" class=\"CitationRef\"\u003e38\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR40\" class=\"CitationRef\"\u003e40\u003c/span\u003e\u003c/sup\u003e. Notably, DL methods enhance estimation accuracy through high-dimensional feature space mapping and nonlinear fitting. Both physics- and data-driven methods offer numerous advantages in ET estimation and have been further refined to address global hydrological challenges\u003csup\u003e\u003cspan citationid=\"CR41\" class=\"CitationRef\"\u003e41\u003c/span\u003e,\u003cspan citationid=\"CR42\" class=\"CitationRef\"\u003e42\u003c/span\u003e\u003c/sup\u003e. However, physics-driven models are currently incapable of comprehensively simulating real-world conditions through high-density formula superposition, which limits their accuracy improvements. Conversely, the performance of data-driven models depends on the richness of the training data. Owing to the limited understanding of physical ET processes and the \u0026ldquo;black-box\u0026rdquo; nature of the utilized models, considerable estimation errors can arise if the data distributions of the training and testing sets exhibit deviations, particularly in data-poor regions. Consequently, developing models that can leverage the insights acquired from data-rich regions to improve performance in data-poor regions is essential, contributing toward data transmission and boosting ET estimation accuracy in Africa and other data-poor regions.\u003c/p\u003e \u003cp\u003eTo address these problems, hybrid methods that integrate physics- and data-driven models have been proposed\u003csup\u003e\u003cspan citationid=\"CR43\" class=\"CitationRef\"\u003e43\u003c/span\u003e\u003c/sup\u003e. These models are typically combined in either series or parallel configurations\u003csup\u003e\u003cspan citationid=\"CR44\" class=\"CitationRef\"\u003e44\u003c/span\u003e,\u003cspan citationid=\"CR45\" class=\"CitationRef\"\u003e45\u003c/span\u003e\u003c/sup\u003e. However, the issue of cross-domain estimation remains unresolved, and comprehensive evaluations of the extrapolation performance of these models remain challenging. To address the \u0026ldquo;black-box\u0026rdquo; nature, explainable methods must be employed\u003csup\u003e\u003cspan citationid=\"CR46\" class=\"CitationRef\"\u003e46\u003c/span\u003e\u003c/sup\u003e. One promising approach in this context is to learn specific information about ET processes in data-rich regions and apply this knowledge to data-poor regions. Notably, the \u0026ldquo;physics-constrained\u0026rdquo; approach introduces strong constraints by incorporating explicit equations through strategies such as adding the energy balance equation to the loss function. Alternatively, the \u0026ldquo;physics-informed\u0026rdquo; approach applies weak constraints by enhancing the ability of the model to perceive certain situations, enabling it to learn common characteristics across different regions. Given the inherent differences in ET trends between data-rich and data-poor regions, perceiving such shared features can improve ET estimation in data-poor regions. The proposed model perceives external environmental factors through an appropriate structural design, alleviates the data imbalance issue, and improves ET estimation accuracy in data-poor regions.\u003c/p\u003e"},{"header":"Results","content":"\u003cdiv id=\"Sec3\" class=\"Section2\"\u003e \u003ch2\u003eHigh accuracy of SAI model in data-poor regions\u003c/h2\u003e \u003cp\u003eWe designed three experimental scenarios to evaluate the extrapolation performance of the models when using unbalanced data (See Method). Notably, the estimation accuracies of the models in the data-rich-region test differed from those in the data-poor-region test (Table. S1). Overall, the RF model demonstrated the highest accuracy in the data-rich-region test (with a root mean square error (RMSE) of 0.62 mm/d and a coefficient of determination (\u003cem\u003eR\u003c/em\u003e\u003csup\u003e\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e\u003c/sup\u003e) of 0.77) owing to strong interpolation capability, which allowed it to accurately estimate values based on data from neighboring points in the data-rich regions. Among DL models, the SA framework demonstrated the lowest accuracy (RMSE of 0.69 mm/d and \u003cem\u003eR\u003c/em\u003e\u003csup\u003e\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e\u003c/sup\u003e of 0.74). Meanwhile, the proposed SAI model demonstrated the highest estimation accuracy (RMSE of 0.64 mm/d and \u003cem\u003eR\u003c/em\u003e\u003csup\u003e\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e\u003c/sup\u003e of 0.76), nearly matching the performance of the RF model. These results indicate that most methods perform well in data-rich regions (such as North America) owing to the abundance of available data, which facilitates data adaptation.\u003c/p\u003e \u003cp\u003eIn the data-poor-region test, test data from Africa, South America, and Siberia were excluded from the training process to evaluate the extrapolation performance (Fig. S1). Unlike the data-rich-region test, the RF model demonstrated the lowest estimation accuracy in the data-poor-region test (RMSE of 1.02 mm/d and \u003cem\u003eR\u003c/em\u003e\u003csup\u003e\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e\u003c/sup\u003e of 0.49). Comparatively, the estimation accuracies of the DL models were higher, with the SAI achieving the highest accuracy (RMSE of 0.76 mm/d and \u003cem\u003eR\u003c/em\u003e\u003csup\u003e\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e\u003c/sup\u003e of 0.68), outperforming both the MTMS (RMSE of 0.80 mm/d and \u003cem\u003eR\u003c/em\u003e\u003csup\u003e\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e\u003c/sup\u003e of 0.67) and SA (RMSE of 0.80 mm/d and \u003cem\u003eR\u003c/em\u003e\u003csup\u003e\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e\u003c/sup\u003e of 0.68). Notably, the inclusion of a physics-informed module substantially improved estimation accuracy by addressing data imbalance issues, particularly in Africa and South America. This module also facilitated the transfer of information gathered from the data-rich regions to the data-poor regions, which is challenging for other learning models.\u003c/p\u003e \u003cp\u003eAccording to the underlying surface classification framework of the International Geosphere-Biosphere Programme (IGBP), we compared the bias distributions (model estimations minus \u003cem\u003ein-situ\u003c/em\u003e observations) of the estimation results. Interestingly, the following conclusions were derived from the data-rich-region test (Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e): The SA model demonstrated the worst performance and underestimated the ET trends of the evergreen broadleaf forest (EBF). Although the RF model achieved the highest accuracy, it considerably overestimated the grassland (GRA) ET. Finally, the estimation accuracy of the MTMS was consistent with that of the SAI, and the physics-informed architecture effectively balanced the estimation accuracy for various underlying surfaces.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eIn the data-poor-region test, the performance of the four models varied substantially (Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e). In particular, the RF model performed poorly in the savannas (SAV) owing to variations in the relationship between input variables and ET during extrapolation based on feature comparison. Meanwhile, both the RF and MTMS models tended to overestimate ET values in the mixed forest (MF). Although the MTMS model adopted a multi-scenario architecture, the environmental characteristics of the MF conflated with those of the other underlying surface types owing to the model\u0026rsquo;s limited perception of external factors. The SA model generally underestimated the ET trends of both the deciduous broadly forest (DBF) and open shrubland (OSH). This is because a uniform model often sacrifices accuracy on specific underlying surfaces to minimize the overall loss function. Conversely, the proposed SAI demonstrated high accuracy across various underlying surfaces. This is because the SAI model learns external information from data-rich regions and adapts it well to data-poor regions.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eTo avoid the contingency of above test results, we conducted additional experiments to support the results (Table. S2) by excluding data from specific regions (i.e., Australia and Asia) as the test sets (see Method). The results of the Australia test were consistent with those derived from the data-poor-region test. Specifically, the SAI model demonstrated the best performance (RMSE of 0.85 mm/d and \u003cem\u003eR\u003c/em\u003e\u003csup\u003e\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e\u003c/sup\u003e of 0.71), followed by the MTMS (RMSE of 0.88 mm/d and \u003cem\u003eR\u003c/em\u003e\u003csup\u003e\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e\u003c/sup\u003e of 0.69), SA (RMSE of 0.99 mm/d and \u003cem\u003eR\u003c/em\u003e\u003csup\u003e\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e\u003c/sup\u003e of 0.63), and RF (RMSE of 1.05 mm/d and \u003cem\u003eR\u003c/em\u003e\u003csup\u003e\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e\u003c/sup\u003e of 0.58). In the Asia test, the SAI and RF models demonstrated the highest and lowest accuracies (RMSEs of 0.88 mm/d and 0.96 mm/d and \u003cem\u003eR\u003c/em\u003e\u003csup\u003e\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e\u003c/sup\u003e values of 0.58 and 0.51), respectively. Interestingly, the SA (RMSE of 0.90 mm/d and \u003cem\u003eR\u003c/em\u003e\u003csup\u003e\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e\u003c/sup\u003e of 0.56) model outperformed the MTMS model (RMSE of 0.91 mm/d and \u003cem\u003eR\u003c/em\u003e\u003csup\u003e\u003cem\u003e\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e\u003c/em\u003e\u003c/sup\u003e of 0.57). This is because the multi-scenario architecture alone is insufficient for the DL models. Overall, the SAI framework consistently achieved higher accuracy across most underlying surfaces compared to the other models. A comparative analysis of the estimation accuracies of the models across various underlying surfaces in the Australia and Asia tests (Fig. S2 and S3) revealed that the RF, SA, and MTMS frameworks exhibited substantial errors for certain underlying surfaces owing to inadequate perception. In contrast, the SAI model delivered reasonable estimations for all types of underlying surfaces.\u003c/p\u003e \u003cp\u003eIn summary, the SAI model transfers the common information learned from data-rich regions to data-poor regions, substantially improving estimation accuracy in several challenging unseen regions such as Africa and South America.\u003c/p\u003e \u003c/div\u003e\n\u003ch3\u003eET prediction mechanism of machine learning approaches\u003c/h3\u003e\n\u003cp\u003eTo gain insights into the black-box nature of the four methods and to comprehend the relationship between input variables and ET, we used SHapley Additive exPlanations (SHAP) values (see Method) to quantify the contributions of different input variables. The results revealed that the RF model was more sensitive to solar-induced fluorescence (SIF) data (Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e). In contrast, the DL methods prioritized incoming shortwave radiation (SW_IN), with site variables being generally more important than remote sensing (RS) variables. This is because \u003cem\u003ein-situ\u003c/em\u003e measurements within the same spatial domain were consistent with the data distribution, as confirmed by a partial dependence plot (PDP) analysis (Method, Fig. S4).\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eOwing to differences in model architectures, static variables (such as digital elevation model (DEM) and clay content (CLAY)) were not directly involved in the ET calculations of the MTMS and SAI models. These variables were instead used to control the data processing weights of the expert and influence layers. Consequently, the average SHAP values of these static variables were low. However, the SHAP values of the static variables considered in the SAI were higher than those in the MTMS owing to the presence of the influence layer, which enhanced the model\u0026rsquo;s ability to perceive external environmental factors. Furthermore, the RF model exhibited extremely low sensitivity to relative humidity (RH) (Fig. S4). Meanwhile, the SA model displayed the least sensitivity to cumulative weekly precipitation (acc_rain7), failing to capture the time-lag characteristics of precipitation. All models failed to capture the seasonal variation in ET (Fig. S4) owing to the lack of consideration for time-series continuity and confusion caused by differences between the northern and southern hemispheres.\u003c/p\u003e \u003cp\u003eWe further analyzed the impact of interactions between two variables on ET, using soil moisture (SM) and leaf area index (LAI) as examples (Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e) owing to their medium SHAP importance. The interactions were categorized into three types: In Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e, the red points correspond to conditions of high SM but low LAI, indicating a negative impact on ET in areas with sparse vegetation even with high SM. When LAI is below approximately 1.2 m\u0026sup2;/m\u0026sup2;, the interaction between SM and LAI negatively impacts ET, with low SM levels exhibiting a smaller negative correlation. When LAI exceeds 1.2 m\u0026sup2;/m\u0026sup2;, higher SM levels exhibit a stronger positive relationship with ET. These findings suggest that the interactions between input variables are more complex than individual variable correlations. The SAI model demonstrates the most pronounced clustering, indirectly reflecting the advantages of its physics-informed architecture.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e\n\u003ch3\u003eUncertainty of the SAI model at basin scale\u003c/h3\u003e\n\u003cp\u003eUsing remote sensing and reanalysis data, we generated global ET estimates using the four models (Fig. S5 shown result from SAI model). Next, these ET estimates were compared with the data of two ET products across 15 global basins (top 5 data-rich basins and top 10 data-poor basins\u003csup\u003e\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e\u003c/sup\u003e, Table. S3, Fig. S6). The selected products, FLUXCOM and GLEAM, are representatives of machine learning and remote sensing approaches, respectively. Given the lack of ground truth values at the basin scale, the three-cornered hat (TCH) method was employed to determine the relative uncertainties\u003csup\u003e\u003cspan citationid=\"CR47\" class=\"CitationRef\"\u003e47\u003c/span\u003e\u003c/sup\u003e. As illustrated in Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e, the uncertainties in the ET estimates of the models range from 1 to 26 mm/month. Overall, the SAI model demonstrates the lowest uncertainty (3.027 mm/month), followed by SA and RF (4.653 mm/month and 5.413 mm/month, respectively). The MTMS model, GLEAM, and FLUXCOM (10‒18 mm/month) exhibit the highest uncertainties. Notably, all models (products) demonstrate lower relative uncertainties in data-rich basins owing to their adaptation to large datasets. However, the SAI model maintains the lowest uncertainty in data-poor basins. This finding aligns with the \u003cem\u003ein-situ\u003c/em\u003e results, demonstrating the ability of the SAI model to transfer knowledge from data-rich regions to data-poor regions.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eGenerally, all models have less uncertainty in data-rich basins than in data-poor basins. For example, in the drought-prone African region and the cold Siberian region, the uncertainty of MTMS and FLUXCOM is very high, especially FLUXCOM shows overestimation in some basins (blue background), while the uncertainty of the SAI model is relatively lower. However, the uncertainties in the Amazon basin and the Congo basin are relatively low across the models, but the consistency is poor (see Fig. S7). Due to the few data available in South America and Africa, ET estimations fluctuate widely and are difficult to determine.\u003c/p\u003e\n\u003ch3\u003eSpatial patterns of ET trends\u003c/h3\u003e\n\u003cp\u003eThe Mann-Kendall trend test was employed to examine the spatial patterns of ET trends from 2000 to 2021 (with most pixel points reaching a 95% level of significance, Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e). Overall, the ET results demonstrated an increasing trend on the global scale; however, this trend varied across different regions. In Siberia and northwestern Canada, a weaker ET growth trend was observed, which is confirmed by greening of the Pan-Arctic and intensified permafrost thawing\u003csup\u003e\u003cspan additionalcitationids=\"CR49\" citationid=\"CR48\" class=\"CitationRef\"\u003e48\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR50\" class=\"CitationRef\"\u003e50\u003c/span\u003e\u003c/sup\u003e. In contrast, the southern forests of Canada exhibited a weaker declining trend\u003csup\u003e\u003cspan citationid=\"CR51\" class=\"CitationRef\"\u003e51\u003c/span\u003e\u003c/sup\u003e. In the Middle East, a strong decline in ET was detected along the southeastern Caspian Sea, potentially attributed to decreasing precipitation\u003csup\u003e\u003cspan citationid=\"CR52\" class=\"CitationRef\"\u003e52\u003c/span\u003e,\u003cspan citationid=\"CR53\" class=\"CitationRef\"\u003e53\u003c/span\u003e\u003c/sup\u003e. Most European regions exhibited an increasing ET trend\u003csup\u003e\u003cspan citationid=\"CR54\" class=\"CitationRef\"\u003e54\u003c/span\u003e\u003c/sup\u003e. However, in the Amazon basin, the increase in ET was not evident, likely owing to human activities such as deforestation\u003csup\u003e\u003cspan citationid=\"CR55\" class=\"CitationRef\"\u003e55\u003c/span\u003e,\u003cspan citationid=\"CR56\" class=\"CitationRef\"\u003e56\u003c/span\u003e\u003c/sup\u003e, Meanwhile, in central Africa, a strong increasing ET trend was observed, although the uncertainty in ET values was high in these areas. In Australia, the eastern coast exhibited an increasing ET trend, while the southern and central regions experienced a strong decline in ET, possibly influenced by changes in the mean sea surface temperature\u003csup\u003e\u003cspan citationid=\"CR57\" class=\"CitationRef\"\u003e57\u003c/span\u003e\u003c/sup\u003e. Additionally, India exhibited a notable increase in ET. These trends are generally consistent with those reported in several previous global studies\u003csup\u003e\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e\u003c/sup\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e"},{"header":"Discussion","content":"\u003cp\u003eAccurate estimations of ET trends are crucial for understanding the water cycle, particularly in regions where unpredictable disasters can impede sustainable development, such as in parts of Africa. However, the uneven global distribution of ET observation sites introduces substantial uncertainties in ET estimates for data-poor regions. Therefore, transferring information from data-dense regions to data-poor areas, such as Africa and South America, is essential to improve ET estimation accuracy.\u003c/p\u003e \u003cp\u003eTo address this challenge, we developed a physics-informed DL model, called the SAI model, for ET estimation. Notably, the SAI framework exhibits high-precision extrapolation capabilities, making it well-suited for estimating ET in data-poor regions. By categorizing the input variables into static and calculation variables and incorporating the influence layer, the SAI model can simulate real-world ET conditions, thereby improving the estimation accuracy. Notably, the SAI model demonstrates improved ET estimation accuracy across various types of land covers, providing high-precision predictions in diverse scenarios. Furthermore, using explainable analysis methods, the black-box nature of the model is unveiled, demonstrating the rationality of its design and further enhancing trust in its results. Finally, based on reanalysis and remote sensing data, generating monthly global results at the 0.25\u0026deg;scale and integrating them to the basin scale. The SAI model has the lowest uncertainty proved by TCH method, especially in data-poor basins.\u003c/p\u003e "},{"header":"Methods","content":"\u003cdiv id=\"Sec9\" class=\"Section3\"\u003e\n \u003ch2\u003eStudy data\u003c/h2\u003e\n \u003cp\u003eData were obtained from FLUXNET 2015 Tier One (\u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://fluxnet.org/data/fluxnet2015-dataset/\u003c/span\u003e\u003c/span\u003e) and the National Tibetan Plateau Data Center (\u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://data.tpdc.ac.cn/home\u003c/span\u003e\u003c/span\u003e). Thirteen variables were selected as model inputs (Table S4): including site-specific observation variables such as wind speed (WS), air temperature (Ta), RH, air pressure (Press), acc_rain, SW_IN, month, and IGBP land cover type. Additional variables included LAI\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e58\u003c/span\u003e\u003c/sup\u003e, SIF\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e59\u003c/span\u003e\u003c/sup\u003e, DEM\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e60\u003c/span\u003e\u003c/sup\u003e, SM\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e61\u003c/span\u003e\u003c/sup\u003e, and soil texture (CLAY, SAND, or SILT)\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e62\u003c/span\u003e\u003c/sup\u003e. Given the lack of SM data for some sites, remote-sensing-derived SM values were used consistently. Furthermore, cumulative precipitation was selected as the input, instead of daily precipitation, to account for time lag, and based on preliminary experiments, we finally selected cumulative weekly precipitation (acc_rain7) as our input variable (Table. S5). The adopted data processing techniques included gap-filling approaches and energy closure corrections using the Bowen ratio method. These methods resulted in a high-quality dataset.\u003c/p\u003e\n \u003cp\u003eGlobal ET pixel scale results were generated, and the relevant FLUXNET 2015 meteorological station data were replaced with ERA5 reanalysis data, and SW_IN was obtained from GLDAS (Table. S6).\u003c/p\u003e\n\u003c/div\u003e\n\u003ch3\u003eRF and SA\u003c/h3\u003e\n\u003cp\u003eRF, the fundamental model in our study, comprises an ensemble of decision trees that vote to make predictions. The RF model is widely used for interpolation and, in some cases, achieves greater accuracy than DL methods. Hence, we established an RF model for comparison. In our experiment, the hyperparameters of the RF model were set using the Python sklearn package. Particularly, the number of decision trees was set as 100, maximum depth was set to None (allowing the node of each tree to be always split), and random seed (random_state) was set to 100.\u003c/p\u003e\n\u003cp\u003eIn recent years, Transformers have emerged as the most prominent DL methods, with the SA mechanism at their core\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e63\u003c/span\u003e\u003c/sup\u003e. Compared to other DL methods such as the convolutional neural network\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e64\u003c/span\u003e\u003c/sup\u003e, which compensates for the locality problem of receptive fields through deeper hidden layers with convolutional kernels, or the long short-term memory frameworks\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e65\u003c/span\u003e,\u003cspan class=\"CitationRef\"\u003e66\u003c/span\u003e\u003c/sup\u003e, which can easily lose information in long time-series data, the SA mechanism directly computes the correlations between among individual input variables to improve accuracy. Hence, an ET estimation model based on the SA mechanism (Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e7\u003c/span\u003ea) was developed for comparison. The principle and experimental architecture of the SA model:\u003c/p\u003e\n\u003cp\u003eFirst, the input data are transformed by the Linear layer operation:\u003c/p\u003e\n\u003cdiv id=\"Equa\" class=\"Equation\"\u003e\n \u003cdiv id=\"FileID_Equa\" class=\"mathdisplay\"\u003e$$\\:\\begin{array}{c}{X}_{hidden}=W\\times\\:{X}_{input}+b \\left(1\\right)\\end{array}$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{X}_{input}\\)\u003c/span\u003e\u003c/span\u003e denotes the input data value, weight W and offset b represent the parameters to be learned, and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{X}_{hidden}\\)\u003c/span\u003e\u003c/span\u003e represents the output obtained after feature extraction.\u003c/p\u003e\n\u003cp\u003eSecond, the output feature vector, as the input, passes through 10 layers of blocks, each incorporating a multi-head attention mechanism, a feedforward operation, and residual connections:\u003c/p\u003e\n\u003cdiv id=\"Equb\" class=\"Equation\"\u003e\n \u003cdiv id=\"FileID_Equb\" class=\"mathdisplay\"\u003e$$\\:\\begin{array}{c}{\\stackrel{\\sim}{X}}_{hidden}=FF\\left(MH\\left({X}_{input}\\right)+{X}_{input}\\right)+MH\\left({X}_{input}\\right)+{X}_{input} \\left(2\\right)\\end{array}$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003ewhere FF represents a feedforward operation designed for feature extraction, and MH denotes the multi-head attention mechanism. Notably, in the residual connection, the feature vector post-operation (FF(\u0026middot;) or MH(\u0026middot;)) is added to the original input vector. The FF structure involves a layer-by-layer accumulation of the linear layer (Eq.\u0026nbsp;1) and rectified linear unit\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\:\\)\u003c/span\u003e\u003c/span\u003e(ReLU) function:\u003c/p\u003e\n\u003cdiv id=\"Equc\" class=\"Equation\"\u003e\n \u003cdiv id=\"FileID_Equc\" class=\"mathdisplay\"\u003e$$\\:\\begin{array}{c}ReLU\\left(X\\right)=\\text{max}\\left(0,X\\right) \\left(3\\right)\\end{array}$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eThe ReLU function introduces nonlinearity into the model through the \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\text{m}\\text{a}\\text{x}\\)\u003c/span\u003e\u003c/span\u003e function. As the core of the Transformer architecture, the SA mechanism enables the model to learn the relationships between input data. The following calculations are performed:\u003c/p\u003e\n\u003cdiv id=\"Equd\" class=\"Equation\"\u003e\n \u003cdiv id=\"FileID_Equd\" class=\"mathdisplay\"\u003e$$\\:\\begin{array}{c}\\stackrel{\\sim}{V}=Attention\\left(Q,K,V\\right)=Softmax\\left(\\frac{Q{K}^{T}}{\\sqrt{{d}_{K}}}\\right)*V \\left(4\\right)\\end{array}$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Eque\" class=\"Equation\"\u003e\n \u003cdiv id=\"FileID_Eque\" class=\"mathdisplay\"\u003e$$\\:\\begin{array}{c}Q={W}_{Q}*X+{b}_{Q} \\left(5\\right)\\end{array}$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Equf\" class=\"Equation\"\u003e\n \u003cdiv id=\"FileID_Equf\" class=\"mathdisplay\"\u003e$$\\:\\begin{array}{c}K={W}_{K}*X+{b}_{K} \\left(6\\right)\\end{array}$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Equg\" class=\"Equation\"\u003e\n \u003cdiv id=\"FileID_Equg\" class=\"mathdisplay\"\u003e$$\\:\\begin{array}{c}V={W}_{V}*X+{b}_{V} \\left(7\\right)\\end{array}$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Equh\" class=\"Equation\"\u003e\n \u003cdiv id=\"FileID_Equh\" class=\"mathdisplay\"\u003e$$\\:\\begin{array}{c}Softmax\\left({z}_{i}\\right)=\\frac{{e}^{{z}_{i}}}{{\\sum\\:}_{i=1}^{n}{e}^{{z}_{i}}} \\left(8\\right)\\end{array}$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{W}_{Q}\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{W}_{K}\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{W}_{V}\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{b}_{Q}\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{b}_{K}\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{b}_{V}\\)\u003c/span\u003e\u003c/span\u003e are similar to the parameters of the linear layer (Eq.\u0026nbsp;1). Q, K, and V represent the query, key, and value of the input data respectively. First, the weight coefficient \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\alpha\\:=Q{K}^{T}\\)\u003c/span\u003e\u003c/span\u003e is calculated according to the query and key to learn the correlation between inputs. Subsequently, the correlation is multiplied by the corresponding value to output the result. The Softmax function maps the original output to values (0,1), ensuring that the cumulative sum of these values is one (satisfying the nature of probability). Eq.\u0026nbsp;4 is divided by \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\sqrt{{d}_{K}}\\)\u003c/span\u003e\u003c/span\u003e to prevent the input value to the Softmax function from being excessively large, which could cause the partial derivative to approach zero, and the result of\u0026alpha;satisfies a distribution with an expectation of zero and a variance of one, similar to normalization. MH slices a vector into different dimensions to capture various patterns through multiple attention mechanisms.\u003c/p\u003e\n\u003cp\u003e\u0026nbsp;\u003c/p\u003e\n\u003cdiv id=\"Sec11\" class=\"Section2\"\u003e\n \u003ch2\u003eMTMS\u003c/h2\u003e\n \u003cp\u003eStudies on global ET trends are typically performed across various underlying surface conditions, similar to recommendation systems in computer vision. To accommodate this, we designed the MTMS model (Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e7\u003c/span\u003eb). Building upon the SA model, we introduced an expert decision layer in each block to create a multi-scenario framework. The input variables were categorized into two groups: variables for calculating ET (calculation variables) and variables for distinguishing underlying surfaces (static variables). Among these, the static variables were controlled by a fully connected layer, called the parameter layer, which determined the weights for data processing in the expert decision layer. The principle of parameter layer is similar to that of FF, differing only in the number of layers and hidden neurons. The final output of the model included an ET estimate and sensible heat flux (H), ensuring that the data processed by the model shared common features of energy flux, thereby preventing overfitting. The loss function was designed as the sum of the mean-square errors (MSEs) of the two variables, with weights of 0.7 and 0.3:\u003c/p\u003e\n \u003cdiv id=\"Equi\" class=\"Equation\"\u003e\n \u003cdiv id=\"FileID_Equi\" class=\"mathdisplay\"\u003e$$\\:\\begin{array}{c}Loss=0.7*\\frac{1}{n}{\\sum\\:}_{i=1}^{n}{\\left({ET}_{output}-{ET}_{truth}\\right)}^{2}+0.3*\\frac{1}{n}{\\sum\\:}_{i=1}^{n}{\\left({H}_{output}-{H}_{truth}\\right)}^{2} \\left(9\\right)\\end{array}$$\u003c/div\u003e\n \u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec12\" class=\"Section2\"\u003e\n \u003ch2\u003eSAI\u003c/h2\u003e\n \u003cp\u003eA physics-informed DL method, called the SAI method (Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e7\u003c/span\u003ec), was proposed herein. Although the MTMS model theoretically enables the recognition of underlying surfaces during training, the expert decision layers may become overly focused on certain channels, potentially degrading the performance of the base SA model. To address this, we leveraged the advantages of the SA mechanism and introduced an influence layer derived from static variables, which directly affects the learning of correlations between input variables. The size of this layer is the same as that of the \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\alpha\\:(=Q{K}^{T})\\)\u003c/span\u003e\u003c/span\u003e matrix in the attention layer (Eq.\u0026nbsp;4). By multiplying the corresponding positions of the two matrices directly, external variables are incorporated into the correlation learning process of the attention mechanism:\u003c/p\u003e\n \u003cdiv id=\"Equj\" class=\"Equation\"\u003e\n \u003cdiv id=\"FileID_Equj\" class=\"mathdisplay\"\u003e$$\\:\\begin{array}{c}{I}_{E}=ReSize\\left(Influence\\left({X}_{static}\\right),Q{K}^{T}\\right) \\left(10\\right)\\end{array}$$\u003c/div\u003e\n \u003c/div\u003e\n \u003cdiv id=\"Equk\" class=\"Equation\"\u003e\n \u003cdiv id=\"FileID_Equk\" class=\"mathdisplay\"\u003e$$\\:\\begin{array}{c}\\stackrel{\\sim}{V}=Attention\\left(Q,K,V\\right)=Softmax\\left(\\frac{Q{K}^{T}\\text{*}{I}_{E}}{\\sqrt{{d}_{K}}}\\right)*V \\left(11\\right)\\end{array}$$\u003c/div\u003e\n \u003c/div\u003e\n \u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{X}_{static}\\)\u003c/span\u003e\u003c/span\u003e represents the value of the static variables, and Resize(A, B) reshapes the size of matrix A to that of matrix B.\u003c/p\u003e\n \u003cp\u003eThe benefits of the SAI framework are twofold: First, it ensures that the learning of correlations between input variables is influenced by static variables from the external environment, thus avoiding the opacity induced by the \u0026ldquo;black-box\u0026rdquo; nature of the model. Furthermore, the interactions between different variables may vary across different underlying surfaces. However, most models tend to maintain constant weights for the correlations of variables once training is completed, leading to poor accuracy. Hence, to improve estimation accuracy, we attempted to enhance the model\u0026rsquo;s awareness of its external environment.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec13\" class=\"Section2\"\u003e\n \u003ch2\u003eHyperparameter settings of the DL model\u003c/h2\u003e\n \u003cp\u003eThe number of hidden layer neurons in the DL structure was set to 48, with \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:FF\\)\u003c/span\u003e\u003c/span\u003e having two or three layers. The number of heads in the multi-head attention mechanisms was three, and the expert layer included six experts. A dropout rate of 0.1 was set to prevent overfitting.\u003c/p\u003e\n \u003cp\u003eDuring training, Batch_size is set as 64. The warm up training method is used, with a learning rate of 0.1 and attenuation factor of 0.98.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec14\" class=\"Section2\"\u003e\n \u003ch2\u003eExperimental Setup\u003c/h2\u003e\n \u003cp\u003eNumerous models have been previously reported for diverse ET estimation tasks, where the training and testing sets are randomly or sequentially divided from the same dataset. The model proposed in this study was aimed at improving extrapolation ability and enhance ET estimation accuracy in data-poor regions. To assess this performance, we designed three experiments:\u003c/p\u003e\n \u003cp\u003eThe data were divided into a training set (including validation set) and a test set for the three experiments (Fig. S8). Experiment 1: The data-rich-region test set was randomly selected from North America, Europe, Asia, and Australia, with numbers consistent with those of poor regions. The data-poor-region test set corresponded to other regions with fewer sites (approximately 3% of the total data). To minimize the contingency of the experimental results, two additional experiments were designed. In Experiment 2, all sites from the Asian region were selected as the test set, while in Experiment 3, all sites from the Australian region were selected as the test set. The remaining data were used for training and validation to further verify the extrapolation of the performance model.\u003c/p\u003e\n \u003cp\u003eSHAP value principle\u003c/p\u003e\n \u003cp\u003eThe SHAP approach, inspired by game theory, is designed to interpret the predictions of machine learning models\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e67\u003c/span\u003e\u003c/sup\u003e. SHAP generates a value for each input feature (named a SHAP value). This value indicates how the feature contributes to the prediction of the outcome (positively or negatively). The principle of SHAP can be summarized as calculating the marginal contribution of an input to the result for a given sample:\u003c/p\u003e\n \u003cdiv id=\"Equl\" class=\"Equation\"\u003e\n \u003cdiv id=\"FileID_Equl\" class=\"mathdisplay\"\u003e$$\\:\\begin{array}{c}{\\varPhi\\:}_{j}={\\sum\\:}_{S\\subseteq\\:N\\backslash\\:\\left\\{j\\right\\}}\\frac{\\left|S\\right|!\\left(N-\\left|S\\right|-1\\right)!}{N!}\\left(v\\left(S\\cup\\:\\left\\{j\\right\\}\\right)-v\\left(S\\right)\\right) \\left(11\\right)\\end{array}$$\u003c/div\u003e\n \u003c/div\u003e\n \u003cp\u003ewhere j denotes the input variable, S represents a subset of the input space, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:(v\\left(S\\cup\\:\\left\\{j\\right\\}\\right)-v(S\\left)\\right)\\)\u003c/span\u003e\u003c/span\u003e denotes the marginal contribution of j to S, and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\frac{\\left|S\\right|!\\left(N-\\left|S\\right|-1\\right)!}{N!}\\)\u003c/span\u003e\u003c/span\u003e signifies the weight. In machine learning, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:v\\left(S\\right)=v\\left(S\\cup\\:\\left\\{{j}_{mean}\\right\\}\\right)\\)\u003c/span\u003e\u003c/span\u003e is usually set to the model result value with the mean value of j.\u003c/p\u003e\u003cp\u003ePartial dependency plot (PDP) principle\u003c/p\u003e\u003cp\u003ePDPs illustrate the dependencies between the objective function (our models) and a set of features while marginalizing other features. These plots represent functional relationships by applying a model to a dataset and varying the value of the feature of interest while maintaining the other feature values constant. This process helps analyze the model output to determine the effect of the feature variables on the model\u0026apos;s prediction results, such as a near-linear, monotonic, or more complex relationship.\u003c/p\u003e\u003cp\u003eGlobal pixel-scale outputs and basin scale validation\u003c/p\u003e\u003cp\u003eBased on reanalysis and remote sensing data (Table. S6), we retrained four models and produced global pixel-scale results. Five data-rich basins and 10 data-poor basins were selected according to the site density in the basin area. The four models are analyzed with FLUXCOM and GLEAM using the TCH method to account for uncertainties, owing to the lack of relative truth values for pixel-scale data. Next, given the differences in the vacancy value positions of different products (or models), we only selected the common pixels in the basin as the test set (the TCH method does not represent the entire basin (Fig. S6). The results of the SAI model were output through partial interpolation and averaging (Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e6\u003c/span\u003e, S6)).\u003c/p\u003e\u003c/div\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003eAcknowledgments\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eFunding:\u003c/strong\u003e this work was funded by the National Natural Science Foundation of China (42171315).\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAuthor contributions:\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003e\u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp;\u0026nbsp;Conceptualization: Jiancheng Wang, Tongren Xu\u003c/p\u003e\n\u003cp\u003e\u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp;\u0026nbsp;Methodology: Jiancheng Wang, Tongren Xu,\u0026nbsp;Sayed M. Bateni\u003c/p\u003e\n\u003cp\u003e\u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp;\u0026nbsp;Investigation: Sayed M. Bateni,\u0026nbsp;Shaomin Liu\u003c/p\u003e\n\u003cp\u003e\u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp;\u0026nbsp;Visualization: Jiancheng Wang,\u0026nbsp;Ziwei Xu\u003c/p\u003e\n\u003cp\u003e\u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp;\u0026nbsp;Supervision: Changhyun Jun,\u0026nbsp;Dongkyun Kim,\u0026nbsp;Xiaoyan Li,\u0026nbsp;Xin Li\u003c/p\u003e\n\u003cp\u003e\u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp;\u0026nbsp;Writing\u0026mdash;original draft: Jiancheng Wang, Tongren Xu\u003c/p\u003e\n\u003cp\u003e\u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp;\u0026nbsp;Writing\u0026mdash;review \u0026amp; editing:\u0026nbsp;Xiaofan Yang, Gangqiang Zhang, Wenting Ming\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eCompeting interests:\u003c/strong\u003e the authors declare that they have no\u0026nbsp;competing interests.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eData and materials availability:\u003c/strong\u003e All data needed to replicate these analyses and the code used to make these main figures are available at https://github.com/mayismine/SAI.git.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\u003cli\u003e\u003cspan\u003eYang Y et al (2023) Evapotranspiration on a greening Earth. 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Neural Information Processing Systems. ; 2017\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[{"identity":"aa066ff9-305c-4d91-b368-7a22445fc023","identifier":"10.13039/501100001809","name":"National Natural Science Foundation of China","awardNumber":"42171315","order_by":0}],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"State Key Laboratory of Earth Surface Processes and Resource Ecology, School of Natural Resources, Faculty of Geographical Science, Beijing Normal University","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"","lastPublishedDoi":"10.21203/rs.3.rs-5150315/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-5150315/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eAs a key component of the water cycle, evapotranspiration (ET) plays a critical role in agricultural management and climate prediction. While numerous long-term observation sites have been established in Europe and North America (data-rich regions), fewer short-term observation sites exist in South America and, particularly, in Africa (data-poor regions). Several machine learning approaches have been developed for ET estimations. However, most existing studies in this field have used training and testing data from the same region, potentially leading to poor extrapolation in unseen areas. This paper proposes a physics-informed deep-learning model that considers external environmental variables, enabling more accurate identification of different underlying surfaces. Our results demonstrate that the proposed model effectively transfers the knowledge acquired from its training on data-rich regions to data-poor regions, thereby mitigating spatiotemporal imbalances in global \u003cem\u003ein-situ\u003c/em\u003e ET observations. Overall, this approach can support the sustainable development of data-deficient regions or countries.\u003c/p\u003e","manuscriptTitle":"Physics-informed deep-learning model for mitigating spatiotemporal imbalances in FLUXNET2015 global evapotranspiration data","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-09-26 08:13:00","doi":"10.21203/rs.3.rs-5150315/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"71b0f660-a11d-459e-8d60-1e44fc2cc6da","owner":[],"postedDate":"September 26th, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":38177075,"name":"Hydrology"}],"tags":[],"updatedAt":"2024-09-26T08:13:00+00:00","versionOfRecord":[],"versionCreatedAt":"2024-09-26 08:13:00","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-5150315","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-5150315","identity":"rs-5150315","version":["v1"]},"buildId":"qtupq5eGEP_6zYnWcrvyt","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}