A Deep Learning Inversion Method for 3D Temperature Structures in the South China Sea with Physical Constraints

Abstract

The South China Sea, a vital marginal sea in tropical–subtropical Southeast Asia, plays a globally significant role in marine biodiversity and climate system dynamics. The accurate monitoring of its thermal structure is essential for ecological and climatic studies, yet retrieving subsurface temperature remains challenging due to complex ocean–atmosphere interactions. This study develops a Convolutional Long Short-Term Memory (ConvLSTM) neural network, integrating multi-source satellite remote sensing data, to reconstruct the Ocean Subsurface Temperature Structure (OSTS). To address the multiparameter complexity of temperature retrieval, physical constraints—particularly the heat budget balance of water bodies—are incorporated into the loss function. Experiments demonstrate that the physics-informed ConvLSTM model significantly improves the temperature estimation accuracy by simultaneously optimizing the physical consistency and predictive performance. The proposed approach advances ocean remote sensing by synergizing data-driven learning with thermodynamic principles, offering a robust framework for understanding the South China Sea’s thermal variability.

1. Introduction

Satellite remote sensing technology plays a crucial role in marine environmental monitoring due to its advantages of synchronicity, a broad coverage, and high-frequency observation capabilities [1,2,3]. However, its observations are limited to sea surface information and cannot directly capture deep ocean data. In contrast, buoy and submersible-based methods can provide high-precision vertical profiles but suffer from spatiotemporal discontinuity. Therefore, integrating multi-source observational data has become an effective approach for studying three-dimensional ocean structures.

The South China Sea, as one of China’s major marginal seas with a maximum depth exceeding 5500 m, features a complex tropical marine environment and highly variable heat structure that significantly influences key processes such as marine ecosystems, ocean circulation, and air–sea interactions [4,5,6,7,8,9,10]. Investigating the three-dimensional temperature structure of the South China Sea not only enhances our understanding of regional ocean dynamics and monsoon systems [11,12] but also provides scientific support for marine resource management. In recent years, significant progress has been made in this field: Mao et al. combined remote sensing and in situ data to construct a dynamic 3D temperature–salinity field using statistical analysis and data assimilation methods [13]; Tang et al. reconstructed the heat and haline structures in the northern South China Sea using satellite observations, revealing mesoscale variability [14]; and Qi et al. improved the inversion of subsurface heat structures through an ensemble machine learning model [15].

Prior to the widespread application of artificial intelligence methods, ocean temperature inversion primarily relied on numerical models, empirical orthogonal function (EOF) analysis, and linear approximation approaches. For instance, Khedouri, E. et al. utilized the correlation between the sea surface height and subsurface temperature to invert heat structures using satellite altimetry data [16], while Chu, P.C. et al. employed parameterized models to estimate the subsurface heat structure based on the sea surface temperature (SST) [17]. Additional studies have explored the application of satellite data in reconstructing interior ocean temperature fields [18,19,20].

With recent advancements in big data and computational technologies, machine learning has gained increasing prominence in oceanographic research. Classical algorithms, such as artificial neural networks (ANNs), have been successfully applied to temperature profile inversion. Ali, M.M. et al. demonstrated that ANNs can effectively estimate subsurface temperatures using surface parameters including the SST, sea surface height, and wind stress [21]. Su, H. et al. implemented support vector machines (SVMs), random forests (RFs), XGBoost, and LightGBM to invert subsurface temperature anomalies in the Indian Ocean and globally [22,23,24,25], further enhancing temporal modeling capabilities through long short-term memory (LSTM) networks [26]. Cheng, H. et al. improved North Pacific temperature inversion by incorporating sea surface velocity (SSV) with backpropagation neural networks (BPNNs) [27].

Recognizing the spatial correlation inherent in oceanographic data, convolutional neural networks (CNNs) in deep learning have been increasingly adopted. Han, M. et al. reconstructed Pacific subsurface temperatures using multi-source sea surface data, though this was constrained by local convolutional operations [28]. To better capture spatiotemporal features, Shi, X. et al. proposed ConvLSTM to establish an end-to-end trainable model [29]. Song, T. et al. further integrated the advantages of the CNN and LSTM, demonstrating their superiority over conventional methods in temperature–salinity field inversion [30], and developed the Convformer model which, while exhibiting an excellent performance in tropical Pacific temperature prediction, still shows limitations in estimating extreme events [31].

Current ocean temperature inversion methods based purely on mathematical models have achieved notable success. However, their reliance on mathematical optimization while neglecting physical principles may lead to physically unrealistic results. Incorporating physical constraints can effectively enhance model performance and improve the rationality of inversion results. For instance, Zhang, Y. et al. found that when inverting temperatures at 100–500 m depths, using 50 m depth data as a constraint—compared to relying solely on satellite surface observations—yielded modest but statistically significant improvements in the inversion accuracy [32], underscoring the critical role of internal ocean physical parameters.

This study employs ConvLSTM networks to invert the three-dimensional temperature structure in the northern South China Sea. By incorporating seafloor physical characteristics and modifying the loss function to include heat budget balance constraints, our optimized model demonstrates a superior performance across 11 depth layers. Notably, it achieves a 12.6% reduction in the RMSE within the mixed layer (19–301 m), validating the effectiveness of physical constraints in enhancing the inversion accuracy.

2. Materials and Methods

2.1. Materials

The study area of this research is between 9.25° N and 18.25° N latitude and between 111° E and 117.5° E longitude (as shown in the red frame in Figure 1). Some of the original data from satellite observations involved in this study include the sea surface height (SSH), sea surface temperature (SST), sea surface winds in the east–west and north–south directions (USSW, VSSW), and sea temperature data (ST) at 11 depth layers (19 m, 47 m, 78 m, 97 m, 147 m, 200 m, 301 m, 508 m, 1046 m, 1516 m, and 2101 m), which are used as labels. In addition, there are some variables introduced in the custom loss function, including the sea surface sensible heat flux (sshf), latent heat flux (slhf), long-wave and short-wave radiation fluxes (str, ssr), as well as the seawater flow velocities in the east–west and north–south directions (U, V). Their sources are shown in

Table 1. Data summary.

Input Variable	Dataset	Source
SSH	AVISO	https://www.aviso.altimetry.fr/en/data.html (accessed on 29 December 2023)
SST	OISST	https://www.ncei.noaa.gov/thredds/blended-global/oisst-catalog.html (accessed on 29 December 2023)
USSW, VSSW	CCMP	https://rda.ucar.edu/datasets/d745001/dataaccess/ (accessed on 29 December 2023)
ST	CMEMS	https://data.marine.copernicus.eu/products (accessed on 29 April 2024)
slhf, ssr, str, sshf	ERA5	https://cds.climate.copernicus.eu/datasets/reanalysis-era5-single-levels%3Ftab%3Dform?tab=download (accessed on 24 June 2024)
U, V	CMEMS	https://data.marine.copernicus.eu/products (accessed on 16 August 2024)

. The data used in this study cover the time range from January 1993 to December 2022, with a time resolution of monthly averages and a spatial resolution of 0.25°.

2.2. Methods

The neural network architecture and hyperparameter configuration adopted in this study are derived from our team’s previous research [32]. In [32], we systematically explored model configurations and optimization strategies, and the selected setup has demonstrated an exceptional performance in capturing complex spatiotemporal patterns in oceanographic data. We employed a ConvLSTM model and ultimately selected a network architecture composed of two ConvLSTM2D layers and one Conv3D layer. To mitigate overfitting, we embedded Dropout regularization layers between the ConvLSTM layers, thereby reducing the model’s excessive reliance on specific neurons. Considering the size of the dataset and to facilitate the subsequent experimental model’s capture of the global perspective within the dataset, we decided not to batch process the dataset during the model training and testing. Additionally, the activation function chosen was the ReLU function, and the optimizer selected was the Adam optimizer. Parameter updates are normalized to ensure similar magnitudes in each update, which helps accelerate convergence, improve training efficiency, and stabilize the training process. The final parameter configuration of the ConvLSTM model in this study is presented in

Table 2. Model parameter settings.

Hyperparameters		Optimal Values
Network parameters	num_layers_ConvLSTM2D	2
	kernel_size_ConvLSTM2D	3 × 3
	activation_ConvLSTM2D	Relu
	num_layers_Conv3D	1
	kernel_size_Conv3D	3 × 3 × 3
	activation_Conv3D	Relu
Optimized parameters	time_step	3
	training_epoch	Customized
	learning_rate	0.001
	optimizer	Adam
Regularization parameter	Dropout	0.3

.

The input data of a single model are the sea surface temperature, sea surface height, and sea surface winds in two directions at time t, time t − 1, and time t − 2. The output is the ocean temperature of a single depth layer at time t. A group of models consists of 11 models for different depth layers, and each model is independent of the others.

To better simulate the actual changes in the water temperature, we treat the water body within the study area as a whole and incorporate the heat budget balance of the entire water column as a physical constraint into the model training process. To achieve this, we first integrate the seawater temperatures of 11 depth layers into a single model for inversion, that is, we splice 11 individual models into one, enabling the model to process data from all 11 depth layers simultaneously. This design allows for a comprehensive calculation of the heat budget losses across the entire water column, ensuring consistent heat flux calculations while maintaining computational efficiency. Regarding the input and output dimensions, the model receives input data where the information from all 11 depth layers is concatenated along the latitude dimension, forming a spatially comprehensive representation of the water column. The model outputs predictions in a corresponding concatenated format, which are then decomposed through an inverse transformation process to calculate the error for each water layer. This approach enables us to maintain the physical relationships between different depth layers, efficiently compute the heat budget for the entire water column, and obtain precise, layer-specific error metrics during the evaluation process. Subsequently, we modify the loss function of this model to reflect the optimization direction of the water body’s heat budget balance during training.

The loss function used by default in the model is the mean squared error between the training target value and the true value. Here, we have defined a new loss function, that is, calculating the difference between the total surface heat flux of the entire water body from one moment to another moment and the change in the total heat content of the water body. This difference is weighted and summed with the aforementioned mean squared error to serve as the loss function of the new model. During the training process, the optimization algorithm will take into account both the data constraints represented by the mean squared error and the newly introduced physical constraints, enabling the model to not only fit the data but also comply with physical laws.

In the specific operation, we, respectively, calculated the cumulative surface heat flux of the entire water body from one moment to another moment, including the sum of the heat fluxes at the sea surface and the four vertical side surfaces (∆T₁, here, we ignored the seabed heat flux), as well as the change in the heat content of the water body itself (∆T₂). Among them, the sea surface heat flux is obtained by integrating sshf, slhf, ssr, and str in the spatial and temporal dimensions; the heat flux at the lateral boundary of the water body is obtained by multiplying the normal flow velocity of the water body at the interface by the heat content; and the change in the heat content of the water body can be calculated from the difference in the seawater temperature between the previous and subsequent moments. Ideally, ∆T₁ and ∆T₂ should be equal. Therefore, the form of the loss function of the model is as follows:

LOSS = a × MSE + b × (∆T₁ − ∆T₂)

(1)

Considering the huge difference in the magnitude between the mean squared error (MSE) and the heat budget, it is necessary to set the weight coefficients a and b to make their importance roughly equivalent. Through experiments, we found that when $a=1.0 ,b={2\times 10}^{(-23)}$, the model inversion effect is the best (for a detailed comparison, see the discussion in Section 4.1). The specific calculation process is shown in Figure 2.

2.3. Experimental Setup

The data time range involved in the experiment of this study is from January 1993 to December 2022 (a total of 360 months). The dataset is randomly divided into two datasets with a ratio of 8:2. The latter is the test set (a total of 72 months), and 20% is randomly selected from the former as the validation set, with the remaining part being the training set.

A comparative experiment was conducted on two types of models: One group used the ConvLSTM model to train a separate inversion model for each depth layer, following the traditional process, which we call the “Traditional model”. The other group designed a multi–deep–layer joint model and trained it with a custom loss function, which we call the “Heat budget constraint model”. Figure 3 shows the technical route flowcharts of the two inversion models, including the steps from dataset construction to model prediction. The orange part represents the Traditional model, and the blue part represents the Heat budget constraint model. After the model training is completed, the performance of the two models is evaluated by calculating the RMSE and R² between the model prediction values and the true label values on the test set.

3. Results

3.1. Results of the Traditional Model

Figure 4 shows the vertical distribution of the evaluation indicators (RMSE, R²) of the inversion results of the Traditional model test set. It can be seen that the RMSE gradually increases within the range of 19–78 m, reaching a maximum value of 1.18 °C at 78 m. After that, as the depth increases, the RMSE gradually decreases. Below 1046 m, with the increase in depth, the change in the RMSE tends to be gentle. The R² shows a trend of first decreasing and then increasing, and the minimum value of 0.81 appears at the 78 m layer, which is in the same position as the maximum value of the RMSE.

Table 3. Seasonal variations in the inversion results of the test set of the Traditional model. The upper part is the RMSE (°C), and the lower part is R².

	19 m	47 m	78 m	97 m	147 m	200 m	301 m	508 m	1046 m	1516 m	2101 m	Avg	Avg Above 301 m
Spring	0.50	0.87	0.98	0.88	0.59	0.44	0.36	0.18	0.11	0.05	0.03	0.45	0.66
Summer	0.63	1.03	1.01	0.91	0.55	0.40	0.31	0.18	0.10	0.05	0.03	0.48	0.69
Autumn	0.59	1.00	1.23	0.99	0.63	0.48	0.40	0.20	0.09	0.04	0.02	0.52	0.76
Winter	0.75	0.97	1.20	1.04	0.68	0.52	0.39	0.19	0.08	0.04	0.02	0.54	0.79
Spring	0.96	0.89	0.84	0.84	0.92	0.96	0.97	0.99	0.99	0.99	0.99	0.94	0.91
Summer	0.93	0.87	0.84	0.86	0.93	0.97	0.97	0.99	0.99	0.99	0.99	0.94	0.91
Autumn	0.94	0.86	0.80	0.83	0.91	0.95	0.96	0.99	0.99	0.99	0.99	0.93	0.89
Winter	0.90	0.86	0.79	0.81	0.90	0.94	0.96	0.99	0.99	0.99	0.99	0.92	0.88

shows the seasonal variations in the evaluation indicators of the inversion results of the test set of the Traditional model. It can be seen that at the 78 m layer, the RMSE of the model first increases and then decreases with the seasonal change. The minimum value is 0.98 °C in spring, and the maximum value is 1.23 °C in autumn. At the 97 m layer, there is a similar trend of change. The RMSE increases with the seasonal change, with the minimum value in spring and the maximum value in winter. In terms of the average value of the 11 depth layers, the RMSE continuously increases from spring to winter. This seasonal variation in errors is likely closely related to the climatic and ocean dynamic processes in the northern South China Sea. In spring, solar radiation gradually strengthens, the water temperature begins to rise, and the climate remains relatively stable without strong seasonal disturbances. This stable environment allows the water temperature inversion model to more accurately capture the relationships between factors, such as solar radiation, monsoons, and the ocean circulation and water temperature, thereby achieving a higher inversion accuracy. In contrast, in winter, solar radiation weakens, the water temperature drops, and the region is strongly influenced by climatic factors such as typhoons and monsoons, leading to more complex ocean circulation and water mass changes. The superimposition of these factors makes water temperature changes more variable and difficult to predict, increasing the challenge for the water temperature inversion model and resulting in a relatively lower model accuracy.

From the above results, it can be seen that there is still room for further improvement in the method of the Traditional model for retrieving the seawater temperature from the sea surface remote sensing data. Especially in the mixed layer region, where its accuracy still needs to be enhanced.

3.2. Results of the Heat Budget Constraint Model

Figure 5 shows the comparison of the accuracies of the inversion results of the test sets of the Traditional model and the Heat budget constraint model from 19 m to 301 m. The vertical distribution patterns of the RMSE (left) and R² (right) of the results of the Heat budget constraint model are similar to those of the Traditional model: the maximum error of the results of the Heat budget constraint model still appears at 78 m, which is 0.96 °C; the corresponding minimum R² is 0.87. However, the inversion accuracy of the data at each layer has been improved to varying degrees (except for the 301 m layer). The most significant improvement is at the 78 m layer, where the RMSE is reduced by 20% and the R² is increased by 7%. It can be seen that the model significantly improves the inversion ability of the sea temperature in the mixed layer.

Table 4. Seasonal variations in the inversion results of the test set of the Heat budget constraint model. The upper part is the RMSE (°C), and the lower part is R².

	19 m	47 m	78 m	97 m	147 m	200 m	301 m	Avg Above 301 m
Spring	0.586	0.699	0.743	0.712	0.515	0.406	0.375	0.577
Summer	0.527	0.884	0.935	0.805	0.519	0.427	0.420	0.645
Autumn	0.580	0.91	1.005	0.841	0.548	0.435	0.373	0.670
Winter	0.529	0.744	0.986	0.842	0.530	0.423	0.384	0.634
Spring	0.935	0.920	0.900	0.893	0.931	0.954	0.945	0.925
Summer	0.945	0.890	0.871	0.878	0.922	0.948	0.928	0.912
Autumn	0.936	0.880	0.856	0.867	0.922	0.947	0.948	0.908
Winter	0.945	0.911	0.845	0.863	0.927	0.951	0.946	0.913

presents the performance of the Heat budget constraint model method on the test set in the form of the RMSE and R² of each layer from 19 m to 2101 m in each season. At the 78 m layer, the RMSE of the model still first increases and then decreases with the seasonal change. The minimum value is 0.74 °C in spring, and the maximum value is 1.01 °C in autumn. In the depth range of 19–301 m, the average root mean square errors of the Traditional model from spring to winter are 0.66 °C, 0.69 °C, 0.76 °C, and 0.79 °C, while the average root mean square errors of the four seasons of the Heat budget constraint model are 0.58 °C, 0.65 °C, 0.67 °C, and 0.63 °C. It can be seen that in the range of 19–301 m, the RMSE of the Heat budget constraint model in all seasons is lower than that of the Traditional model. The improvement is the most significant in winter (the error is reduced by 20.3%), followed by spring (12.1%).

To systematically evaluate the applicability of the Heat budget constrained model across different seasons, we compared the average seawater temperatures of the entire water column in the study area during winter and summer. The results show significant seasonal differences in model performance: in winter, the prediction error of this model was significantly lower than that of the Traditional model, particularly in the depth range from the mixed layer to the thermocline, where the simulated temperature structure showed a higher consistency with in situ measurements. This may be attributed to the relatively stable ocean stratification in winter, which facilitates a more accurate characterization of thermal budget physical constraints. In summer, the model errors were comparable to those of the Traditional model. It is hypothesized that complex dynamic processes in summer lead to increased horizontal temperature gradients, making it more challenging for the model to capture transient thermal budget balances. Figure 6 shows the temperature performance and error distributions of the measured sea temperatures, Traditional model predictions, and Heat budget constrained model predictions for the entire region in winter and summer.

To validate the Heat budget constrained model’s ability to characterize key temperature structures in the study area, especially the simulation accuracy of core elements such as the mixed layer and thermocline, we selected the single Point A (115° E, 15° N), which is close to the central position, to analyze the temperature structure in the upper-middle layer (above 300 m). This site is located at the geographic center of the study area, and its temperature structure is typical and representative and capable of effectively reflecting the regional average characteristics. The analysis shows that the thickness and morphology of the mixed layer and thermocline basically match the original data, and the model’s simulation results for the winter mixed layer depth and summer seasonal thermocline are largely consistent with the in situ measurements, with error patterns consistent with the regional average results. Figure 7 shows the temperature performance and error distributions of the measured sea temperatures at the single Point A, as well as the predictions from the Traditional model and the Heat budget constrained model for winter and summer.

To validate the universality and limitations of the Heat budget constrained model across different geographic locations, we further analyzed the temperature prediction errors at multiple single points, such as Point B (116° E, 16° N), Point C (113° E, 16° N), and Point D (116° E, 12° N). These sites cover different latitude zones and dynamic environments within the study area, enabling a comprehensive assessment of the model’s performance under complex oceanic conditions. The analysis reveals that the Heat budget constrained model exhibits a higher universality in winter, particularly performing well near the thermocline and mixed layer, which confirms the enhancing effect of thermal budget constraints on sea temperature inversion. This result indicates that in the relatively stable winter environment, the model can effectively suppress prediction biases through physical constraints. The summer scenario is generally consistent with the above analyses of the entire region and Point A, where errors show no significant difference from those of the Traditional model, and the overall optimization of the model is primarily reflected in winter.

Next, let us look at the changes brought about by the Heat budget constraint model method to the horizontal distribution of the inversion errors. Figure 8 shows the horizontal distribution of the RMSE of the two methods in the depth layer from 19 m to 301 m. Through comparison, we find that the errors in Figure 8b are significantly improved in the northern part, especially in the northwestern region where the errors are relatively high in Figure 8a. We find that the Traditional model method has concentrated areas with relatively large errors in the central and western parts at depths from 47 m to 147 m (d, g, j, and m). Although the Heat budget constraint model shows improvements and a reduction in errors in the same areas at the same depth layers (e, h, k, and n), the problem is still quite significant compared with other areas. This indicates that the root cause of the errors still exists, and the Heat budget constraint model method has not fundamentally subverted or reshaped the basic structure of the model but mainly carried out a quantitative optimization and adjustment of the results. In other words, although the improvement has brought about certain positive effects, the core problems of the model have not been completely solved. The lower part of Figure 8c,f,i,l,o shows the differences in the RMSE between the two methods. The images show that the RMSE values have decreased in almost all areas.

4. Discussion

4.1. A Comparison of the Performance of the Heat Budget Constraint Model Method with the Research Results of Others

In related studies of the same region, the integrated method proposed in Reference [33] combines the outputs of three base models, namely eXtreme Gradient Boosting (XGB), Random Forest (RF), and LightGBM (LGB), and uses a clustering algorithm for fusion optimization to establish an effective temperature retrieval model. According to the Root Mean Squared Error (RMSE) distribution within the depth range of 10–1150 m from 2017 to 2021, shown in Figure 6a of Reference [33], the maximum model error also occurs near the depth of 78 m, and the average RMSE across the entire water depth is 0.639 °C. Although the Traditional model constructed in this study adopts different data processing procedures and dataset division strategies, the final retrieval accuracy is at a similar level to the results of Reference [33] (with an average RMSE of 0.67 °C and a relative difference of less than 5%). This, to some extent, verifies the applicability of the algorithm framework we used in this scenario and also reveals the potential limits of the inversion capabilities of pure mathematical methods.

The new model developed by further introducing the heat budget constraint mechanism in this study demonstrates a superior performance. These improvements mainly stem from the introduction of the physical constraint mechanism, which enables the model to better capture the essential characteristics of the oceanic thermal processes.

4.2. The Influence of the Randomness of the Dataset Division on the Results

Since there are only 360 data samples (360 months) in the entire experiment, and the test set contains only 72 data samples, before drawing conclusions, we need to rule out the situation where “the improvement of the model is merely a coincidence brought about by the division of the data samples”. To this end, we repeated the comparative experiment between the Traditional model and the Heat budget constraint model 10 times, and each repeated experiment adopted a different random division of the data samples. The conclusions we presented in Section 3 are the average results of the above 10 sets of repeated experiments.

Figure 9 intuitively shows the distribution of the RMSE of these two models in the depth layer of 19–301 m in the ten repeated experiments. It can be seen that, compared with the Traditional model, the average RMSE value of the Heat budget constraint model in the depth range of 19–301 m is generally reduced, especially near the mixed layer, and the reduction range of the error is more obvious. The reduction in the error level brought about by the Heat budget constraint model is real and not a coincidence caused by the randomness of the dataset division.

At the same time, it can be seen that in the 19–97 m layer with a large error, the box of the box plot is relatively long, which reflects the huge differences between different data samples, so that the results obtained by training the model with different sample combinations are significantly different. This reflects that the temporal variation law of the temperature structure of the upper mixed layer is complex and difficult to be described by a simple model, which is also one of the reasons why it is difficult for various inversion models to reduce the error in this depth area. This is also the reason why it is necessary to carry out such research by repeating experiments to eliminate the influence of the randomness of the dataset division.

4.3. The Influence of the Selection of the Weighting Coefficient in the Loss Function on the Model Accuracy

In the loss function we designed, it includes the mean squared error between the model output value and the true value (this is the default loss function form of the ConvLSTM model) and the heat budget balance term calculated from the temperature at the time t output by the model and the true temperature value at time t − 1. There is a huge difference in magnitude between the two. Under the model grid framework of this study, they differ by approximately 10²² times. Therefore, selecting appropriate weighting coefficients for the two so that the mathematical constraints and physical constraints both play a role in the model training is the key to the success of the improved model. In order to find the appropriate weighting coefficients, we carried out a set of sensitivity tests to test the changes in the model inversion accuracy brought about by different weighting coefficients. Similarly, in order to avoid the influence of the randomness of the dataset division on the results, 10 repeated experiments were carried out for each group of weighting coefficients, and the average results were taken for comparison. Figure 10 shows the comparison of the RMSE of the Heat budget constraint models with different weighting coefficient combinations on the test set. It can be seen that the model effects in case3 and case4 are quite similar. We give priority to the model with a lower error in the mixed layer (47, 78, 97 m). By calculation, the average RMSE of the two cases in the mixed layer is 0.88 °C and 0.89 °C, respectively. Then, we adopt the third weighting coefficient combination, that is, we define the model loss function as Equation (1).

4.4. The Analysis of the Reasons for the Failure of the Heat Budget Constraint Model in the Deep Sea Ocean

The inversion results of the test set of the Heat budget constraint model are improved compared with those of the Traditional model in the water layers shallower than 300 m. The results at the 301 m layer are comparable to those of the Traditional model, but the errors within the depth range from 508 m to 1516 m far exceed those of the Traditional model. Specifically, at the 508 m, 1046 m, and 1516 m layers, the RMSE of the model reaches 0.76 °C, 0.95 °C, and 1.03 °C, respectively. Figure 11 shows the horizontal distribution of the RMSE of the inversion results of the test sets of the Traditional model and the Heat budget constraint model in these three water layers. It can be seen that compared with the upper ocean, the horizontal distribution of the RMSE is relatively more uniform within the study area, while the errors suddenly decrease near the edge of the area.

We analyzed the possible reasons for this abnormal increase in errors. First of all, when calculating the part of the water body heat budget in the loss function, due to the discretization of the data, a certain amount of systematic deviation will be introduced. Especially when we calculate the lateral boundary heat flux, we only use the flow velocity at the central depth of the grid point to represent the average flow velocity of the entire depth range of the grid. Since the CMEMS data have fewer vertical layers in the middle and lower layers of the ocean, the depth range represented by each grid point can reach several hundred meters, and the introduced error cannot be ignored. Coupled with the scarcity of observations in the deep ocean, the flow velocity data of the deep ocean in the reanalysis data themselves are questionable, which will further increase the calculation error of the lateral boundary heat flux. This part of the systematic deviation is introduced into the loss function and cannot be eliminated but will only be distributed among various output variables. Since the oceanic processes affecting the temperature of the deep seawater generally have a longer time scale than those in the upper ocean, that is, the temperature of the deep seawater is more stable, it has a lower tolerance for errors in the rate of temperature change over time, but a higher tolerance for temperature errors. Therefore, during the training process, the variables representing the temperature of the deep seawater are assigned a more systematic deviation. This leads to an increase in the inversion error of the deep seawater temperature compared with the traditional method. For the grid points near the edge, since their temperature values are themselves used in the calculation of the lateral boundary heat flux in the loss function, they have a lower tolerance for temperature errors, so they are assigned less systematic deviations during the training process, as shown in Figure 11.

Secondly, since the overall heat budget of the water body within the grid is introduced into the loss function, we have to combine the temperatures of all water layers into the same model for calculation. Therefore, when calculating the loss function represented by the mean squared error during the training process, there will inevitably be a tendency to sacrifice the accuracy of variables with smaller errors in order to reduce the errors of variables with larger errors. Specifically in this study, the joint training of sea temperatures of multiple water layers will reduce the errors of the water layers with originally large errors (the upper mixed layer) and increase the errors of the water layers with originally smaller errors (the deep layers). This tendency has, to a certain extent, improved the performance of the model in the upper mixed layer and exacerbated the amplification of the inversion errors in the deep layers.

Therefore, when using the Heat budget constraint model method, we only use the inversion results of the layers above 300 m and discard the deep-layer results that contain more systematic deviations and have their errors amplified by the joint training of multiple water layers. For these water layers, we still use the inversion results obtained by the traditional method.

5. Conclusions

In this study, focusing on the problem that the remote sensing inversion models of sea temperatures below the surface layer generally have relatively large errors in the upper mixed layer, a new inversion algorithm was designed. By adding the overall heat budget balance as a physical constraint into the loss function of the AI model, the inversion accuracy of the seawater temperature in the upper mixed layer was successfully improved.

We used the CMEMS reanalysis sea temperature product as the training data and designed the experimental process based on the ConvLSTM model. The sea temperatures of multiple depth layers were jointly trained, and the mean squared error between the output results and the true values was weighted and summed with the difference between the overall change in the heat content of the water body in the region and the heat flux at the water body boundary. This sum was used as the loss function of the model for model training. The results show that, compared with the traditional ConvLSTM method, the method with the addition of physical constraints has achieved remarkable results. In the depth range of 19–301 m, the overall RMSE has been reduced by 12.6%, especially at a depth of 78 m where the error is the largest, with a reduction of 20%. It has significantly improved the ability to use the AI method to invert the subsurface sea temperature through remote sensing data.