1. Introduction
The Nakdong river estuary, a representative brackish water zone in which ecological diversity and economic values coexist, provides domestic, agricultural, and industrial water to Busan and Yeongnam regions. The construction of the estuary bank in 1987 prevented salinity intrusion and flood damage by blocking the circulation of seawater and freshwater, but it also caused new ecological problems, such as a decline in migratory bird habitats and fish species diversity (Kim et al., 2018). In response, the Korean government implemented a pilot opening of the estuary bank in 2017 and later expanded it to full opening policies, such as keeping the bank continuously open during spring tides.
This has changed salinity diffusion and inflow patterns in the Nakdong river estuary, which requires a quantitative analysis based on sophisticated predictions. In particular, high-salinity seawater may infiltrate upstream areas when the estuary bank is opened during high tide, which can damage nearby intake stations and farmland, severely affecting the local economy. Such damage may become worse in complex scenarios, such as spring tides, torrential rainfall, and floodgate opening. Therefore, a systematic prediction model that can accurately predict salinity levels in response to changing estuarine environments is required.
Hence, research has been conducted on various physics-based models to analyze salinity distribution and intrusion phenomena in estuaries. As representative cases, numerical models, such as the three-dimension hydrodynamic and sediment transport model (ECOMSED) and the environmental fluid dynamics code (EFDC), simulate salinity distribution changes in estuaries using physical and hydrological variables, such as the tidal level, discharge, water level, and salinity. They exhibit strength in implementing interactions between seawater inflow and freshwater discharge (Jeong et al., 2010; Beak et al., 2021). These three-dimensional numerical models have been used to reproduce the salinity characteristics of estuarine environments that are affected by such factors as tides, wind, and water level differences. They have exhibited high interpretive capacity in domestic and overseas estuary cases.
Meanwhile, salinity intrusion curve models in simpler forms have been presented, including one-dimensional approximation models. They calculate salinity levels that infiltrate the upstream areas of an estuary based on the salinity, freshwater discharge, and distance from the estuary. They are used as decision-making tools in the policy planning stage owing to the simple calculation structure and high applicability. In particular, such models are favorable for estimating the salinity change tendency in a simple structure by reflecting the distance-salinity relationship.
As described, physics-based models have played a key role in quantitatively analyzing and interpreting salinity intrusion that occurs under hydraulic and environmental conditions, such as estuary bank opening, discharge control, and tidal levels. However, most existing physical models are highly sensitive to initial conditions and parameter settings for various external factors, and their long computation time and complexity have been consistently cited as limitations for real-time decision-making support.
To overcome such limitations, researchers have conducted data-oriented predictive studies based on deep or machine learning. Woo et al. (2022) improved salinity prediction performance using long short-term memory (LSTM), a time-series prediction algorithm. Lee et al. (2022) achieved high prediction speed and high-performance prediction results based on such models as XGBoost and light gradient boosting machine (LGBM). However, these studies focused on simple predictions that primarily utilize historical data and still have limitations in response to policy changes (e.g., estuary bank opening) or rapid salinity extrapolation interval scenarios.
This paper considers the benefits and limitations of the physics-based approach and data-based approach in a balanced manner and proposes a physics-informed machine learning (PIML)-based predictive approach that combines both approaches. The proposed methodology designs the loss function of a machine learning model that integrates the salinity calculated from the physics-based prediction model with the predicted values of the machine learning model and utilizes a method of adjusting the weights of the two models according to the characteristics of the training section. This ensures high predictive performance in both sections where the physics-based model exhibits strength and sections in which the data-based model showed higher performance. The proposed PIML approach is a technique with convergent predictive capabilities that can simultaneously reflect complex hydrological and environmental factors in areas with frequent environmental changes by external factors, such as the Nakdong river estuary. The results of this study are expected to be used as basic data for predicting salinity intrusion and evaluating estuary bank operations in the future.
2. Physical Model for Nakdong River Salinity Prediction
2.1. Data Used
In this study, the data of the Nakdong river estuary from January 2022 to June 30, 2022 were used to verify the proposed model. Table 1 shows the collected data.
Further preprocessing was also performed to ensure the consistency and reliability of the data. First, the data of all variables were standardized on an hourly basis. Missing data for each time period were interpolated by applying the linear interpolation method.
2.2. Salinity Intrusion Curve Model
The salinity according to the distance in a continuously opened estuary is predicted using Eq. (1) (Savenije, 1986).
where S is the salinity at the estuary (practical salinity units (psu)), and Sf is the salinity of freshwater. Generally, the salinity of freshwater can be assumed to be 0 psu. S0 is the salinity at the estuary inlet (e.g., 30 psu), which is generally the salinity of seawater. K is Van der Burgh’s coefficient, which is a constant that represents the salinity distribution characteristics in the estuary. α is a specific constant used in the model, which is related to longitudinal and transverse diffusion. Qf is the discharge of the river (negative value), which represents the amount of freshwater introduced into the estuary. D0 is the longitudinal and transverse diffusion coefficient at the estuary inlet, which represents the degree of salinity diffusion at the estuary inlet. A0 is the cross-sectional area of the estuary inlet, which represents its physical size. x is the distance from the estuary, which represents the distance from the estuary inlet to a specific point.
This equation considers various physical elements in predicting salinity in the estuary. It is possible to predict changes in salinity along the estuary through such variables as the discharge of the river (Qf ), the cross-sectional area of the estuary inlet (A0 ), and salinity distribution characteristics in the estuary (K ). In particular, this model well reflects nonlinear changes in salinity according to the distance by expressing the change in salinity according to the distance x using the exponential function
exp ( x a )
2.3 Optimization of Physical Coefficients Using a Genetic Algorithm
Genetic algorithms are optimization techniques to find optimal solutions inspired by the natural selection process. Because they do not require a function to be differentiable, they have been widely used in complex optimization problems (Katoch et al., 2021). Genetic algorithms consider a possible solution as a chromosome, and a set of such chromosomes is referred to as a population. Each chromosome consists of several genes. These are visually expressed in Fig. 1.
Genetic algorithms explore entities with better solutions using operators, such as selection, crossover, and mutation. The crossover operation is the process of generating a new solution based on two chromosomes. The selection operation is the step of selecting two chromosomes to be used in the crossover operation. They are selected probabilistically by quantifying the fitness of the solutions through fitness values. The mutation operation enhances the diversity of solutions by randomly modifying the values of entities to prevent genetic algorithms from falling into local optima. Fig. 2 shows the procedure of a genetic algorithm. The initial population is randomly generated, and the entities evolve through selection, crossover, and mutation operations. This process is repeated until the termination conditions are satisfied (Katoch et al., 2021).
Fig. 3 shows the visualization results that optimized K and α, which are the physical coefficients of the salinity intrusion curve, using the salinity given for upstream points (7.5 and 9 km) and the Nakdong river bridge (5.5 km upstream). Fig. 4 compares the salinity prediction results based on the optimized physical model with the observed salinity from April to July in 2022. The physical coefficients were optimized over time using the salinity data during the period. From the intrusion curve function derived based on this, the salinity at the 5.5km upstream point (Nakdong river bridge) was calculated.
In this study, the salinity values observed from upstream points at 7.5 and 10 km were utilized in addition to the observation sites in Table 1 to efficiently reflect the physical characteristics of the estuary. The coefficients of the physical salinity intrusion curve model (e.g., Van der Burgh’s coefficient K and α the diffusion coefficient ) were repeatedly optimized for the salinity observed from the 7.5 and 10 km points to coincide with the values predicted by the physical model at each time step. Coefficient optimization was performed though a genetic algorithm, and the following procedure was applied at each time step. The salinity observed at the 7.5 and 10 km points was compared with the values predicted by the physical model, and physical coefficients were explored to minimize the sum of squares of prediction errors at the two locations. The initial, maximum, and minimum values of each coefficient were empirically determined and used, and the optimal combination of coefficients was derived by repeatedly applying the selection, crossover, and mutation operations of the genetic algorithm.
When the prediction results of the optimized physical model were examined, the predicted values did not differ significantly from the observed values and responded sensitively, showing high predictive performance in sections with sharp salinity increases. However, sections with no significant salinity change, i.e., sections where salinity remains relatively stable, an overestimation tendency was observed compared with the observed values. This shows that the physical model alone cannot ensure consistent reliability in environments with large salinity changes and diverse change patterns. Therefore, an approach is required to respond to various scenarios predicted in estuaries with complex environmental changes.
3. Nakdong River Salinity Prediction Data Model
3.1. Model Used
In this study, LSTM, a time-series deep learning model, was used to predict the salinity at the Nakdong river bridge based on a previous study by Woo et al. (2022). LSTM, a type of recurrent neural network (RNN), is a technique with proven effectiveness in time-series analysis in which the information from past time points significantly affects prediction (Lipton et al., 2015). Because the salinity of the Nakdong river estuary and bridge changes continuously and nonlinearly owing to various factors (e.g., discharge and weather conditions), the benefits of LSTM that can reflect time-series and change patterns together are highlighted. This study implemented a prediction structure combined with the physics-based model using the data-based prediction of the PIML framework and data-based analysis that utilizes the existing LSTM model. Fig. 5 shows the internal structure of the LSTM network. LSTM has a special cell structure that consists of input, forget, and output gates. This structure receives the input of each time point (xt), previous output (ht – 1 ), and previous cell state ( Ct – 1 ), and transmits the cell state ( Ct ) and output ()ht while effectively adjusting them.
In particular, the forget gate learns parts to be maintained and discarded from the past information, and effectively removes unnecessary information at distant locations in time. The input and output gates precisely adjust the storage of new information and its transfer to the next step to help LSTM in learning both long- and short-term dependencies inside complex time-series data. This causes less sensitivity to the vanishing gradient problem compared with typical RNN and results in strength in time-series prediction.
The key parameters and experimental conditions of the LSTM model used in this study are presented in Table 2. All prediction experiments were performed based on the values, which are the results used in a previous study by Yang et al. (2025).
3.2. Sliding Window Technique
In this study, the sliding window technique was applied to effectively reflect the seasonal variability and anomalous patterns of the Nakdong river estuary data in time-series data and the real-time change characteristics of the data. This method sequentially divides the entire time series into training and validation sections in a certain period unit to repeatedly perform training and evaluation while moving the window at constant intervals. This procedure can minimize the overfitting problem for training data that may occur during the use of the fixed training period, and it can reflect seasonality and the unique time-series characteristics of the estuary to the model more rapidly (Norwawi, 2021).
In the experiment described in section 3, the data of 1,440 h (60 days) in the training section were set as training data and 24 h (one day) that followed as test data. Training and validation were then sequentially performed for each window. In the PIML experiment described section 4, different training sections were used depending on the experimental purpose. Related details will be described in sections for the results of each experiment.
3.3. Comparison of Results of the Physical and Data Analysis Model
As shown in Fig. 6, the machine learning model shows relatively high predictive performance in interpolation intervals where the trend of salinity does not vary significantly, but the predicted salinity cannot respond quickly to salinity surges in extrapolation intervals.
Data-based machine learning models, such as LSTM, can effectively track salinity changes in interpolation intervals in which sufficient historical data and clear trends exist. However, they cannot rapidly respond to new environmental changes without historical data in extrapolation intervals, such as the salinity surge after April 15, and exhibit an increase in prediction error. Furthermore, because the data-based model does not contain physical causality, the reliability and consistency of prediction are further reduced in scenarios where the observed data pattern exceeds the scope of learning or data quality is degraded. Therefore, a single machine learning model can be effective in prediction under normal conditions, but it has limitations in predicting salinity in the estuary with frequent environmental changes, such as rapid salt inflow and policy changes.
As described, the data-based model and physics-based approach have their own strengths, but ensuring consistent reliability and accuracy under all environmental conditions using only one model is difficult. Therefore, this study aimed to propose an integrated prediction framework that overcomes the limitations of a single model and maximizes the benefits of the two approaches in the estuary with frequent environmental changes.
4. LSTM-Based PIML
4.1 Physics-informed Loss Function
In this study, predictions were performed using a method of deriving new predicted values by combining the predicted values obtained through the physical model and the loss function of the machine learning model. Physical constraints were applied to the loss function to address the extrapolation problem of the machine learning model and improve generalization performance.
The basic equation of the physics-informed loss function is expressed as follows.
where
λML is the loss weight of the machine learning model, and Ytrue is the actual value or observed value. Ŷ is the value predicted by the model. λphy is the loss weight of the physical model. If it increases, the physical model is more reflected in the overall loss function. In particular, a high value of the physics-informed loss function within the loss function indicates that the predicted value significantly deviates from the physics-based model, i.e., the exponential salinity relationship. In contrast, a reduction in the value of the physics-informed loss function causes the model to follow the pattern of the traditional physical intrusion curve. This can ensure the physical consistency of the model. Yphysics is the value calculated by the physical model, and R represents an additional normal term or penalty term. In this process, λML is set as 1 – λphy. Based on this, the loss weights of the machine learning and physical models are adjusted. Here, R is the parameter of the model, i.e., an L2 normalization term for the weight vector w. It inhibits overfitting by preventing the excessive growth of the weight value in the learning process. This can improve the generalization performance of the model.
An efficient PIML model must be constructed by properly adjusting λ in the structure of Eq. (2). Hence, two methods for setting λ are presented in this paper.
The first method involves applying a fixed value (e.g., 0.3, 0.5, and 0.7) for the value of λphy. The use of a fixed value can maintain a balance between the physical and machine learning models in the loss function. For example, the loss of the physical and machine learning models can be considered equally if the λphy value is set to 0.5. This method is relatively easy to implement and enables determining an appropriate weight by attempting various λphy values in the initial experimental stage. However, the use of a fixed value may not be optimal depending on the characteristics of the data and the machine learning model used, and adjusting the value dynamically according to data pattern changes is difficult.
The second method involves the use of the dynamic weight. The dynamic weight enables each item of the loss function to be dynamically adjusted according to individual data. λx,phy of the dynamic weight can be defined by Eq. (4).
The dynamic weight assigns individual observations for each training data point rather than considering the weight of the entire training data. Here, x represents individual observations. The two loss functions used to obtain λphy in Eq. (4) are defined as follows. Lossx,ML (Ytrue, Ŷ) is the machine learning model loss at the individual observation x. It represents the difference between the actual and predicted values. Lossx,phy (Ytrue, Ŷ) is the physical model loss at the individual observation x, which represents the difference between the value calculated by the physical model and predicted value. The loss functions used to calculate the two loss functions can be modified according to the model and data characteristics.
This method automatically adjusts the weight according to the loss ratio between the physical and machine learning models in the training section. For example, when the loss of the machine learning model is relatively large for a specific observation, the λphy value increases to increase the influence of the physical model. In contrast, when the loss of the physical model is large, the λphy value increases to increase the influence of the machine learning model. The main benefit of this method is that flexible predictions are possible because the weight can be dynamically adjusted according to the characteristics of the data and the performance of the model.
4.2 Experimental Procedure of the Physics-informed Machine Learning Model
The sequence of constructing a PIML model using the loss function proposed in section 2.1 is as follows. First, the window size (N ) is set. Next, the physical coefficients of the given physical model are optimized using a genetic algorithm, and the predicted values of the physical model are obtained. They are used to train the machine learning model. When prediction is complete, the window is moved through i+1. This process is repeated to perform modeling and obtain predicted values. Here, T is the step size, which represents the size of the window movement, and is the number of repetitions. For each repetition, the window moves and the model sequentially repeats training and prediction. If the size of the training data is smaller than N +T ×i, the algorithm is terminated because the number of data that can be extracted from the training data is smaller than. This process applied the moving window method considering changes in data distribution over time. Predictions were performed by moving the range of training data sequentially as shown in Fig. 7.
4.3 Nakdong River Salinity Prediction Using PIML
In this study, overall intervals were divided into extrapolation intervals with sharp salinity surges and interpolation intervals with relative stable salinity changes, and predictive performance in each interval was compared. In addition, the prediction results of the LSTM and physical models were compared and Eq. (1) was used as the physics-informed loss function to evaluate prediction accuracy according to the weight ratio between the physical and machine learning models. For the loss function weight λphy that adjusts the influence of the physical model, fixed values of 0.3, 0.5, and 0.7 and the adaptive method that dynamically adjusts the weight were applied. Based on this, an attempt was made to analyze which models exhibit high performance in each interval.
4.4 Experimental Results
In this section, predictive performances in extrapolation intervals (estuary bank opening overlaps with salinity surges) and interpolation intervals (stable salinity maintained) are compared through Tables 3–5 and Fig. 7. Table 3 shows the prediction results for extrapolation intervals with salinity surges. In these intervals, PIML with the highest proportion of the physical model (λphy = 0.7) exhibited the lowest prediction error and high performance compared to LSTM.
In contrast, as shown in Table 4 for interpolation intervals (relative stable salinity changes), the PIML that used adaptive λphy exhibited the lowest error, and the LSTM and PIML that used a low proportion of the physical model exhibited relatively high performance. In these intervals, the prediction error tended to gradually increase as the proportion of the physical model increased. This appeared to be owing to the nature of the physical model that overpredicts salinity compared to the actual salinity.
Table 5 shows performance in overall intervals. The adaptive PIML, which dynamically adjusts the combination ratio of the two models, recorded the lowest average prediction error, thereby verifying that it can flexibly respond to prediction requirements for each interval and each environment. Fig. 8 compares the observed salinity with adaptive PIML prediction results in overall intervals.
In combining the two prediction methods according to the characteristics of the data and the environment, the experiment results showed that higher predictive performance was obtained as the physical model was more reflected for intervals with sharp salinity surges. We also confirmed that the adaptive PIML framework, which dynamically adjusts the weight for each training, exhibits stable performance in salinity prediction. These showed that the reliability and predictive power of the model can be improved by reflecting physical laws through an increase in the proportion of the physical model for external environment changes and applying the data model for stable scenarios.
5. Conclusions
In this study, a physics-informed machine learning (PIML) framework that combines the physics-based and data-based models was developed to enhance salinity prediction reliability by reflecting various physical and environmental elements in the Nakdong river estuary. The framework was verified through experiments. The strengths and limitations of the data analysis and physical models were analyzed for each environment. The need for model combination measures was highlighted to overcome the limitations.
In the research, the prediction results of long short-term memory (LSTM), a time-series deep learning model, and the physics-based model (salinity intrusion curve) were compared with the actual salinity. For intervals with relatively stable changes in salinity, LSTM exhibited high performance by effectively learning the salinity patterns. In contrast, for intervals with sharp salinity surges, the physics-based prediction exhibited higher performance as the model that reflects physical laws. However, a single model showed limitations in predicting long-term environmental changes in a consistent and reliable manner for both models. The two models should be combined through PIML to address this problem.
In extrapolation intervals with sharp salinity surges, prediction accuracy significantly improved as the weight of the physics-based model increased (Table 3). This is because the physical model effectively respond to rapid environmental changes by reflecting hydraulic and environmental laws. In contrast, the data-based LSTM model exhibited higher performance for interpolation intervals with relatively stable salinity changes. As the proportion of the physical model increased, the error tended to increase owing to overprediction (Table 4). The strengths and limitations of each model were clearly observed in extrapolation and interpolation intervals.
As for performance in overall intervals, the adaptive PIML model, which dynamically adjusts the weights of the two models, recorded the lowest average prediction error (Table 5). This dynamic weight-based combination technique has high practical value and academic significance in that it reflects both rapid environmental changes and stable fluctuations in complex hydraulic environments, such as the Nakdong river estuary, and effectively maximizes predictive performance for each interval. This verified that reliable results can be derived by fully reflecting physical laws through an increase in the influence of the physical model for intervals with rapid environmental changes and performing predictions with focus on the data-based model for stable environments.
These results highlight the need for an approach that overcomes the limitations of a single model and maximizes the synergy between the models in hydraulic environments that involve both rapid environmental changes and stable intervals, such as the Nakdong river estuary. They also present directions for salinity prediction in estuaries by distinguishing the importance and performance advantages of each model for each interval.
This study had some limitations. First, the proposed experiments could not reflect all policy changes, such as actual estuary bank opening and closing. The current model cannot utilize all the complex patterns of floodgate operations (e.g., estuary bank opening width and rate) as input variables, resulting in inherent limitations in prediction stability for policy changes or exceptional scenarios. Second, because the physics-based model and machine learning combination structure was focused on “weight-based loss function integration,” the two models could not be structurally integrated. In other words, a hybrid framework that features dynamic feedback of prediction results rather than a simple combination is the future challenge. Third, overfitting for specific datasets or short-term scenarios may have occurred in the process of optimizing the coefficients of the physical model using a genetic algorithm, and its empirical evaluation is insufficient.
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