In this study, we analyzed a 1 km stretch of the Catu River, located in a watershed covering 208.5 km² and encompassing the municipalities of Alagoinhas, Catu, and Pojuca ( Figure 1 ). Measurements were conducted between the coordinates 12˚10'05.1"S, 38˚24'23.8"W and 12˚11'04.6"S, 38˚24'11.9"W, with altitudes ranging from 40 to 255 meters. The average slope of the basin is 30%.

Meteorological data were obtained from the National Institute of Meteorology (INMET), using measurements from the station located in Alagoinhas, BA, with latitude −12˚14'86" and longitude −38˚50'57". The station is situated at an altitude of 47.56 meters and provides records from January 1, 2009, to January 1, 2017, with a sampling resolution of 10 minutes.

The meteorological variables included in the study are air temperature (Air Temp), Wind Speed, relative humidity (Humidity), evaporation and precipitation. Table 1 summarizes the training data used for predictive model calibration.

To validate the model, a temperature measurement campaign of the Catu River was conducted between January and December 2023. Data were collected using DS18B20 sensors coupled to an Arduino microcontroller, positioned 5 cm from the water surface. Measurements followed the same sampling frequency as that adopted by the National Institute of Meteorology (INMET, Brazil), ensuring compatibility with the available meteorological data.

The collected data were compared with the predictions generated by the trained PINN-LSTM networks, allowing for an assessment of the model’s accuracy in the real context of the watershed. The data used and historical records are available at https://www.kaggle.com/datasets/corcova/caturiverbahia/data .

Although this study focuses on the middle section of the Catu River and uses data limited to the period 2016-2017, this choice was made based on the availability and reliability of observed river temperature records. Future work will expand the dataset to include other stretches and longer time series to improve model generalization.

As shown in Figure 2 , the PINN-LSTM forecasting model follows a structured pipeline:

First, meteorological and hydrological data, including air temperature, wind speed, humidity, evaporation, and precipitation, are preprocessed for consistency, normalization, and handling missing values.

Next, the Physics-Informed Neural Network (PINN) is trained to learn the governing thermal dynamics of the river, providing physically consistent estimates $y_{\text{pinn}\_\text{pred}}$. This output is incorporated into the final hybrid model training.

The Long-Short-Term Memory (LSTM) network captures temporal dependencies and nonlinear patterns, integrating both observational data and PINN constraints. The final model is evaluated using RMSE, NSE, and R², ensuring accuracy across different climatic and hydrological conditions.

Once validated, the model is deployed for inference, enabling real-time river temperature predictions. Figure 2 summarizes the complete PINN-LSTM model pipeline.

The Physics-Informed Neural Network (PINN) used in this study is built upon the governing advection-dispersion-reaction equation that models river water temperature dynamics. The equation integrates advective heat transport, longitudinal thermal dispersion, and source/sink terms representing atmospheric and hydrological exchanges:

$\frac{\partial T_w}{\partial t}=-U\frac{\partial T_w}{\partial x}+D_L\frac{{\partial }^2{T}_w}{\partial x^2}+R_h+R_i,$ (1)

where:

The PINN architecture approximates the solution $T_w\left(x,t\right)$ and minimizes the residual of the governing PDE, which is defined as:

${ℛ}_{\text{PDE}}=\frac{\partial T_w}{\partial t}+U\frac{\partial T_w}{\partial x}-D_L\frac{{\partial }^2{T}_w}{\partial x^2}-R_h-R_i.$ (2)

This residual is evaluated at a set of collocation points using automatic differentiation. The neural network is trained to minimize this residual along with prediction errors based on available data.

Boundary and Initial Conditions:

${\frac{\partial T_w}{\partial n}|}_{\partial \Omega }=q_0\left(x,t\right),$ (3)

where $\partial \Omega$ is the boundary domain and $q_0$ the known flux.

Loss Function:

The complete loss function for the hybrid PINN-LSTM model combines the physical constraint, prediction accuracy, and LSTM forecast loss:

${ℒ}_{\text{total}}={\lambda }_1{ℒ}_{\text{PDE}}+{\lambda }_2{ℒ}_{\text{LSTM}}+{\lambda }_3{ℒ}_{\text{data}},$ (4)

where ${\lambda }_i$ are weighting coefficients, ${ℒ}_{\text{PDE}}$ corresponds to the residual of the PDE, ${ℒ}_{\text{LSTM}}$ represents the sequential prediction error from the LSTM, and ${ℒ}_{\text{data}}$ is the supervised loss from observed data. This formulation ensures that the predictions are consistent with both the observed data and the physical dynamics of the river system.

The Physics-Informed Neural Network (PINN) employed in this study was designed to estimate the thermal dynamics of the Catu River by embedding the underlying advection-dispersion-reaction equation into its architecture. The network consists of:

Figure 3 shows the architecture of the network, including a symbolic representation of the differential constraints.

The training process uses automatic differentiation to compute the residuals of the governing PDE:

${ℛ}_{\text{PDE}}=\frac{\partial T_w}{\partial t}+U\frac{\partial T_w}{\partial x}-D_L\frac{{\partial }^2{T}_w}{\partial x^2}-R_h-R_i,$

which are incorporated into the loss function. The model is optimized via stochastic gradient descent (AdamW), minimizing a composite loss consisting of the PDE residual, temporal prediction error from the LSTM, and observational data mismatch. This architecture ensures both physical consistency and high predictive capability.

The proposed hybrid model integrates a Physics-Informed Neural Network (PINN) with a Long-Short-Term Memory (LSTM) network. The physically-informed prediction $y_{\text{pinn\_pred}}$, generated by the PINN module, is used as input to the LSTM. This integration allows the model to extract temporal features and nonlinear dependencies from time-series data that already respect physical constraints.

The LSTM architecture consists of three bidirectional layers-two with 64 neurons and one with 32 neurons-followed by a dense output module with layers of 64, 32, and 1 neuron, respectively. A temporal window of seven days was adopted (time_steps = 7), enabling weekly pattern analysis and sensitivity to abrupt thermal variations. Several topologies and hyperparameters were evaluated, and only the best configuration is presented in this work.

The internal structure of the LSTM cell is defined by the following set of equations:

$\begin{array}{l}{i}_t=\sigma \left(U_i{h}_{t-1}+W_i{x}_t+b_i\right)\\ f_t=\sigma \left(U_f{h}_{t-1}+W_f{x}_t+b_f\right)\\ {\tilde{c}}_t=\text{tanh}\left(U_c{h}_{t-1}+W_c{x}_t+b_c\right)\\ c_t=f_t\odot c_{t-1}+i_t\odot {\tilde{c}}_t\\ o_t=\sigma \left(U_o{h}_{t-1}+W_o{x}_t+b_o\right)\\ h_t=o_t\odot \text{tanh}\left(c_t\right)\end{array}$ (5)

where:

Figure 4 illustrates the overall LSTM pipeline, showing the PINN prediction being processed through sequential LSTM and dense layers. The internal cell computation matches the mathematical formulation above.

The Physics-Informed Neural Network (PINN) employed in this study approximates the solution to the Advection-Dispersion-Reaction Equation (1):

During training, the PINN learns a continuous temperature function ${\hat{T}}_w\left(x,t\right)$ satisfying:

The trained PINN generates a continuous function:

${\hat{T}}_w\left(x,t\right)={\text{NeuralNetwork}}_{\theta }\left(x,t\right)$ (6)

where $\theta$ represents network parameters. Practically, this continuous function is sampled at discrete points corresponding to observational and meteorological data intervals, resulting in a discrete time series:

${\hat{T}}_{\text{PINN}}\left(t_1\right),{\hat{T}}_{\text{PINN}}\left(t_2\right),\cdots ,{\hat{T}}_{\text{PINN}}\left(t_n\right)$ (7)

This discretized series serves as input for the LSTM model. Specifically, the LSTM takes sequences of the simulated temperature data (e.g. time window of 7 days):

$\left[{\hat{T}}_{\text{PINN}}\left(t_i\right),{\hat{T}}_{\text{PINN}}\left(t_{i+1}\right),\cdots ,{\hat{T}}_{\text{PINN}}\left(t_{i+6}\right)\right]\to T_w\left(t_{i+7}\right)$ (8)

This integration is conceptually justified because the PINN provides physically-informed, realistic temperature estimations based on governing physical laws, while the LSTM model refines predictions by capturing temporal patterns observed historically.

The boundary conditions implicitly applied within the PINN framework are Dirichlet conditions at $x=0$:

$T_w\left(0,t\right)=T_r\left(t\right)$ (9)

where $T_r\left(t\right)$ represents observed temperature data at the initial spatial point. This condition is enforced by minimizing the discrepancy between predicted and observed temperatures at $x=0$.

Thus, the PINN-LSTM hybrid approach effectively combines theoretical physical modeling (PINN) and empirical temporal pattern recognition (LSTM), leveraging strengths of both methods to enhance prediction accuracy for river temperature forecasting.

To evaluate the performance of the PINN-LSTM hybrid model in predicting the thermal behavior of the Catu River, three commonly used metrics were adopted: the coefficient of determination ( $R^2$), the Mean Absolute Error (MAE), and the Root Mean Squared Error (RMSE). These metrics quantify how well the model predictions ${\tilde{u}}_i$ approximate the observed values $u_i$ across all $n$ data points.

The formulas are defined as follows:

$R^2=\frac{{{\displaystyle \sum }}_{i=1}^n{\left({\tilde{u}}_i-\bar{u}\right)}^2}{{{\displaystyle \sum }}_{i=1}^n{\left(u_i-\bar{u}\right)}^2}$ (10)

$\text{MAE}=\frac{1}{n}\underset{i=1}{\overset{n}{{\displaystyle \sum }}}\left|u_i-{\tilde{u}}_i\right|$ (11)

$\text{RMSE}=\sqrt{\frac{1}{n}\underset{i=1}{\overset{n}{{\displaystyle \sum }}}{\left(u_i-{\tilde{u}}_i\right)}^2}$ (12)

where:

The R² metric assesses the proportion of variance in the observed data explained by the model, with values closer to 1 indicating better fit. MAE represents the average magnitude of the errors without considering their direction, while RMSE penalizes larger errors more heavily due to squaring, making it particularly sensitive to outliers.

Extreme temperature events in the Catu River are typically characterized by monthly peaks exceeding 21˚C, often associated with seasonal heating and hydrological stress conditions. These events are critical for ecological balance, water quality, and system-level modeling. Therefore, evaluating the model’s capability to predict these upper-bound temperature scenarios is essential.

The comparison between observed and predicted peak temperatures, as illustrated in Figure 7 , demonstrates that the PINN + LSTM model is capable of approximating the seasonal dynamics of monthly maxima. The predicted values follow the overall trend of real peaks, particularly during the warmer months when temperature variability is higher. However, systematic deviations are observed in several periods.

Notably, there is a consistent offset between observed and predicted peaks, with an average deviation of approximately 2˚C. This level of divergence is within the

acceptable range for natural measurement uncertainty in environmental monitoring, especially considering the complexity of river temperature dynamics influenced by meteorological, hydrological, and topographic factors.

During the summer and late spring months, where extreme peaks are more pronounced, the model reproduces the temperature elevation pattern but tends to slightly underestimate the peak amplitude. In contrast, during cooler months, the difference between prediction and observation is reduced, indicating better alignment in low-variability scenarios.

The results suggest that while the model captures the general structure of monthly extremes, high-frequency and short-term peak magnitudes remain partially attenuated. This behavior is consistent with temporal smoothing effects induced by the LSTM architecture and the regularization from physics-based constraints.

A sensitivity analysis was performed to investigate how variations in individual meteorological inputs affect the LSTM model’s output for river temperature. This analysis was conducted by computing the partial derivatives of the model’s predictions with respect to each input variable using automatic differentiation.

The results, shown in Figure 8 , reveal that different variables exhibit distinct levels of influence on the model output. Variables related to atmospheric moisture, such as dew point and humidity, display higher sensitivity, suggesting that the model registers stronger gradients in temperature predictions when these inputs are perturbed. This may be associated with the model’s internal representation of near-surface heat exchange mechanisms.

Evaporation also demonstrates a measurable impact, indicating a potential role in modulating temperature through latent heat processes. Meanwhile, rainfall and wind speed contribute lower gradient responses, reflecting reduced direct sensitivity

under the current temporal and spatial resolution.

These sensitivity patterns vary across variables and likely depend on the seasonal dynamics, temporal scale of the input data, and the autoregressive [34] nature of the LSTM architecture, which may amplify or dampen certain features based on their historical behavior in the input sequences.

To complement the gradient-based sensitivity analysis, an ablation study was conducted to examine the dependency of the model on individual meteorological variables. In this approach, each input feature was systematically removed by zeroing its values in the test set while keeping the remaining variables unchanged. The resulting change in Mean Squared Error (MSE) was computed to quantify the relative importance of each variable to the predictive performance of the model.

The outcomes of this procedure are illustrated in Figure 9 . Variables associated with atmospheric moisture-particularly humidity and dew point-produced the largest increases in error when removed, indicating that these features provide essential information for capturing the thermal behavior of the river system. Wind speed and evaporation also led to noticeable error increases, suggesting that they contribute relevant dynamic cues to the model’s internal representation of energy exchange processes.

Precipitation, on the other hand, showed minimal influence on model performance. Its removal did not result in a meaningful increase in prediction error, and even led to a slight improvement in some cases. This behavior may be explained by the regional climatic characteristics: the study area experiences relatively low rainfall throughout the year, with precipitation concentrated in brief seasonal periods. As a result, the temporal resolution and dynamics of precipitation events may not align strongly with short-term river temperature fluctuations, reducing its predictive

value in this specific configuration.

The ablation study reveals that meteorological variables such as humidity and dew point play significant roles in river temperature prediction. However, other important drivers, including river discharge, riparian vegetation cover, and riverbed morphology, were not incorporated in this study. Future work should aim to integrate these factors to enhance the model’s predictive capabilities.