Multi-Objective Hyperparameter Optimization Improves the Interpretability of LSTM Rainfall–Runoff Models

Abstract

Rainfall–runoff modeling is a key challenge in hydrological research. Despite the extensive application of long short-term memory (LSTM) networks in rainfall–runoff modeling, our understanding of the influence of different hyperparameter configurations on various hydrograph components, as well as the linkages between hydrological concepts and LSTM architectures, remains elusive. Here, we integrated Multi-Objective Particle Swarm Optimization (MOPSO) with LSTM hyperparameter optimization by targeting the root mean square error of the overall hydrograph (RMSE_all), high-flow (RMSE_high) and low-flow (RMSE_low) dynamics, and water volume deviation (Dv). The MOPSO-LSTM framework was applied to the upstream catchments of the Miyun Reservoir in Beijing, China. At a lead time of 1d, the optimal solution achieved an NSE of 0.920 in the Chaohe River Basin, with a minimum RMSE_all of 0.848 m³/s, RMSE_high of 2.081 m³/s, RMSE_low of 0.382 m³/s, and Dv of 0.002%. However, as the lead time increased to 3 and 7 days, the maximum NSE declined to 0.747 and 0.560, respectively, with process-related metrics deteriorating more substantially than water balance-related metrics. The Baihe River Basin performed better, with maximum NSE and KGE values of 0.949 and 0.970 at a lead time of 1d. Clear trade-offs among different evaluation objectives were further identified, particularly the competitive relationship between RMSE_high and RMSE_low, as well as the coupling between RMSE_low and Dv. SHAP (Shapley additive explanation) and partial dependence plots (PDPs) were used to quantify and interpret the effects of hyperparameters on model performance, and the results showed that learning rate, number of units, and lookback window served as the most influential hyperparameters. Moreover, optimization preferences resulted in distinct hyperparameter configurations, where Dv-oriented solutions favored smaller learning rates, longer lookback windows, and larger batch sizes than RMSE-oriented solutions. Compared with the Chaohe River Basin, the larger Baihe River Basin favored LSTM configurations with longer lookback windows, more hidden units, higher learning rates, and lower dropout rates, which was associated with the hydrological memory of the catchment. Overall, this study provides a novel multi-objective LSTM optimization framework, improving the understanding of LSTM hyperparameters and offering practical guidance for hydrological prediction and water resource management.

1. Introduction

Rainfall–runoff modeling and forecasting provide critical support for water resource management and flood regulation [1]. However, in recent years, the intensification of climate change, land use changes, and human activities have contributed to pronounced non-stationarity, stochasticity, and spatiotemporal variability in rainfall–runoff processes, thereby increasing the difficulty of forecasting [2]. Physically based models, derived from rigorous hydrological and mathematical laws, exhibit strong physical consistency and reliability. However, they depend on considerable calibration efforts and expert knowledge, are subject to uncertainties in parameter estimation, and are constrained by rigid model structures that limit their ability to capture complex hydrological behaviors [3]. In contrast, machine learning models learn hydrological relationships between meteorological inputs and streamflow directly from data without explicit representation of hydrological processes, featuring convenient implementation and flexible architectures, and a strong capability to capture nonlinear processes in complex systems. Accordingly, they have been extensively adopted to rainfall–runoff modeling with promising performance and have become the hotspot of hydrology research [4,5].

The long short-term memory (LSTM) network with a multi-gate architecture, designed specifically for time-series modeling, addresses the vanishing gradient problem encountered by conventional RNNs when learning long-term dependencies, and its use has proven to be effective in hydrological forecasting [6]. For example, Frederik et al. [7] applied LSTM based on the CAMELS dataset to simulate streamflow processes across 241 catchments with diverse hydrological conditions in the United States, showing that the accuracy of simulated data was comparable to that of well-established process-based models. Martel et al. [8] employed LSTM to simulate streamflow under climate change scenarios across more than 1000 catchments in the United States, showing that it significantly outperformed conceptual hydrological models in terms of both the KGE and NSE metrics. However, machine learning models involve a large number of hyperparameters, and thus model performance critically depends on appropriate hyperparameter configuration. Neurons constitute the hidden layers and progressively transform and transmit information through the network during training [9]. The number of neurons depends on the complexity of the problem, as an excessive or insufficient number of hidden layers or neurons may lead to overfitting or underfitting [10]. Learning rate is a key hyperparameter of the Adaptive Moment Estimation (Adam) optimizer, which is widely used due to its simplicity and rapid convergence. However, setting it too small decelerates training, while too large a value risks divergence or suboptimal convergence [1]. In practice, learning rate tuning is more efficient on a logarithmic scale, which facilitates exponential decay and enables stable, systematic exploration of the parameter space. Batch size is dependent on the memory capacity of computational units (CPU/GPU) and controls the training speed. Larger batches accelerate training but risk degrading model quality when too large, while overly small batches may impede loss function convergence [11]. The dropout rate specifies the proportion of neurons randomly deactivated during training. It helps prevent overfitting and reduce generalization error by balancing training error, though its effectiveness depends on dataset characteristics and interactions with other hyperparameters [11]. Still, interpreting machine learning model results and understanding how hyperparameters influence model performance remain central challenges in hydrology [12].

In current practice, hyperparameter tuning for hydrological machine learning models relies largely on manual trial-and-error, a strategy that is not only experience-intensive but also prone to suboptimal outcomes. Some studies have gone further by adopting random search methods for LSTM hyperparameter optimization [13]. Nevertheless, they remain constrained by the use of a single performance metric, while multi-objective optimization approaches are still limited, particularly those considering different components of hydrological processes, which inevitably compromises the model’s capacity to balance streamflow dynamics with water balance consistency [14]. NSE is one of the most popular evaluation metrics, and its importance is often taken for granted [15]. However, it is particularly sensitive to high-flow extremes while being relatively insensitive to systematic overestimation or underestimation, which often leads to a model’s poor performance in low flows and substantial water balance bias [16]. For instance, Xu et al. [14] used a particle swarm optimization algorithm with NSE as the sole objective function to tune several hyperparameters, and their results demonstrated that though NSE remained around 0.7 for longer lead times, the percent bias approached 50%. Amirmasoud et al. [1] applied random search and tree-structured Parzen estimators to tune hyperparameters of DNN, CNN, and LSTM models with RMSE over the full simulation period as the objective and found that different hyperparameter configuration strategies yielded varying performances under different rainfall intensities, hindering the simultaneous representation of both high- and low-flow conditions. Therefore, multi-objective optimization of the LSTM model should be further explored to obtain optimal hyperparameters across different aspects of model performance.

As for the interpretation of hydrological modeling results, a few studies have employed SHAP and PDP methods to quantify the contributions of parameters to model performance [12,17,18]. However, they have mainly focused on the relationship between hyperparameters and overall predictive performance, while the influences of hyperparameter configurations on different hydrograph components, including overall streamflow dynamics, high-flow and low-flow processes, and water balance preservation, across multiple lead times remain largely unexplored. Multi-Objective Particle Swarm Optimization (MOPSO) is a representative heuristic multi-objective optimization algorithm [19,20], and it has demonstrated effectiveness in hydrological parameter optimization [21]. To address this gap, we speculate that further interpretation of the MOPSO-LSTM optimization results by using SHAP and PDPs may help establish linkages between hyperparameter configurations and hydrological behavior, particularly the responses of different components of the hydrograph. Theoretically, this framework can underpin practical water resource management, as distinct objectives call for tailored model architectures and hyperparameter configurations. For instance, reservoir operation and water supply management tend to focus on medium- to long-term inflows and drought conditions associated with low flows, whereas flood control is primarily concerned with short-term high-flow events and flood risks.

Our study focusses on the upstream catchments of Miyun Reservoir, which is an important surface drinking water source for Beijing, China. The upstream runoff directly affects flood safety, water supply security, and ecological regulation in the capital region. At present, process-based hydrological models are relatively well established in those basins, with most studies focusing on models such as SWAT [22] and HSPF [23]. However, in recent years, with changing precipitation regimes and intensified human activities in this region, the non-stationary characteristics of upstream inflows have become increasingly pronounced [22]. Traditional process-based models show limited capability in responding to complex rainfall conditions, whereas the application of machine learning models in this region remains relatively scarce. We therefore established a MOPSO-LSTM framework in this region to (1) characterize the trade-offs among different optimization objectives in LSTM rainfall–runoff modeling; (2) identify variations in optimal hyperparameter configurations under different lead times and optimization objectives; and (3) interpret the relationships between hyperparameter configurations and the representation of different hydrological behaviors.

2. Materials and Methods

2.1. Case Study and Dataset

The upstream watershed of the Miyun Reservoir (115.4°~117.5° E, 40.4°~41.6° N) is located in the northern part of Beijing, with elevations mostly ranging from 500 to 2000 m and a predominantly mountainous terrain. This region lies within a continental monsoon climate zone, characterized by cold and dry weather in winter, but it is rainy and hot in summer. The long-term mean annual temperature is 10.5 °C, and the mean annual precipitation is approximately 500 mm, with 80% of the rainfall occurring from July to September. Xiahui and Zhangjiafen hydrological stations serve as their respective control stations [24]. Based on the 30 m resolution digital elevation model (DEM) downloaded from the Geospatial Data Cloud (https://www.gscloud.cn/), we used the ArcSWAT 2012.10.26 tool to delineate the watershed boundaries and extract the river networks. Subsequently, the catchment area, trunk stream length, catchment perimeter, mean elevation, and mean slope of the Chaohe River and Baihe River Basins were calculated, as summarized in Table 1.

Table 1. Summary of the basic characteristics of the Chaohe River and Baihe River Basins.

Item Number	Physical Properties	Description	Chaohe River Basin	Baihe River Basin
1	Area (km2)	Catchment area	5517.21	9208.43
2	River length (km)	Length of trunk stream	217.64	290.19
3	Perimeter (km)	Catchment perimeter	656.18	790.94
4	Elevation_mean (m)	Catchment mean elevation	782.86	1047.66
5	Slope_mean (°)	Catchment mean slope	16.85	18.19
6	BFI	Baseflow Index	0.78	0.74
7	ACF(1)	1-day-lag autocorrelation function of streamflow	0.89	0.95
8	ACF(3)	3-day-lag autocorrelation function of streamflow	0.67	0.81
9	ACF(7)	7-day-lag autocorrelation function of streamflow	0.46	0.66

Figure 1 demonstrates the study area and the location of hydrometric stations and rainfall gauged stations. Observed daily streamflow data from the Xiahui hydrometric station and daily precipitation data from 10 rainfall gauged stations in the Chaohe River Basin were collected from 2000 to 2017. The data from 2000 to 2012 were used for calibration, while those from 2013 to 2017 were used for validation. To further evaluate the applicability of LSTM in the upstream watershed of the Miyun Reservoir, observed daily streamflow data at the Zhangjiafen hydrometric station and concurrent daily precipitation data from 17 rainfall stations in the Baihe River Basin were also collected from 2000 to 2010. The data from 2000 to 2006 were used for calibration, and those from 2007 to 2010 were used for validation. The split-ratio of the calibration and validation dataset was approximately 7:3. Further, based on the historical streamflow records, the baseflow index (BFI) and streamflow autocorrelation functions (ACFs) at lag times of 1, 3, and 7 days were calculated for the two basins, as summarized in Table 1. Detailed descriptions of the calculation methods for BFI and ACFs are provided in Supplementary Material Texts S1 and S2, respectively.

Figure 1. Overview of the study area. (a) Distribution of the Chaohe River Basin, Baihe River Basin, rain gauged stations, and hydrometric stations. The inset in the upper-right corner shows the location of the upstream watershed of the Miyun Reservoir within China. (b) Elevation distribution of the upstream watershed of the Miyun Reservoir. The inset in the upper-right corner shows the location of the upstream watershed relative to Beijing.

2.2. Long Short-Term Memory (LSTM) Rainfall–Runoff Model

Long short-term memory (LSTM) neural networks, as a special type of Recurrent Neural Network (RNN), address the problems of vanishing and exploding gradients during training by replacing the traditional recurrent units with memory cell units. Figure 2 shows the structure of the memory cell unit in the hidden layer of LSTM [25]. An LSTM consists of an input layer, one or more hidden layers, and an output layer. By introducing a forget gate, input gate, output gate, and cell state, the memory cell in the hidden layer can selectively remember and forget information from the input data. Specifically, the input gate controls the extent to which the current input information updates the memory state. The forget gate regulates the retention and discarding of historical information, thereby determining how long information is preserved within the memory cell. The output gate controls the contribution of the current memory state to the model output, enabling the effective modeling of long-term dependencies in sequential data [26]. In this study, precipitation data and historical runoff are used as input features, while streamflow at the Xiahui and Zhangjiafen stations is used as the output variable to train the LSTM prediction model. To investigate differences in model performance across lead times, the lead times were set to 1, 3, and 7 days, respectively. To further mitigate overfitting, the LSTM model was trained for a maximum of 200 epochs with an early stopping strategy based on validation loss, using a patience of 20 epochs (Δ < 0.5%).

Figure 2. The internal operation of the LSTM block [25].

2.3. Hyperparameter Optimization Based on Multi-Objective Particle Swarm Optimization (MOPSO)

Hyperparameter optimization can effectively improve the performance of deep learning-based hydrological models. For instance, Amini et al. employed a random search approach to optimize nine hyperparameters, including the number of hidden units, lookback, learning rate, activation function, and other training-related parameters, and evaluated the performance of multivariate deep learning-based rainfall nowcasting models using NSE and RMSE. Their sensitivity analysis further identified lookback, learning rate, and batch size as the most influential hyperparameters [1]. Xu et al. applied the particle swarm optimization (PSO) algorithm to optimize three key hyperparameters (batch size, lookback, and number of units) of LSTM models for rainfall–runoff forecasting with lead times ranging from 1 to 12 h. The corresponding search ranges were set as [24, 128], [4, 8], and [32, 256], respectively, with a population size of 30 and 100 iterations [14]. Huseyin et al. employed PSO to optimize the hyperparameters of a hybrid Bayesian–ConvLSTM model for hydrological time-series forecasting, including convolutional filter learning rates (0.001, 0.01, and 0.1) and LSTM units (32, 64, and 128), using a swarm size of 5 and a maximum of 5 iterations [27]. Building upon previous findings, we selected the most relevant hyperparameters and determined their search spaces by considering both the recommendations from previous studies and the characteristics of the rainfall–runoff time series in the study area.

Multi-Objective Particle Swarm Optimization (MOPSO) is an evolutionary optimization algorithm that solves optimization problems involving multiple conflicting objectives by constructing a Pareto-optimal set, which consists of a group of non-dominated solutions representing different trade-offs among the objectives [28]. In this study, an improved MOPSO algorithm incorporating the crowding distance strategy was adopted. An external archive was introduced to preserve elite particles, while particles with larger crowding distances were preferentially retained to maintain population diversity. A linearly decreasing inertia weight was further employed to achieve a dynamic balance between global exploration and local exploitation. To identify the optimal hyperparameter configuration for the LSTM model, MOPSO was used to optimize the major hyperparameters, with the hydrological performance metrics evaluated on the validation dataset serving as the objective functions. Following previous studies, the swarm size and the maximum number of iterations were set to 40 and 100, respectively [14]. The inertia weight was linearly decreased from 0.9 to 0.4, while both the cognitive and social learning factors (c₁ and c₂) were set to 1.5. To evaluate the convergence behavior and optimization performance of MOPSO, the hypervolume (HV) and inverted generational distance (IGD) metrics were calculated at each iteration. Detailed descriptions and calculation of these metrics are provided in Supplementary Material Text S3. Based on the multi-objective relationships between hyperparameter configurations and different aspects of hydrological processes, this study develops a novel MOPSO–LSTM framework for rainfall–runoff forecasting. All training–optimization experiments were conducted on a PC running Windows 11, equipped with an NVIDIA GeForce RTX 4060 Laptop GPU (8 GB VRAM). The deep learning framework was implemented using PyTorch 2.5.1 with CUDA 12.4 and cuDNN 9.1.0. The detailed settings of the MOPSO-LSTM method are summarized in the following subsection.

2.3.1. Decision Variables

Five key hyperparameters were selected as decision variables: batch size, number of hidden units, lookback window, dropout rate, and learning rate. Based on the characteristics of the rainfall–runoff time series, the LSTM model was configured with a single hidden layer [7,29,30,31]. Batch size, defined as the number of samples processed in each training iteration, was optimized over a range of 8 to 256 with an interval of 8. The number of units was searched over integers ranging from 10 to 200 with an interval of 10. The learning rate, which controls the step size during gradient descent optimization of the loss function, was selected from [0.0001, 0.001, 0.01]. The dropout rate ranged from 0.1 to 0.5 with an interval of 0.1 [1]. The lookback window, defined as the number of historical hydro-meteorological input time steps used to predict streamflow at the next time step, was searched over integer values from 1 to 40 [32].

2.3.2. Objective Functions

To comprehensively evaluate the performance of the LSTM rainfall–runoff model in simulating different flow regimes and water balance, the objective functions of MOPSO include the root mean square error of the entire flow series (RMSE_all), the root mean square error of high-flow conditions (RMSE_high), the root mean square error of low-flow conditions (RMSE_low), and water volume deviations (Dv). Here, RMSE_high and RMSE_low represent the high-flow and low-flow segments of the hydrological process, respectively. The high-flow segment (0~10% flow exceedance probabilities) represents rapid catchment responses to precipitation events, while the low-flow segment (10%~100%) represents the remaining flow conditions, which are mainly related to long-term sustainability and controlled by slower hydrological components, such as baseflow [33,34]. The high-flow quantile threshold was selected based on previous studies, which demonstrated that a 10% exceedance probability threshold is effective for identifying high-flow components and evaluating model performance under high-flow conditions [35,36,37]. Accordingly, in this study, “low-flow” denotes the remaining flow conditions after excluding the highest 10% high-flow segment, corresponding to the 10~100% flow-exceedance range.

The formulations are detailed as follows:

$${RMSE}_{all}=\sqrt{{\displaystyle \frac{1}{N}}{\sum }_{i=1}^N{{(Q}_i-\widetilde{Q_i})}^2}$$

(1)

where ${RMSE}_{all}$ (m³/s) represents the root mean square error for the entire flow series, $Q_i$ (m³/s) denotes the observed daily mean discharge on day i, $\widetilde{Q_i}$ (m³/s) signifies the predicted discharge by LSTM on day i, and N is the total number of time steps.

$${RMSE}_{high}=\sqrt{{\displaystyle \frac{1}{N_h}}\sum _{i:Q_i\ge Q_{90}}{{(Q}_i-\widetilde{Q_i})}^2}$$

(2)

where ${RMSE}_{high}$ (m³/s) represents the root mean square error for high-flow conditions. High-flow conditions were defined as discharge values greater than or equal to the 90th percentile ($Q_{90}$). $N_h$ represents the number of samples with $Q_i\ge Q_{90}$.

$${RMSE}_{low}=\sqrt{{\displaystyle \frac{1}{N_l}}\sum _{i:Q_i<Q_{90}}{{(Q}_i-\widetilde{Q_i})}^2}$$

(3)

where ${RMSE}_{low}$ (m³/s) represents the root mean square error for low-flow conditions. Low-flow conditions were defined as discharge values less than the 90th percentile ($Q_{90}$). $N_l$ represents the number of samples with $Q_i<Q_{90}$.

$$Dv={\displaystyle \frac{|{\displaystyle {\sum }_{i=1}^N}\widetilde{Q_i}\Delta t-{\displaystyle {\sum }_{i=1}^N}{Q}_i\Delta t|}{{\displaystyle {\sum }_{i=1}^N}{Q}_i\Delta t}}\times 100\mathrm{\%}$$

(4)

where Dv (%) represents the total water volume deviation, and Δt represents the time step, which is 1 day.

2.3.3. Evaluation Metrics

Additionally, the commonly used Nash–Sutcliffe efficiency (NSE) and Kling–Gupta efficiency (KGE) metrics were used to comprehensively evaluate model performance, which are calculated as follows:

$$NSE=1-{\displaystyle \frac{{\displaystyle {\sum }_{i=1}^N}{(Q_i-\widetilde{Q_i})}^2}{{\displaystyle {\sum }_{i=1}^N}{(Q_i-\overline{Q})}^2}}$$

(5)

where $Q_i$ (m³/s) means the observed daily mean discharge on day i; $\overline{Q}$ is average observed discharge (m³/s); $\widetilde{Q_i}$ (m³/s) represents the predicted discharge on day i; and N is the total number of time steps.

$$KGE=1-\sqrt{{\left(r-1\right)}^2+{\left(\alpha -1\right)}^2+{(\beta -1)}^2}$$

(6)

where r represents the Pearson correlation coefficient between simulated and observed values, α is the ratio of the mean of simulated values to the mean of observed values, and β is the ratio of the standard deviation of simulated values to the standard deviation of observed values [38].

Furthermore, Pearson correlation analysis was employed to evaluate the relationships among the above evaluation metrics for the predicted streamflow processes of the Pareto front solutions. The workflow of the MOPSO-LSTM approach is shown in Figure 3.

Figure 3. Schematic diagram of MOPSO-LSTM. The text in blue indicates the specific parameters or metrics involved in each step.

2.4. SHAP and PDP for Interpretation of LSTM Performance

The Shapley additive explanation (SHAP) method was adopted to quantify the effects of different hyperparameters on LSTM rainfall–runoff model performance. SHAP is derived from Shapley value theory in cooperative game theory and explains complex machine learning models by computing the marginal contribution of each input feature to the model output [18,39]. When dealing with large datasets and high-dimensional hyperparameter spaces, evaluating every possible hyperparameter configuration to construct a dataset for SHAP analysis was unnecessary and computationally inefficient [40]. Therefore, we employed Latin hypercube sampling (LHS) to sample the hyperparameter space and construct a representative dataset, which has been combined with SHAP analysis to explore variable contributions in complex parameter spaces in previous studies [41]. LHS promotes approximately uniform coverage of the marginal distribution of each hyperparameter, thereby achieving broad coverage of the hyperparameter space with a relatively small number of samples [42]. An LSTM model was independently trained and evaluated for each sampled hyperparameter configuration. The sampled hyperparameters were then used as the input features, while the corresponding model evaluation metrics were used as the output features in the subsequent surrogate-model-based SHAP analysis. For the sample i ($S_i$), the predicted value of the metric is $P_i$, and the average value of this metric is $P_{base}$. The hyperparameter k is $H_{i,k}$. Then, the SHAP value is calculated as follows:

$$P_i=P_{base}+f\left(H_{i,1}\right)+f\left(H_{i,2}\right)+\cdots +f\left(H_{i,j}\right)$$

(7)

where $f\left(H_{i,k}\right)$ represents the contribution of hyperparameter k to $P_i$. By calculating the mean absolute SHAP values, the influence of batch size, units, lookback window, dropout rate, and learning rate on model performance can be quantitatively evaluated. A larger SHAP value indicates a greater contribution of the hyperparameter to model performance.

Furthermore, partial dependence plots (PDPs) were used to reveal how each attribute affects the prediction results of the LSTM model. PDPs quantify the change in model outputs by varying the target feature while holding all other non-target features constant. In the PDP analysis, the hyperparameter spaces of the learning rate and dropout rate were refined by adding intermediate values to better capture local variations in response patterns. The refined learning rate values were [1 × 10⁻⁴, 3 × 10⁻⁴, 5 × 10⁻⁴, 7 × 10⁻⁴, 1 × 10⁻³, 3 × 10⁻³, 5 × 10⁻³, 7 × 10⁻³, 1 × 10⁻²], and the refined dropout rate values were [0.1, 0.13, 0.15, 0.17, 0.2, 0.23, 0.25, 0.27, 0.3, 0.33, 0.35, 0.37, 0.4, 0.43, 0.45, 0.47, 0.5].

$$K{PD}_M\left(x_S\right)=E_{X_C}\left[f_M\left(x_S,X_C\right)\right]=\int f_M\left(x_S,X_C\right)dP\left(X_C\right)$$

(8)

where $x_S$ denotes the hyperparameter of interest, $X_C$ represents the set of all other hyperparameters$,f_M\left(x_S,X_C\right)$ is the response function that maps hyperparameters to metric M, and $P\left(X_C\right)$ is the model’s overall feature probability density [43].

3. Results

3.1. MOPSO of the LSTM Model Under Different Lead Times

During the calibration and validation periods, the daily discharge at the Xiahui station in the Chaohe River Basin ranged from 0.071 to 99.8 m³/s and from 0.16 to 33.4 m³/s, with mean values of 2.97 and 3.41 m³/s, respectively. For lead times of 1, 3, and 7 days, the MOPSO-LSTM model obtained 43, 59, and 98 Pareto-optimal solutions after 36.46 h, 42.90 h, and 54.35 h of training and optimization, respectively. The HV and IGD convergence curves, as well as computational time, are presented in Supplementary Material Figures S1–S3. The distributions and ranges of the four metrics on the Pareto front are shown in Figure 4 and Figure 5, respectively. For lead times of 1, 3, and 7 days, the validation RMSE_all ranged from 0.848 to 1.083 m³/s, 1.504 to 1.700 m³/s, and 1.985 to 2.157 m³/s, respectively. The corresponding RMSE_high ranged from 2.081 to 2.824 m³/s, 3.678 to 4.660 m³/s, and 5.213 to 6.208 m³/s, while RMSE_low ranged from 0.382 to 0.775 m³/s, 0.629 to 1.272 m³/s, and 0.699 to 1.425 m³/s, and the Dv ranged from 0.002% to 6.913%, 0.002% to 7.667%, and 0.003% to 8.887%, respectively. In addition, the commonly used performance metrics NSE and KGE were also evaluated. For lead times of 1, 3, and 7 days, NSE ranged from 0.869 to 0.920, 0.677 to 0.747, and 0.480 to 0.560, respectively, while KGE ranged from 0.838 to 0.950, 0.629 to 0.827, and 0.510 to 0.686.

Figure 4. Pareto fronts obtained from MOPSO-LSTM model with lead times of 1d (a), 3d (b), and 7d (c). The four-dimensional Pareto front is visualized with RMSE_all, RMSE_high, and RMSE_low on the X-axis, Y-axis, and Z-axis, respectively, and Dv is represented by the color mapping. The point S_all and the vertical lines projected onto the X-Y-Z-axes represent the Pareto solution with the minimum RMSE_all. Similarly, S_high and S_low indicate the Pareto solutions with the minimum RMSE_high and RMSE_low, respectively. The coordinates of these points are expressed as RMSE_all, RMSE_high, RMSE_low, and Dv.

Figure 5. The ranges of hydrological performance metrics in the Pareto-optimal solution sets under different lead times. (a) RMSE_all, (b) RMSE_high, (c) RMSE_low, (d) Dv, (e) NSE, and (f) KGE.

It can be seen that model performance deteriorated as the lead time increased. Specifically, from 1 to 3 days, the mean values of RMSE_all, RMSE_high, and RMSE_low across the Pareto-optimal solutions increased by 64.06%, 67.86%, and 56.67%, respectively. Additionally, NSE and KGE decreased by 19.21% and 16.92%, respectively. From 3 to 7 days, the mean values of RMSE_all, RMSE_high, and RMSE_low increased by 30.05%, 37.22%, and 10.95%, respectively. Meanwhile, NSE and KGE decreased by 26.16% and 20.87%, respectively. However, Dv changed only slightly, with average Dv values of 1.037%, 1.560%, and 1.601% at lead times of 1, 3, and 7 days, respectively. Compared with the transition from 1 to 3 days, the decreases in NSE and KGE were slightly greater from 3 to 7 days, while the increases in RMSE_all, RMSE_high, and RMSE_low were approximately halved. However, the increase in Dv was substantially reduced.

3.2. Multi-Objective Trade-Offs Among Hydrological Evaluation Metrics

From the Pareto fronts, the preference-based solutions for RMSE_all, RMSE_high, RMSE_low, and Dv were selected and denoted as S_all, S_high, S_low, and S_Dv, respectively. The predicted and observed hydrographs of different solutions are shown in Figure 6, and the corresponding parameter settings are listed in Table 2. S_all mainly exhibited overestimation of low-flow conditions and underestimation of high-flow conditions. S_high yielded lower RMSE_high than other solutions, indicating superior performance in high-flow prediction. At a lead of 1 day, S_high accurately predicted the flood peaks on 17 July 2013 and 25 July 2016, with observed peak discharges of 30.0 m³/s and 30.6 m³/s, and simulated peak values of 28.13 m³/s and 30.76 m³/s, respectively. In comparison, S_all, S_low, and S_Dv produced peak discharges of 25.39 m³/s, 23.82 m³/s, and 21.13 m³/s for the 2013 event, and 27.63 m³/s, 23.09 m³/s, and 27.56 m³/s for the 2016 event, respectively. S_low showed better performance in simulating low-flow conditions but generally underestimating high-flow events. S_Dv exhibited limited variation in Dv across different lead times, indicating relatively stable water balance performance.

Figure 6. Comparison of predicted and observed streamflow hydrographs in the Chaohe River Basin for the preference-based solutions for RMSE_all, RMSE_high, RMSE_low, and Dv. Lead times of (a) 1d, (b) 3d, and (c) 7d.

Table 2. Performance metrics and corresponding hyperparameter values for different preference-based solutions under different lead times in the Chaohe River Basin.

Lead Time (Day)	Objective Preference	Performance Metric						Hyperparameters
		RMSEall (m3/s)	RMSEhigh (m3/s)	RMSElow (m3/s)	Dv (%)	NSE	KGE	Batch Size	Units	Lookback Window	Dropout Rate	Learning Rate	Epochs 1
1	RMSEall	0.848	2.168	0.512	0.427	0.920	0.883	104	200	15	0.1	0.01	111
	RMSEhigh	0.992	2.081	0.775	0.514	0.890	0.941	48	200	19	0.2	0.001	44
	RMSElow	0.957	2.762	0.382	3.184	0.898	0.900	24	190	27	0.3	0.01	49
	Dv	1.029	2.617	0.629	0.002	0.882	0.894	88	30	37	0.5	0.001	95
3	RMSEall	1.504	4.242	0.676	1.007	0.747	0.743	32	10	1	0.4	0.01	34
	RMSEhigh	1.600	3.678	1.141	2.229	0.714	0.806	72	70	12	0.3	0.01	24
	RMSElow	1.516	4.345	0.629	3.133	0.743	0.739	40	60	1	0.4	0.01	62
	Dv	1.672	4.463	0.912	0.002	0.688	0.629	216	130	1	0.5	0.001	61
7	RMSEall	1.985	5.572	0.909	3.529	0.560	0.600	96	180	8	0.4	0.001	60
	RMSEhigh	2.053	5.213	1.256	7.437	0.529	0.658	64	170	21	0.2	0.001	25
	RMSElow	2.099	6.208	0.699	7.728	0.508	0.521	96	180	24	0.2	0.01	24
	Dv	2.083	5.716	1.042	0.003	0.515	0.565	88	80	31	0.2	0.0001	57

¹ The number of epochs was determined using the early stopping strategy. Please see Section 2.2 for details.

Furthermore, Pearson correlation analysis was conducted to examine the relationships between the different evaluation metrics, as shown in Figure 7 and Supplementary Material Table S1. At a lead time of 1 day, RMSE_all showed a significantly positive correlation with both RMSE_high and RMSE_low. Across all lead times, RMSE_low and RMSE_high exhibited significant negative correlations. In particular, RMSE_low was also predominantly negatively correlated with NSE. In contrast, Dv mainly demonstrated negative correlations with RMSE_all and RMSE_high but positive correlations with RMSE_low. Additionally, KGE showed a significant negative correlation with RMSE_high and a significant positive correlation with RMSE_low across all lead times. These indicate clear trade-offs among different evaluation metrics, suggesting that the traditional practice of evaluating model performance using a single metric is highly limited.

Figure 7. Heatmap of Pearson correlations among six hydrological evaluation metrics. Lead time of (a) 1d, (b) 3d, and (c) 7d. “*” indicates statistically significant correlations (p < 0.05). Only the lower triangular matrix is shown, as the upper triangular matrix is identical due to symmetry.

Distinctive patterns of hyperparameter combinations were observed under different lead times and optimization preferences. Learning rate generally declined with increasing lead times, decreasing from 0.001~0.01 for the 1-day and 3-day lead times to 0.0001~0.01 for the 7-day lead time. Among different preference strategies, the S_Dv solutions consistently adopted smaller learning rates compared with the RMSE preference-based solutions. Batch size primarily varied among different optimization preferences. S_Dv tends to adopt larger batch sizes; for example, at a lead time of 3 days, the batch size reaches 216, whereas the maximum batch size among RMSE preference-based solutions was 72. The lookback window was relatively large for both S_DV and S_low at the lead times of 1 day and 7 days, whereas smaller values were selected at the lead time of 3 days. Similarly, the number of units was larger for the RMSE preference-based solutions compared with S_Dv at the lead times of 1 day and 7 days, while an opposite tendency is observed at the 3-day lead time. The dropout parameter reached its highest value of 0.5 in the D_V preference-based solution.

3.3. Interpretability Analysis of the LSTM Model Using SHAP and PDP

The effects of hyperparameters on model performance with different lead times were interpreted using the SHAP method, as shown in Figure 8. Overall, the learning rate was consistently identified as the most influential hyperparameter across nearly all lead times, with a particularly strong effect on RMSE_low. For the lead time of 1 day, lower learning rates tended to increase RMSE_all, RMSE_high, and RMSE_low while reducing Dv. In contrast, for the lead times of 3 days and 7 days, lower learning rates generally tended to decrease RMSE_all, RMSE_high, and Dv. As lead time increased, the importance of the lookback window became more pronounced. For lead times of 3 days and 7 days, the contribution of lookback window ranked first or second, whereas for the 1-day lead time, batch size showed relatively higher importance. Batch size exhibits moderate importance, with its influence varying across optimization objectives and slightly higher importance observed for RMSE_low and Dv. Similarly, units exhibited lower importance. However, for RMSE_high at a lead time of 1 day, it became the most important hyperparameter, and the dropout rate consistently showed the lowest importance among all hyperparameters.

Figure 8. SHAP summary plots and the importance ranking for the 5 LSTM hyperparameters across Chaohe River Basin. Lead times of (a) 1d, (b) 3d, and (c) 7d. In the SHAP plots, each dot represents one LSTM hyperparameter configuration, with color gradients showing feature magnitudes (red: high; blue: low). The SHAP value axis indicates feature positive and negative contributions to test performance metrics. In the bar plots, feature importance is defined as the mean absolute SHAP value (mean |SHAP value|) of each hyperparameter.

Partial dependence plots (PDPs) were further applied. For clarity, only the effects of the hyperparameters on NSE are presented in the main text (Figure 9), while the corresponding results for the other evaluation metrics are provided in the Supplementary Material Figures S4–S7. For a lead time of 1 day, NSE increased with increasing learning rate and gradually plateaued at higher learning rates. In contrast, for the 3-day and 7-day lead times, the highest NSE was achieved at intermediate learning rates, while both excessively high and low learning rates reduced model performance. Similar response patterns were observed for the RMSE-based metrics. By contrast, Dv exhibited an initial decrease followed by an increase at the 1-day lead time, whereas a monotonic increasing trend was observed at the 3-day and 7-day lead times, indicating that the learning rate resulted in poorer Dv performance. Compared with the learning rate, batch size had a weaker effect on NSE, resulting in a smaller variation in the objective values. For units, NSE generally increased initially and then decreased across all lead times. Among all hyperparameters, the lookback window exhibited the strongest dependence on lead time. At a lead time of 1 day, NSE decreased with increasing lookback window. At a lead time of 3 days, NSE first decreased and then increased, gradually reaching a plateau when the lookback window exceeded approximately 10~15 days. In contrast, at a lead time of 7 days, NSE increased continuously with increasing lookback window, suggesting that incorporating longer historical input sequences improves predictive performance for longer lead times. For the dropout rate, NSE changed slightly across all lead times, indicating the limited effects of this hyperparameter on model performance.

Figure 9. Partial dependence plots of LSTM hyperparameters for NSE under different lead times. (a) 1d, (b) 3d, and (c) 7d. The X-axis represents hyperparameter values, and the Y-axis represents the simulated performance metric under averaged effects of other hyperparameters.

3.4. Application and Comparative Analysis of the MOPSO-LSTM in the Baihe River Basin

To evaluate the applicability of the MOPSO-LSTM model in the upstream Miyun Reservoir, the model was further applied to the spatially adjacent Baihe River Basin. The predicted streamflow under different preference solutions in the Pareto front is shown in Figure 10. At the Zhangjiafen station in the Baihe River Basin, the daily streamflow ranged from 0.077 to 94.2 m³/s in the calibration period and 0.39 to 108 m³/s in the validation period, with average values of 5.23 m³/s and 6.56 m³/s, respectively, approximately twice those observed at the Xiahuai station in the Chaohe River Basin. The optimization results showed that RMSE_all, RMSE_high, RMSE_low, and Dv in the Baihe River Basin were generally 2~3 times higher than those in the Chaohe River Basin, which was mainly attributed to the higher streamflow at this site. While NSE and KGE were even better in the Baihe River Basin, reaching maximum values of 0.949 and 0.970 at a lead time of 1d, 0.756 and 0.829 at a lead time of 3d, and 0.632 and 0.729 at a lead time of 7d, respectively. These results suggest that the MOPSO-LSTM achieved better performance in the Baihe River Basin.

Figure 10. Comparison of predicted and observed streamflow hydrographs in the Baihe River Basin for the preference-based solutions for RMSE_all, RMSE_high, RMSE_low, and Dv. Lead times of (a) 1d, (b) 3d, and (c) 7d.

The optimal hyperparameter configurations are provided in Table 3. Notable differences in optimal hyperparameter configurations were observed between the Baihe and Chaohe River Basins. Specifically, compared with the Chaohe River Basin, LSTM models for the Baihe River Basin tended to adopt longer lookback windows, larger numbers of hidden units, and higher learning rates, together with lower dropout rates. For the Pareto-optimal solution sets across all lead times, the mean lookback window, number of hidden units, and learning rate were 28 days, 144, and 0.0067, respectively, for the Baihe River Basin. These values were consistently higher than the corresponding values of 18 days, 119, and 0.0039 for the Chaohe River Basin. The mean batch sizes were similar between the two basins, with values of 94 and 99 for the Baihe and Chaohe River Basins, respectively. However, the mean dropout rate was lower for the Baihe River Basin (0.26) than for the Chaohe River Basin (0.33).

Table 3. Performance metrics and corresponding hyperparameter values for different preference-based solutions under different lead times in the Baihe River Basin.

Lead Time (Day)	Objective Preference	Performance Metric						Hyperparameters
		RMSEall (m3/s)	RMSEhigh (m3/s)	RMSElow (m3/s)	Dv (%)	NSE	KGE	Batch Size	Units	Lookback	Dropout Rate	Learning Rate	Epochs 2
1	RMSEall	1.99	5.02	1.26	3.68	0.95	0.95	16	200	40	0.4	0.001	69
	RMSEhigh	2.12	4.41	1.68	0.89	0.94	0.97	48	100	19	0.2	0.01	31
	RMSElow	2.40	6.81	1.10	2.28	0.93	0.95	136	120	28	0.2	0.01	67
	Dv	2.51	6.58	1.48	0.0017	0.92	0.92	80	70	25	0.3	0.001	53
3	RMSEall	4.38	11.86	2.37	1.67	0.75	0.80	208	10	2	0.1	0.001	102
	RMSEhigh	4.60	10.77	3.26	12.29	0.73	0.79	56	160	35	0.3	0.01	25
	RMSElow	4.75	13.69	2.05	3.72	0.71	0.73	48	200	40	0.4	0.01	42
	Dv	4.79	12.45	2.86	0.0004	0.71	0.80	128	90	40	0.4	0.0001	200
7	RMSEall	5.35	13.20	3.51	4.77	0.63	0.73	48	200	35	0.1	0.01	39
	RMSEhigh	5.61	13.18	3.95	9.39	0.59	0.68	112	140	22	0.2	0.01	78
	RMSElow	6.43	18.58	2.72	10.98	0.47	0.50	64	170	31	0.2	0.01	25
	Dv	6.63	18.30	3.38	0.0020	0.44	0.52	160	200	40	0.1	0.0001	47

² The number of epochs was determined using the early stopping strategy. Please see Section 2.2 for details.

4. Discussion

4.1. Capability of MOPSO-LSTM in the Upper Miyun Reservoir Catchment

The proposed MOPSO-LSTM framework provides a multi-objective hyperparameter optimization strategy for LSTM-based rainfall–runoff modeling, enabling the simultaneous consideration of different components of the hydrograph. MOPSO employs an external archive to preserve non-dominated solutions and uses crowding distance to maintain solution diversity during the optimization process [20]. In this study, the proposed framework explored a hyperparameter space containing 372,000 possible configurations using 40 particles and 100 iterations, with a computational time of approximately 36~55 h. The computational burden mainly resulted from LSTM model training rather than the MOPSO algorithm itself. Notably, the MOPSO framework achieved relatively stable convergence at approximately 60 iterations, indicating that satisfactory solutions could be obtained earlier. The computational cost of the proposed framework is comparable with that reported in previous LSTM hyperparameter optimization studies. For example, Hosseini et al. applied random search to optimize LSTM rainfall–runoff models by exploring approximately 5,400,000 possible hyperparameter configurations. Their experiments involved 10 random seeds and 1000 runs across 40 catchments, using hourly data from 2000 to 2015, with NSE or KGE as optimization objectives, and required approximately 32 days of computation [44].

The MOPSO-LSTM rainfall–runoff model has achieved favorable performance in the upstream basins of the Miyun Reservoir. Specifically, the preference-based RMSE_high accurately reproduced flood peaks at a lead time of 1 day, and Dv preference-based solution consistently provided excellent predictions of total runoff volume across all lead times, with Dv remaining below 0.03%. Several physically based hydrological models have previously been applied to simulate streamflow in this region. For instance, Zhang et al. applied the HSPF model to simulate streamflow at three hydrological stations (Dage, Gubeikou, and Xiahui), using monthly streamflow data from 1990 to 1995 for calibration and data from 1996 to 2000 for validation. The NSE values during the validation period were 0.78, 0.60, and 0.76 at the three stations, respectively [23]. Xu et al. applied the SWAT model to simulate daily streamflow at Zhangjiafen and Xiahui stations, using data from 1986 to 1988 for calibration and 1989 to 1991 for validation. By incorporating 3″ × 3″ DEM data, meteorological data from seven temperature stations and 75 rain gauged stations, and land-use information, they achieved NSE values of 0.67 and 0.60 for daily streamflow simulations at Xiahui and Zhangjiafen, respectively, during the calibration period, while higher NSE values were obtained for monthly runoff simulations (0.91 and 0.97, respectively) [45]. In comparison, the proposed MOPSO-LSTM model yielded even superior predictive performance, with NSE values reaching 0.920 at Xiahui and 0.950 at Zhangjiafen at a lead time of 1 day. This superiority may be attributed to the ability of LSTM models to better capture the complex nonlinear dynamics of hydrological processes than traditional physically based models, which generally rely on simplified process representations and conceptual assumptions, whereas LSTM models directly learn nonlinear relationships, long-term dependencies, and latent spaciotemporal patterns from observed data [46]. It is also worth noting that this superior performance may be partly attributed to the inclusion of historical runoff as an input variable in LSTM models, which is not typically used in process-based models.

A further comparison between the two catchments revealed that LSTM in the Baihe River Basin achieved better predictive performance than the Chaohe River Basin across all lead times. Although the RMSE-related metrics and Dv at the Zhangjiafen station were approximately two to three times those at the Xiahui station, this difference primarily arose from the fact that the mean streamflow at Zhangjiafen is nearly twice that at Xiahui. Nevertheless, the NSE and KGE values were consistently higher in the Baihe River Basin. The better LSTM performance in the Baihe River Basin may be attributed to its larger catchment area (approximately 67% larger than the Chaohe River Basin) and longer trunk stream (approximately 33% longer), which may enhance flow regulation effects and catchment memory. The BFI, an indicator of catchment memory, was higher than 0.7 in both basins with only minor differences, indicating strong contributions from groundwater and other slow-response flows [47]. Furthermore, the ACF values at 1-, 3-, and 7-day lags were consistently higher in the Baihe River Basin, suggesting stronger memory of streamflow series associated with either strong storage–surface connections or surface water retention [48]. Since autocorrelation is the basis of LSTM model prediction, and is informative for sequence-based models, the higher ACF values in the Baihe River Basin may partly explain the improved predictive performance of the LSTM [49]. Previous studies have also shown that LSTM models can achieve superior performance in catchments with larger drainage areas and slower hydrological responses than in catchments characterized by flashy runoff responses and predominantly diurnal variability [50,51]. In larger catchments, LSTM models are better able to capture the long-term dependencies arising from watershed storage and release processes, and their performance generally improves with increasing discharge and stronger seasonal variability [29,50]. For example, Hunt et al. developed LSTM rainfall–runoff models for 10 catchments in the United States, with drainage areas ranging from 300 to 14,000 km², and found that the LABW4 catchment, with an area of 10,123 km² and an elevation of 1987 m, achieved better performance than the other catchments. Interestingly, its catchment characteristics were similar to those of the Baihe River Basin [50]. Li et al. investigated 49 hydrological stations along the Yangtze River in China, covering catchment areas from 50 to 1,200,000 km², and identified a positive correlation between the logarithm of catchment area and KGE, indicating that LSTM performance generally improved with increasing catchment size, while the model was substantially degraded or even failed in very small catchments [29]. Clark et al. applied LSTM rainfall–runoff modeling across Australian catchments ranging from 4 to 119,000 km² and found that catchments within the largest 20% by area generally exhibited higher NSE values, demonstrating a greater average predictive advantage [52].

However, as the lead time increased, the overall model simulation performance deteriorated, with a more obvious decline from 1-day to 3-day lead times than from 3-day to 7-day lead times. Moreover, process-based metrics such as RMSE, NSE, and KGE exhibit greater degradation than volumetric metrics. This degradation has also been observed in previous studies. For example, in the Yangtze River Basin, 78%, 30%, and 20% of catchments achieved a KGE greater than 0.5 at 1-, 7-, and 15-day lead times, respectively [29]. In the Feilaixia Basin in Southern China, NSE decreased from 0.88 to 0.22, while RMSE increased from 438 m³/s to 1088 m³/s as lead time increased from 1 to 7 days [53]. Similar patterns have also been observed in most catchments in the United States [54], the Red River system in northwestern Vietnam [55], and the Madeira River Basin in northern Brazil [56]. As lead time increases, the LSTM model increasingly relies on historical discharge for prediction, while future precipitation remains unknown, thus leading to a weakened rainfall–runoff relationship. When the lead time exceeds the hydrological response time, the model struggles to capture future runoff regime shifts based solely on historical characteristics. The progressive incorporation of previously predicted discharge into the input sequence further amplifies error accumulation [54]. However, in this study, the LSTM model showed only minimal bias increases in water volume estimation beyond a 3-day lead time. This likely reflects the loss of precipitation-driven runoff dynamics, yielding smoother hydrographs that approach mean flow conditions. Under such circumstances, a low Dv value may not necessarily indicate accurate prediction but rather a reduced ability to reproduce runoff variability. The concurrent deterioration of other process-oriented metrics suggests an overall degradation of LSTM model performance.

4.2. Linking Multi-Objective Machine Learning Hyperparameter Optimization with Watershed Hydrology

Across all lead times, the learning rate was the most influential hyperparameter, controlling the parameter update dynamics during training [1,57]. Its optimal value decreased with increasing lead time, reflecting the increasing difficulty of learning nonlinear rainfall–runoff relationships under longer hydrological response lags in long-term prediction, which requires more conservative updates to ensure stable convergence. In addition, compared with RMSE preference-based solutions, Dv preference-based solutions consistently adopted smaller learning rates, likely because Dv is dominated by low-flow conditions, which occupy a larger proportion of the time series in this region and require more stable updates to preserve water-balance accuracy. For the lookback window, its importance grew with increasing lead time, and both Dv and RMSE_low preference-based solutions selected longer lookback horizons. This is because streamflow in the study area is dominated by low-flow conditions, and for long lead-time forecasting, a long lookback window can provide more contextual information for establishing relationships between inputs and outputs, as well as capturing long-term dependencies essential for hydrological process reconstruction. However, this may increase computational cost [12]. Batch size varied clearly across different optimization priorities. Dv and RMSE preference-based solutions favored relatively large and small batch sizes, respectively. This is mainly because reproducing the detailed characteristics of the streamflow process requires a smaller batch size, which enables more refined parameter updates and better captures local variations. In contrast, a larger batch size tends to produce smoother optimization and faster convergence, although it may overlook localized peak-flow dynamics [1,12,58]. Units moderately affected model performance by reflecting the ability of LSTM to represent hydrological memory. Larger units improve the learning of complex nonlinearities and long-lag responses, but excessive size may reduce generalization. Dropout rate was less important than above other hyperparameters but crucial for regularization. It improves generalization, but excessively high values may impair the model’s ability to learn fine-grained patterns, which is also reflected in the SHAP and PDP analysis results [12].

By linking multi-objective hyperparameter optimization with the prediction ability of hydrograph and water balance, we have enhanced our understanding of the implicit physical hydrological concepts embedded in machine learning models, indicating that the internal cell state contains dynamic information related to these physical processes [59]. In particular, SHAP and PDP analysis provide complementary global and local interpretability of LSTM hyperparameters and highlight the distinctive key factors affecting high-flow, low-flow, and runoff volume predictions [43]. Moreover, different hyperparameter configurations of two watersheds may also reflect basin properties, with learning rate and lookback specifically indicating catchment scale, hydrological memory, and storage capacity. A previous study of multiple subbasins revealed significant positive correlations of catchment lag time and concentration time with drainage area, trunk stream length, and perimeter [60]. Considering the larger area, longer trunk stream length, and greater perimeter of the Baihe River Basin discussed above, we assume that the Baihe River Basin has longer lag time and concentration time than the Chaohe River Basin. Accordingly, the larger lookback window identified in the Pareto-optimal solutions of the LSTM rainfall–runoff model for the Baihe River Basin, compared with the Chaohe River Basin, may to some extent support the interpretation that the lookback window reflects hydrological memory.

4.3. Implications of Multi-Objective Hyperparameter Optimization for Water Resource Management Practice

Previous studies have commonly employed a single metric as the evaluation criterion for hydrological models, yet such an approach inherently sacrifices other aspects to match a particular facet of hydrological processes [61]. Our findings corroborate this notion. Across all lead times, RMSE_high and RMSE_low exhibited a significant negative correlation, indicating a pronounced trade-off between the simulation performance of high-flow and low-flow conditions. Nevertheless, under a 1-day lead time, RMSE_all showed significant positive correlations with both RMSE_high and RMSE_low, suggesting that the overall error is influenced by the model’s capability to simulate both high-flow and low-flow processes. In contrast, the negative correlations of Dv with RMSE_all and RMSE_high but positive correlations with RMSE_low indicate that the overall water balance deviation in the Chaohe River Basin is more substantially affected by prediction bias in low-flow processes. The Chaohe River Basin has a typical temperate monsoon climate, with precipitation and runoff highly concentrated in the flood season and sharp hydrographs [62]. Since high-flow events are short-lived, their contribution to cumulative streamflow is limited, whereas low-flow conditions dominate most of the year and account for a larger proportion of the total streamflow volume.

Different objective preferences correspond to different hyperparameter configurations. Therefore, in practical applications, model structures should be selected according to specific hydrological scenarios [63]. After obtaining the Pareto front for different objectives using the MOPSO-LSTM farmwork, for a single management task with a clearly defined objective, the solutions can be sorted in ascending order according to each objective to identify the hyperparameter configurations that optimize RMSE_all, RMSE_high, RMSE_low, and Dv, respectively. For flood control, priority should be given to Pareto solutions that exhibit stronger prediction capability for high-flow processes. Given the growing water use competition and rising demand for low-flow predictions, drought forecasting should target low-flow conditions and solutions with superior performance in this range [64]. For water resource allocation, long-term reservoir operation, and water supply management, models that perform better in terms of Dv are more appropriate. It should be noted that the selection of solutions based on water-balance criteria should be considered together with process-related performance, although such solutions generally maintain relatively small volume deviations even at longer lead times. Furthermore, for comprehensive management tasks, a comprehensive multi-criteria decision-making framework, such as the entropy-weighted Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) method, can be applied to identify the most suitable solutions from the Pareto set by integrating optimization outputs, objective information content, and decision-maker preferences, thereby balancing specific preferences with other performance metrics [65]. Future work may introduce additional metrics related to event duration, timing, and frequency to better support aquatic ecosystem management applications [66]. Thus, compared to single-objective optimization, multi-objective hyperparameter optimization better addresses diversified needs in hydrological prediction, offering greater flexibility and reliability in hyperparameter selection for the LSTM model under diverse operational contexts.

Finally, we would like to point out that, although this study provides insights into the relationships between hyperparameters and different components of the hydrograph, it also acknowledges certain limitations that should be addressed in future research. This study focused on the upstream basins of the Miyun Reservoir, where hydrometric stations are relatively sparse. Future studies should extend the analysis to a larger number of catchments covering diverse climatic conditions, elevations, and basin sizes and incorporate longer hydrological time series to assess the applicability and generalizability of the proposed framework. Moreover, catchment response time and concentration time computed from physical properties or observed should be directly compared with the optimized lookback window, and their relationships should be further quantitatively interpreted through SHAP and PDP analyses to provide deeper insights. Furthermore, we may assume that when the LSTM hyperparameter configuration achieves an optimal understanding of physical hydrological concepts, its representation becomes more consistent with those of process-based models. This, in turn, may substantially improve model prediction performance and provide better generalization and extrapolation capabilities under extreme conditions [42]. Additionally, to address the degradation of LSTM performance with increasing forecast lead times, future studies could explore the incorporation of attention mechanisms and additional catchment-related input variables (e.g., land use, temperature, and other hydroclimatic characteristics) to improve model predictive capability, while cross-validation strategies could be introduced to further evaluate its generalization performance under extreme climatic conditions.

In terms of multi-objective optimization algorithms, MOPSO is a resource-intensive algorithm with most of the computational cost arising from LSTM training rather than the optimization algorithm itself. Future studies focusing on hyperparameter optimization frameworks should test various multi-objective optimization algorithms with machine learning models. Beyond benchmark random search, more advanced hyperparameter space exploration methods than traditional grid search, such as surrogate-based Multi-objective Tree-structured Parzen Estimator (MOTPE) [67] and probabilistic surrogate-based multi-objective Bayesian optimization (MOBO) [68], should also be considered for comparison. Under consistent experimental settings, such as the same hyperparameter space, iterations, and termination criteria, different frameworks can be compared in terms of computational efficiency and optimization performance to provide more informative guidance for multi-objective hyperparameter optimization. Additional hyperparameters, such as the number of hidden layers and activation functions, could be incorporated to fully assess the capabilities of deep learning models.

5. Conclusions

The MOPSO-LSTM model with a lead time of 1d achieved maximum NSE values of 0.920 and 0.949 for the Chaohe and Baihe River Basins, respectively. As the lead time extended to 3 and 7 days, RMSE-related metrics declined substantially, whereas Dv remained relatively low, reaching a minimum of 0.003% at a lead time of 7 days. The learning rate, number of units, and lookback window were the most crucial hyperparameters governing model performance. Notable trade-offs existed between RMSE_high and RMSE_low, while a synergistic relationship existed between RMSE_low and Dv, with different objectives favoring distinct hyperparameter configurations for different objectives. Compared with the Chaohe River Basin, the larger Baihe River Basin favored LSTM configurations with longer lookback windows, more hidden units, higher learning rates, and lower dropout rates. This study contributes to new insights into the linkage between machine learning model architecture and hydrology, as well as provides direct modeling support for water resource management.