Rainfall–Runoff (R–R) modelling plays a fundamental role in hydrological science, enabling the prediction of how precipitation transforms into streamflow (Beven, 2012). This predictive capability is essential for a range of applications, including flood forecasting, water resource management, and environmental protection (Hunt et al., 2022). Over the past decades, R–R modelling has advanced through the development of physics-based hydrological models, particularly conceptual and distributed models (Clark et al., 2015). These advances have been supported by improvements in hydrological data availability and computational power (Zhang et al., 2025). Typically, R–R models comprise two core components: runoff generation and runoff routing. Runoff generation refers to the partitioning of rainfall into surface runoff or subsurface flow, while runoff routing represents the transport and temporal distribution of this water through river networks (Beven, 2012). Developing these physical models often requires extensive parameterization and iterative calibration. This challenge is compounded by high-resolution versions, which demand significant data handling. Furthermore, the transferability of such models remains a challenge, as models calibrated for one catchment often perform poorly in ungauged or data-scarce basins (Arsenault et al., 2023).

Recent reviews indicate a notable shift from purely physics-based models towards data-driven approaches, especially deep learning (DL) models (Tripathy and Mishra, 2024). This shift is driven by the increasing availability of streamflow and meteorological datasets. Among these DL methods, the Long Short-Term Memory (LSTM) neural network has gained widespread adoption due to its effectiveness in capturing complex temporal dependencies inherent in hydrological time series (Kratzert et al., 2018; Anderson and Radić, 2022). Most initial DL applications have treated catchments as lumped systems, where meteorological variables are spatially averaged to predict runoff at the outlet. However, with the growing availability of high-resolution spatial datasets (Brocca et al., 2024), more sophisticated deep learning architectures that integrate both spatial and temporal features such as Convolutional Neural Networks (CNNs) combined with LSTMs have emerged (Anderson and Radić, 2022). These models aim to enhance predictive accuracy by leveraging spatial patterns alongside temporal sequences (Li et al., 2023).

Despite these advancements, these deep learning approaches focus solely on runoff generation and do not explicitly model the runoff routing component (Wang et al., 2024). Including runoff routing is crucial because it accounts for flow delays, attenuation, and connectivity within river systems. Neglecting routing can lead to significant inaccuracies, such as overestimation or underestimation of peak flows and misrepresentation of flow dynamics, particularly in large or complex basins (Cortés-Salazar et al., 2023; Baste et al., 2025). For instance, Cortés-Salazar et al. (2023) demonstrated that adding an explicit routing scheme improved the Kling–Gupta efficiency of daily streamflow from 0.64 (without routing) to 0.81 (with the best routing scheme). Some efforts attempt to address this by integrating upstream hydrological information into LSTM models or combining LSTM outputs with physically based routing models (Yu et al., 2024; Yang et al., 2025). While these approaches improve spatial realism, the routing component itself is not inherently learned within the DL framework. This limitation primarily stems from these models' inability to incorporate river network topology in a physically meaningful way.

Graph Neural Networks (GNNs) offer a promising solution to this challenge by explicitly modeling graph-structured data, making them well-suited for representing river network topology (Sun et al., 2021). In the context of hydrology, the river system can be naturally represented as a graph, where nodes correspond to subbasin outlets or gauge locations, and edges, the links that connect these nodes and represent river channels, capture the connectivity of the network. The key strength of GNNs lies in their ability to propagate information across the graph structure through what are known as edges, allowing each node to learn from its upstream and downstream neighbors. This formation flow mimics the physical process of runoff routing, enabling the model to learn spatial dependencies within the river network. Several recent studies have explored GNNs in R–R modeling, treating them as spatiotemporal modules within DL frameworks and highlighting their potential. These models typically combine GNNs with LSTMs or other recurrent architectures to capture both spatial and temporal dynamics. For example, Sun et al. (2022) utilized GNNs to capture physics-based connectivity, demonstrating that graph-based data fusion can serve as an effective surrogate for process-based models. Similarly, Deng et al. (2023) addressed the non-Euclidean structure of river networks using spatiotemporal graph convolutions to capture upstream-downstream correlations. Beyond surface water, Gai et al. (2023) applied GNNs to simulate spring discharge by modeling the complex subsurface connectivity of karst systems. More recently, Wang et al. (2025) showed that optimizing graph topologies, specifically transforming tree-like networks into dense graphs can accelerate flood warnings by capturing long-range dependencies. These models typically combine GNNs with LSTMs or other recurrent architectures to capture spatiotemporal dynamics, with a primary focus on improving representations of spatial variability in inputs or learning latent inter-basin correlations, rather than explicitly modeling the flow-routing process along river networks.

Given the importance of runoff routing, our hypothesis is that incorporating GNNs as a dedicated module for runoff routing will improve runoff prediction. Furthermore, most existing GNN studies in hydrology have been limited to small-scale catchments with relatively few sub-basins (Sun et al., 2022; Gai et al., 2023; Deng et al., 2023). These studies often do not fully exploit the potential of GNNs to represent complex physical routing processes across vast networks. Thus, the explicit use of GNNs for runoff routing in large and complex river basins remains an open and underexplored area in current DL frameworks.

In this study, we aim to address this gap by developing a novel model that integrates LSTM networks with GNNs. The proposed LSTM-GNN architecture leverages the temporal modeling capabilities of LSTMs for each subbasin, while a GNN component explicitly models runoff routing through the river network. The model is applied to predict daily river discharge across the Danube River Basin, a large and topologically complex catchment. The specific objectives of this study are: (1) to evaluate the performance of different GNN architectures such as Graph Convolutional Network (GCN), Graph Attention Network (GAT), Graph SAmple and aggreGatE (GraphSAGE), and Chebyshev Spectral Graph Convolutional Network (ChebNet) as routing modules; and (2) to assess the contribution of the routing component by comparing the proposed LSTM-GNN model against a baseline LSTM model that lacks explicit spatial routing.

The Upper Danube River Basin (Danube) in Central Europe is chosen as the study area due to its extensive geographical coverage, inherent hydrological complexity, and the rich availability of associated datasets (Fig. 1). It spans about 170 000 km2 and crosses or borders nine countries (Austria, Germany, Switzerland, Slovakia, Czech Republic, Hungary, Liechtenstein, Italy, and Slovenia). The basin's terrain ranges from high Alpine headwaters down to lowland plains. The basin experiences a broad range of subbasin-level average annual temperatures from -4.14 to 10.45 °C and receives annual precipitation varying significantly across its subbasins from 650 to 2068 mm (Muñoz-Sabater et al., 2021), reflecting its diverse climatic zones. This strong physiographic and climatic gradient produces a wide diversity of catchment characteristics and highly variable streamflow patterns across the basin.

The hydrological dataset for this study comes from LamaH-CE (Large-Sample Data for Hydrology in Central Europe) (Klingler et al., 2021). LamaH-CE provides time series of streamflow and meteorological variables, along with static catchment descriptors for Danube. In total LamaH-CE covers 859 gauged basins over the Danube. We focus on a subset of 530 gauges that have continuous daily streamflow records from 1 January 1987 to 31 December 2017. These 530 subbasins span a very wide range of sizes from a few square kilometers up to over 2500 km2 and include diverse topographic, land cover, and hydrologic conditions. The intricate dendritic structure of the Danube river system, along with the dense network of gauging stations, provides an ideal setting to investigate the role of explicit runoff routing in R–R modeling, particularly through graph-based approaches. Three daily meteorological and hydrological variables provided in the LamaH-CE dataset including precipitation, soil moisture (fraction of water in topsoil layer 0 to 100 cm depth), and 2 m air temperature are derived from the ERA5-Land reanalysis (Muñoz-Sabater et al., 2021) and serve as the dynamic inputs for the R–R model. Crucially, these dynamic inputs are spatially averaged over the entire upstream catchment contributing to each gauge, providing a single representative value per basin per day (Klingler et al., 2021). In addition to these time-varying forcings, LamaH-CE offers 59 static catchment attributes for each of the 530 selected subbasins. These static descriptors capture essential physical and environmental features, including topography, climatological norms, hydrological signatures, land cover classifications, vegetation indices, soil characteristics, and geological formations. Similar to the dynamic variables, these static attributes are pre-processed and provided as basin-averaged values.

We propose a novel LSTM–GNN model to predict the R–R process by jointly capturing local runoff generation and basin-scale flow routing within a unified framework. In contrast to traditional lumped models that treat the catchment as a single unit, our approach partitions the basin into multiple hydrologically connected subbasins, each represented as a node in a graph, with nodes corresponding exclusively to gauged subbasin outlets. At each subbasin, an LSTM unit processes the time series of meteorological inputs including precipitation, soil moisture, and 2 m air temperature to model the temporal evolution of runoff. The output of each LSTM serves as a latent embedding, a vector representation that summarizes the subbasin's runoff response and hydrological state. These node-level embeddings are passed into a GNN, which models spatial interactions across the river network. The river system is represented as a directed graph, where edges reflect downstream flow connections between subbasins. Through messages passing along this graph structure, the GNN propagates and aggregates information from upstream to downstream nodes, enabling explicit modeling of runoff routing and flow accumulation consistent with real-world hydrological connectivity. Importantly, unlike some existing LSTM–GNN models that incorporate historical streamflow as an input (e.g., Deng et al., 2024; Wang et al., 2025), our model excludes streamflow observations. While such data can enhance prediction accuracy in gauged basins, it is inherently unavailable in ungauged regions. By relying solely on meteorological inputs, our framework remains applicable to both gauged and ungauged settings.

As mentioned above, three dynamic variables and 59 static variables are used as input features. All input features are normalized using a positive_robust_log transform for precipitation and streamflow, and min–max scaling [0,1] for other variables, before being used into the models. The historical observed streamflow records serve as the target output (ground truth) for training, validation, and testing. For temporal modeling, a sliding window of 180 d of past data is used as the input sequence, and the model learns to forecast the discharge for the next day. The dataset consists of daily records from 1 January 1987 to 31 December 2017. The dataset was divided into 70 % training and 15 % validation samples selected randomly, and the remaining 15 % (the last part of the time series) was used for testing. To address the inherent class imbalance in hydrological data where extreme discharge events are rare but critically important for flood prediction, we implement a targeted data augmentation strategy. We identified extreme discharge events by selecting the top 2.5 % of maximum discharge values from each subbasin. These events were then augmented by creating four additional copies, increasing their overall representation in the training dataset from 2.5 % to approximately 10 %. This augmentation approach ensures that the model receives sufficient exposure to high-discharge patterns during training, improving its ability to predict flood events while maintaining the overall temporal structure of the time series data. The augmentation is applied only to the training set to prevent data leakage into the validation and testing phases. All models are implemented in PyTorch and trained on a GPU (NVIDIA A100 40GB) to accelerate computation, given the long time series and model complexity.

This study set out with the hypothesis that integrating a GNN-based routing component into a R–R model would yield improved prediction accuracy over a standalone LSTM model. The results strongly support this hypothesis. Across all performance metrics including NSE, KGE, CC, and RMSE the LSTM–GNN model significantly outperformed the baseline LSTM, which lacked explicit routing. In other words, explicitly modeling runoff routing via a GNN led to more accurate streamflow predictions. Our findings are consistent with Cortés-Salazar et al. (2023) demonstrated that including routing in a physical model significantly improved performance (KGE increased from 0.64 without routing to 0.81 with routing). Similarly, Kraft et al. (2025) showed that incorporating routing into lumped LSTM baselines across Switzerland yielded substantial performance gains (KGE improvements of 24 %–62 %). The analysis further revealed that the benefits of GNN-based routing vary across the river network, with the largest performance improvements occurring at stations with larger upstream contributing areas and stronger network connectivity. To illustrate this, we analyzed a cluster of three stations within a connected sub-network of the Isel River system: Matreier Tauernhaus (Station 532) on the Tauernbach River, Waier (Station 530) on the Isel River, and Bruehl (Station 533) also on the Isel River (Fig. 9), with a combined contributing catchment area of approximately 518.4 km2. These stations are arranged along a flow path from upstream to downstream, with Station 530 representing a major junction that aggregates flow from multiple tributaries. At the upstream site (Station 532), where flow is primarily driven by local precipitation and routing effects are limited, the performance difference between LSTM and LSTM–GAT was small, and the baseline model slightly outperformed the GNN-enhanced model. However, at the downstream junctions (Stations 530 and 533), the LSTM–GNN achieved much higher NSE values (0.92 vs. 0.82 at Station 530; 0.90 vs. 0.76 at Station 533). Hydrograph comparisons confirm that the GNN-based model reproduced peaks more accurately, improving both magnitude and timing, whereas the baseline LSTM systematically underestimated high-flow events. These results demonstrate that GNN-based routing is particularly effective in downstream locations where hydrological signals accumulate from multiple upstream areas. Importantly, performance gains were also observed at gauges with only two or three upstream connections, suggesting that even relatively sparse networks can benefit from explicit routing.

At the basin scale, the largest improvements occurred in large, lowland subbasins, and catchment size was positively correlated with performance gains. This indicates that larger basins, where extensive flow accumulation and routing dominate, particularly benefit from GNN-based modeling. This addresses a well-known limitation of LSTM models in large river basins, where their lack of explicit routing can hinder performance. Conversely, in high-slope headwater basins and at the most upstream gauges, the baseline LSTM often performed equally well or better, reflecting the fact that hydrological response in these locations is primarily driven by local precipitation and rapid runoff processes, with minimal routing influence.

Among the different LSTM–GNN configurations tested, the model using a GAT achieved the best performance. It delivered the highest NSE, KGE, and correlation values, outperforming GCN, GraphSAGE, and ChebNet. The improved results of the GAT-based model likely stem from its ability to assign adaptive weights to different upstream neighbors during message passing. In hydrological terms, this means the model can learn which tributaries or upstream catchments exert a stronger influence on downstream flow, rather than treating all upstream nodes equally. This adaptivity is particularly important in heterogeneous networks such as the Danube Basin, where tributaries differ greatly in size, slope, and hydrological response. Our results align with Deng et al. (2024), who also reported that attention-based graph architectures outperform others in hydrological prediction tasks. Together, these findings highlight that capturing heterogeneous upstream contributions is essential for accurate routing, making the LSTM–GAT framework the most effective among the tested models.

To provide a practical perspective on model efficiency, we also recorded the average training time per epoch for each architecture. The baseline LSTM required approximately 56 s per epoch, whereas the LSTM–GNN models ranged from 68 s for LSTM–GAT and 69 s for LSTM–GCN to 73 s for LSTM–GraphSAGE and 79 s for LSTM–ChebNet. As expected, the LSTM–GAT required roughly 20 % longer training time than the baseline, reflecting the additional computational cost of graph-based message passing. However, this increase remains modest relative to the performance gains achieved. Moreover, inference time, relevant for real-time or operational applications, was comparable across all models, indicating that the GNN-based extensions do not introduce substantial computational overhead during prediction.

This study introduced a novel LSTM–GNN framework for R–R modeling that explicitly integrates runoff generation and runoff routing within a unified deep learning architecture. By leveraging LSTMs to capture local temporal dynamics and GNNs to model spatial dependencies across the river network, the proposed approach addresses a major limitation of existing data-driven hydrological models: the absence of physically consistent flow routing. Applied to the Upper Danube River Basin, our model demonstrated significant improvements over a baseline LSTM model that neglects explicit routing. Among the tested GNN variants, the GAT emerged as the most effective, achieving the highest mean NSE (0.61), KGE (0.65), and CC (0.84), while reducing RMSE by approximately 35 % compared to the baseline. These enhancements were particularly pronounced in downstream stations with high network connectivity and large contributing areas, where routing effects dominate hydrological responses, underscoring the value of adaptive message passing in capturing heterogeneous upstream influences.

The findings affirm our hypothesis that integrating GNN-based routing enhances predictive accuracy, especially in complex, large-scale basins like the Danube, where flow accumulation and delays play a critical role. This approach not only improves streamflow forecasting but also advances the physical interpretability of deep learning models by aligning their structure with real-world hydrological processes. While we did not specifically test predictions in completely ungauged basins, the model's design, relying solely on meteorological and static catchment data without the need for past flow observations, enables such applications in principle. Future work should evaluate the model's performance in ungauged basins to validate its generalizability and assess its potential for water resource management, flood risk assessment, and climate change adaptation in data-scarce regions.

Despite these advancements, opportunities for further refinement remain. Future work should explore evaluating the model across diverse global basins to validate its generalizability. Future work could also investigate transfer-learning strategies such as regional pre-training followed by limited subbasin fine-tuning, which may improve performance in hydrological outliers and enhance model transferability to unseen basins. Incorporating dynamic edges in the GNN model using variables like soil moisture could be another approach to test. In addition, hybrid comparisons that combine graph-based routing with simple process-based runoff or routing schemes could help further clarify the complementary roles of physical and data-driven approaches. Ultimately, this study highlights the transformative potential of graph neural networks in hydrological modeling, paving the way for more spatially aware and accurate predictions in an era of increasing environmental challenges.