For our model-intercomparison study, we assembled streamflow observations, catchment-averaged meteorological data, and static catchment attributes for a large sample of 918 catchments in Central Europe (Fig. ). The study domain covers the European Alps and the lowland regions around them, specifically the three major river basins originating in the Central Alps: the Danube, Rhine, and Rhône. It includes catchments in southern Germany (Bavaria and Baden-Württemberg), Austria, Switzerland, and eastern France (Rhône and Rhine basins). Streamflow data for these regions were collected from national agencies in Austria (Austrian Ministry of Sustainability and Tourism), Switzerland (Federal Office of the Environment, FOEN), and France (Banque HYDRO). Regional offices provided the streamflow data for the German states of Bavaria (Bavarian State Office for the Environment) and Baden-Württemberg (State Institute for the Environment Baden-Württemberg). We had to exclude Italy from our analysis because none of the streamflow time series collected for northern Italy (Piemonte, Lombardia, Valle d'Aosta) had data records for the cold period that were sufficiently long for model training.

We selected near-natural catchments only, as we wanted to assess how well the different model types represent the link between climatic variables and streamflow. To identify these catchments, we used information on reservoir locations and characteristics collected from national and regional agencies (Austria: , Switzerland: Swiss Federal Office for Energy, Germany: , France: , see , for details). We limited ourselves to catchments with a degree of regulation (reservoir volume relative to annual streamflow volume) below 10 %. Following , we also excluded catchments which are smaller than 50 km² or have anomalous streamflow signatures (mean specific discharge smaller than 0.1 mm d⁻¹ or larger than 10 mm d⁻¹). We also excluded catchments with insufficient data records (fewer than 10 years of streamflow data in the 1960–1990 period), too many missing values (more than 10 %) or with problems in the data, such as multiple observations for a single day or too many repetitive values (more than 50 %).

We extracted a set of catchment-averaged daily meteorological variables from the E-OBS v29.0 gridded dataset at a resolution of 0.1° , namely precipitation, minimum, mean, and maximum daily temperature, potential evapotranspiration, and incoming shortwave radiation. The aggregation was performed using area-weighted means within catchment boundaries with the Python Xagg library . Additionally, we compiled SWE data for Switzerland and Austria as reference datasets for evaluating the snowpack simulated by the conceptual and hybrid models. For Austria, we used the SNOWGRID-CL dataset, which represents SWE over Austria on a 1 km × 1 km grid and covers the time period from January 1961 until 24 April 2024. It was created using a 2-layer energy balance model with data assimilation of snow height measurements. For Switzerland, we used the OSHD-CLQM dataset which is based on a simple temperature index model with quantile mapping and is available on a 1 km × 1 km grid from September 1961 until September 2021.

From the observed streamflow, the catchment-averaged meteorological variables, and other data sources, we derived a set of hydroclimatic catchment characteristics (Table ). Catchment characteristics describing the land-use, geology, topography, soil composition and glacier characteristics were derived from the HydroATLAS dataset using the procedures proposed in the CARAVAN dataset .

In this study, we considered three types of models and compared them with respect to four key metrics for hydrological climate impact assessments (see Fig. a for an overview of the workflow). We compared (1) their absolute accuracy, (2) their generalizability to a warmer climate (i.e. change in accuracy between the cold training period and the warmer testing period) and (3) their ability to simulate extreme events in ungauged basins, and (4) their ability to simulate observed streamflow signals. The three types of models are: (1) a conceptual process-based model, specifically the HBV model, (2) a data-driven model, specifically an LSTM, and (3) a hybrid model, specifically the differentiable hydrologic δ-HBV model. To compare accuracy and generalizability metrics between two hydrological models, we used the sign test , which determines the statistical significance of differences between models across catchments. For metrics where larger scores correspond to better accuracy (such as the NSE and KGE), we used the one-sided sign test, for all other metrics (such as biases), we used the two-sided sign test.

All models were calibrated with data from the 1960–1990 period, which we used as a reference period representing cold climate conditions. To ensure consistency between the three models, we used the Nash-Sutcliffe efficiency (NSE; ) to calibrate all models. We chose the NSE because it is the most commonly used objective function when training LSTMs for streamflow modelling. Specifically, we used the classical NSE on a catchment level for the HBV model and the smooth-joint NSE proposed by for the LSTM and hybrid models. The HBV model was calibrated for each catchment using 1000 iterations of the DDS Algorithm . For predictions in ungauged basins, we used the regionalization scheme described in Sect. . The two deep learning models were trained using the Adam algorithm . It is common practice to standardize the input and output data of deep learning models to improve the stability of the optimization procedure . We therefore standardized the static catchment attributes and input time series passed to the LSTMs to have zero mean and unit variance, while the inputs to the conceptual model were left unchanged. Following , the specific discharge time series were divided by the 95th streamflow quantile across all catchments when training the stand-alone LSTM. Together with the ReLU activation function, this ensures positive predictions.

To determine the best hyper-parameters for the optimization of the LSTM and hybrid models, we performed a small grid search using a validation set consisting of 128 catchments. The validation set consisted of catchments which had more than 5 years of streamflow data in the 1960–1990 period but did not meet our requirement of having at least 10 years of data in that period. These catchments were not used for training or any further analyses. The set of hyper-parameters determined with this approach can be found in Table . With these hyper-parameters, we trained an ensemble of models using spatial cross-validation by randomly splitting the catchments into 5 equally sized folds. For each target fold, the remaining 4 folds were used to train 5 models with different random initializations. Thus, we trained 25 LSTMs and 25 hybrid models in total. We then averaged the predictions of the 5 models to obtain a single ensemble prediction for every fold. Every catchment was thus held out exactly once from training.

To evaluate the generalizability of the different models across different climatic conditions, we quantified their performance for a variety of metrics under a cold, warm and warmest period (Fig. a). This allowed us to assess the change in accuracy when predicting streamflow in a warmer climate, for which the models were not trained. We chose these periods based on the decadal mean temperature (Fig. b). During the cold period (1960–1990), the mean temperature was relatively constant and lower than in later decades. Therefore, it was chosen as the training period. For the warm period (testing), we used only decades for which the decadal mean is +1 °C above that of the period 1960–1990, resulting in the testing period 2000–2023, to ensure a substantial warming effect between the training and testing periods. To assess the effect of even stronger warming on model accuracy and generalizability, we also considered a second evaluation period, namely the “warmest” period, consisting of the 5 warmest years during the warm period for each catchment. These three scenarios (cold, warm and warmest) are distinct in terms of their climatic conditions and streamflow responses (Fig. c–g). In terms of the median across catchments, the mean temperature increased by 1.4 °C between the cold and the warm period and by 2.0 °C between the cold and the catchment-specific warmest periods. The largest differences between the warm and warmest periods are observed in the temperature and snowfall variables, while potential evaporation and precipitation only show small changes.

In addition to the out-of-time evaluation, given by the three different periods, we also evaluate the models for the prediction of streamflow in ungauged basins (i.e. out-of-sample) in a total of 918 catchments. Specifically, we evaluated the models on the subset (fold) of catchments which was not used to calibrate or train the models. The combination of the out-of-time and out-of-sample evaluation is necessary to limit the influence of overfitting on the training data. This enables us to compare the model simulations across climatic periods without relying on training data. The HBV model was also evaluated in gauged catchments to evaluate the effectiveness of the regionalization scheme and provide a reference model typically used for climate change impact analyses. We refer to this model as the in-sample HBV as opposed to the regional HBV. To evaluate the overall model accuracy, we used the following evaluation metrics (see also Table ): the NSE and the Kling-Gupta efficiency (KGE; ) as well as its components, namely bias (βKGE), variance (αKGE), and correlation (r) . Please note that we evaluated the hybrid model separately for the 1960–1990 and 2000–2023 periods to allow the model to adapt the static parameters to the climate conditions of the different periods, resulting in two distinct parameter sets. For the 5 warmest years, we did not conduct a separate evaluation and the static parameters were thus the same as those of the warm period.

In addition to evaluating general model accuracy, we evaluated the models with respect to how well they simulate extreme events such as peak flows, low flows, and drought events (i.e. seasonal low flow anomalies). Specifically, we considered the following flow metrics for each catchment: the high flow percentage bias (HFPB), the low flow percentage bias (LFPB), and the drought volume percentage bias (DVPB). The HFPB was defined as the percentage bias for days during which the observed streamflow exceeded the observed 99th quantile, that is HFPB=∑t(Q^t-Qt)∑tQt⋅100 where the index t runs over only days with Qt larger than the 99th quantile. The LFPB was defined using the annual minimum flow rate averaged over a 7 d window (NM7Q) NM7Q=min⁡t=0,…,T-617∑j=06Qt+j. We computed the NM7Q for each hydrological year for the observed and simulated streamflow time series. To obtain a catchment-level metric, we considered the percentage bias of the yearly NM7Qs defined as LFPB=∑yNM7Q^y-NM7Qy∑yNM7Qy⋅100 with simulated NM7Q^y and observed NM7Qy for the hydrological year y. Finally, we considered the DVFP metric based on drought events defined using a seasonally varying threshold approach . The observed streamflow data was smoothed by taking running averages and a seasonally varying threshold was determined for each day of the year as the 20th percentile for a 30 d window around that day. This threshold was used to detect drought events as periods during which the observed streamflow lies below the threshold for at least 30 consecutive days. The DVPB for a catchment was then defined as the percentage bias between simulated and observed flow volumes during drought events DVPB=∑e∈E(V^e-Ve)∑e∈EVe⋅100 where E is the set of drought events detected for that catchment, e is an individual event and Ve,V^e are the observed and modelled flow volumes for that event.

As an additional measure of generalizability and to compare the sensitivity of the models, we compared the changes in mean flow, the NM7Q and the annual 99th streamflow percentile between the two testing periods (warm and warmest) and training period (cold) in observations with the simulated change signals by the three models. Further, we calculated catchment-by-catchment differences of these change signals. We tested for significant differences among models with a two-sided sign test.

Next, we compared model accuracy and generalizability for snow- vs. rainfall-dominated catchments across the different model types to assess whether these two aspects depend on the fraction of snowfall. We defined snow-dominated catchments as catchments where snowfall makes up more than 30 % of precipitation (during the cold period) and compared model accuracy in these catchments to the one of rainfall-dominated catchments (less than 30 % snow fraction). Snowfall was defined as precipitation falling on days with air temperatures below 0 °C. We excluded catchments with significant glacier area (larger than 2 % of the total catchment area according to RGI v2.0; ) from this model evaluation since the HBV and hybrid models do not explicitly account for glaciers. In order to understand potential model differences in terms of model accuracy and generalizability in snow-dominated catchments, we looked at how the HBV and hybrid models represent snow. As the LSTM does not explicitly model SWE, we disregarded this model for this analysis. We compared the snow-water equivalent (SWE) modelled by the HBV and hybrid models to observed SWE values derived from gridded SWE datasets for Austria and Switzerland. We focused on these two countries because there are no gridded observational SWE datasets of comparable quality for France and Germany.

For a general comparison of model performance across the three model types, we first focus on model accuracy in ungauged basins during the cold (training) period (Fig. , Table ). Out of the three models, the LSTM has the highest accuracy, with a median KGE (NSE) of 0.76 (0.79). The hybrid and conceptual regional HBV models show a lower accuracy of 0.72 (0.71) and 0.60 (0.56). The regionalization of the parameters of the HBV model results in a reduction in accuracy by 0.15 (0.10) as compared to the in-sample HBV from a median KGE (NSE) of 0.75 (0.66) to 0.60 (0.56).

To assess how well the different models represent streamflow under warmer conditions, we compare general model accuracy across the three model types in terms of KGE when applying the models under warmer climatic conditions than those observed during training. Model accuracy varies for all models depending on the period considered and decreases more strongly with the degree of extrapolation (Fig. a). During the cold period, the LSTM performs significantly better than the hybrid model (p value <0.05 for a one-sided sign test). It also performs significantly better than the HBV model during all periods. For the extrapolation periods (warm and warmest), the difference in accuracies between the LSTM and hybrid model are statistically insignificant (p value >0.05 for both alternatives of the one-sided sign test). At the same time, the hybrid model has a significantly higher accuracy than the HBV model in all periods.

To compare the generalizability of the three models, we now investigate the per catchment change in accuracy between the cold period and the two extrapolation periods (warm and warmest; Fig. b). The LSTM shows a large median change in KGE (NSE) -0.05 (-0.03) when evaluating it on the warm instead of the cold period, while the hybrid and HBV models show a smaller and similar change of -0.02 (0.00) and -0.02 (0.00). This results in median KGE (NSE) scores of 0.71 (0.75) for the warm period for the LSTM, 0.70 (0.71) for the hybrid model, and 0.58 (0.54) for the conceptual HBV model evaluated in ungauged basins. Considering only the 5 warmest years in each catchment leads to even stronger accuracy reductions with respect to the cold period. These are again strongest for the LSTM as compared to the other models, with changes of -0.06 (-0.04) in the KGE (NSE) for the LSTM and of -0.03 (-0.02) and -0.02 (-0.01) for the hybrid and conceptual models, respectively. For this warmest period, the median KGE score of the hybrid model approaches that of the LSTM (0.69 for both), indicating that they both perform equally well under strong extrapolation in terms of absolute accuracy. While the change in accuracy of the regional HBV model is similar to that of the hybrid model, its absolute accuracy (KGE of 0.56) still lies clearly below the one of the LSTM (KGE of 0.69). Even though it generalizes well, the HBV model still has a lower absolute accuracy during extrapolation to very warm conditions than both types of deep learning models. The change in accuracy is not significantly different between the HBV and hybrid models (p value >0.05 for a two-sided sign test) for both periods, while there is a significant difference between the change in accuracy of the LSTM and that of the other two models (p value <0.05 for a two-sided sign test). The accuracy of the in-sample HBV model is comparable to that of the LSTM and hybrid models for the warm and warmest periods with a median KGE (NSE) score of 0.69 (0.63) and 0.68 (0.61), respectively (see Table ). In terms of generalizability, the in-sample HBV shows a larger reduction in accuracy between the cold period and the extrapolation periods than the HBV and hybrid models, with a median change in KGE (NSE) of -0.05 (-0.02) and -0.05 (-0.03) for the warm and warmest period respectively. This is comparable to the change in KGE (NSE) of the LSTM model.

Looking at the different components of the KGE (Table ), we find that the change in KGE of the LSTM is largely explained by reductions in the bias and variability terms. The simulations derived with the LSTM and hybrid models most closely match the observed mean flow in the training period, with bias terms βKGE of 0.99 and 1.01 respectively (values above and below 1 indicate an overestimation and underestimation of the mean flow, respectively; see Table , Fig.  in Appendix C). In the warm period, the bias of the LSTM more strongly increases (βKGE of 0.94), than the one of the hybrid and HBV models (βKGE=0.97). In the warmest period, the hybrid and HBV models but not the LSTM match the observed mean flow even better than in the warm period (bias terms of 0.99, 1.00, 0.95). Considering the median bias across all catchments, the LSTM underestimates the mean flow during the warm and warmest periods by 6 % and 5 %, respectively.

Overall, the LSTM experiences the strongest accuracy reduction between the cold period and the two extrapolation periods among the three model types considered. In absolute terms, the LSTM is more accurate than the regional HBV during extrapolation in all metrics but the bias term (βKGE), while it is on par with the hybrid model with respect to most evaluation metrics. In terms of the NSE metric, the LSTM is the most accurate model in all periods. The three model types do not generalize equally well to warmer climate conditions in all catchments. We find that particularly the hybrid and HBV models generalize less well in catchments with higher mean specific discharge than in catchments with smaller specific discharge (Sect.  and Fig. a). Such high mean specific discharge is associated with higher elevation catchments with larger snowfall fractions (Sect. ).

To assess how model accuracy and generalizability for the three model types compare for extreme events, we investigate the ability of the models to represent high and low flows across the three climate periods (Fig. , Table ). In general, all models tend to underestimate flood peaks during the cold training period (Fig. a). During this period, the LSTM is more accurate in representing flood peaks than the hybrid model (HFPB of -8.2 % and -11.8 %, respectively). The hybrid model in turn is more accurate than the regional HBV model (-17.7 %) and the in-sample HBV (-13.5 %, not shown in figure). While the LSTM and hybrid model show similar accuracies in the warm period (-13.7 % and -13.3 %), the hybrid model (-10.7 %) has a notably higher accuracy than the LSTM (-13.8 %) in the 5 warmest years. While having a lower accuracy than the hybrid model when extrapolating strongly, the LSTM still has a higher accuracy than the regional HBV during both testing periods, which has percentage biases of -18.5 % and -16.4 % for the warm period and the 5 warmest years, respectively. Similarly, the LSTM has a higher accuracy during extrapolation than the in-sample HBV, which has percentage biases of -15.9 % and -16.6 % for the warm period and the 5 warmest years.

To compare model accuracy and generalizability for low flows, we on the one hand focus on absolute low flows and on the other hand consider drought events representing seasonal flow anomalies. While some model types tend to overestimate low flows, others underestimate them (Fig. b–c). The LSTM mostly overestimates absolute low flows with median LFPB of 9.9 %, 8.6 % and 11.8 % for the cold, warm and warmest periods (Fig. b). In contrast, both the hybrid and HBV model underestimate low flows with LFPBs for the cold, warm, and warmest period of -12.1 %, -10.9 % and -10.3 % for the hybrid model and -10.0 %, -5.6 %, -5.0 % for the HBV. While the HBV model has a median across catchments closest to zero for the warm and warmest periods, it also has the largest interquartile range of all models for all periods. Considering the median across all catchments, the hybrid and HBV models are thus more accurate at modelling absolute low flows during the warm and warmest period than during the cold period. Similarly, the LSTM also models low flows more accurately in the warm period than in the cold period. For drought events defined as seasonal low flow anomalies, the three model types behave in a more consistent way than for absolute low flows as all of them tend to overestimate streamflow during drought events (Fig. c). Among the three model types, the hybrid model has the highest accuracy in all periods with median percentage biases of 9.5 %, 4.9 %, 8.8 % for the cold, warm, and warmest periods. The HBV model can compete with the hybrid model in the cold period, but has considerably lower accuracy in the warm and warmest periods with median DFPB of 8.8 %, 10.9 % and 15.5 %. The LSTM is the least accurate model in the cold period, but performs better than the HBV model in the warm and warmest periods with percentage biases of 10.4 %, 9.1 %, and 9.7 %. Similarly to when considering absolute low flows, the HBV model again shows the largest accuracy spread across catchments among the three models. For droughts, only the regional HBV model shows decreasing accuracy in the extrapolation periods compared to the cold period. The LSTM has similar percentage biases in all periods, while the hybrid model is better at representing droughts in the warm period than in the cold period.

Next, we look at generalizability from the perspective of trend preservation. For each catchment, we compare the absolute delta in the change signal between the warm (warmest) and cold period simulated by the models to the one derived from observations. Good generalizability is given when models capture the magnitude and direction of change detected in observations, both at regional and catchment scale. Overall, observations show a decreasing trend in daily mean streamflow (MQ) in the warm period compared to the cold period (-4.6 %), with even stronger decreasing signals for the warmest years (-9.4 %) (see Fig. a). All models are able to capture the overall decreasing trend in MQ in the region, and all show a stronger signal in the warmest compared to warm period. The LSTM shows stronger trends in both periods (warm: -8.5 %; warmest: -12.6 %) compared to observations and the other models, and has a median catchment level signal bias of -3.8 % pts in the warm period and -2.8 % pts in the warmest period (Fig. d). The Hybrid model overestimates the regional signal in the warm period as well (signal: -7.6 %, bias: -3.0 % pts), but compares well with observations in the warmest period (signal: -10.1 %, bias: -0.2 % pts). The HBV model shows signals in the warm period that are comparable to those in observations (signal: -5.0 %; bias: -0.2 % pts), but slightly underestimates signals in the warmest period (signal: -7.4 %; bias: +2.0 % pts). Similarly, high flows (the mean 99th percentile of daily streamflow in a year) are decreasing in the observations and most catchments (warm: -3.6 %, warmest: -9.7 %; Fig. b). Again, all models generally agree with observations on a decrease in high flows, as well as even lower high flows in the warmest period compared to the cold period. In general, the LSTM again overestimates the decreasing streamflow signals in the warm period (signal: -7.0 %, bias: -4.7 % pts) and warmest period (signal: -13.4 %; bias: -5.5 % pts) (Fig. b, e). The Hybrid and HBV model show comparable signals to observations in both periods (Fig. b) and show catchment level signal biases that are below 1.5 percentage points (Fig. d). For low flows (mean annual NM7Q) the observations show a small overall change in the warm period (-1.1 %), but a stronger increase in low flows in the warmest period (-9.2 %; Fig. c). In the warm period, the Hybrid model compares best with observed low flow signals followed by the LSTM, while HBV shows slightly positive low flow signals. However, the LSTM shows a considerably lower signal spread across catchments than observations and the other models. In the warmest period, the LSTM shows the best match with observed low flow signals followed by the Hybrid model, while the HBV model shows a considerable positive bias in the signals. From the perspective of trend generalizability, the Hybrid model performs best among the three models followed by the HBV model, with the exception of low flows. The LSTM generally shows the same direction of change as observations, but has a slightly enhanced sensitivity compared to observations and the other models.

As previously mentioned, the hybrid and HBV models generalize worse in catchments with large mean specific flows than in catchments with lower specific flows (Sect.  and Fig. ). The catchments with large specific flows are often located at high elevations and strongly snow influenced. Since both types of models explicitly parametrize snow, the question arises how such models generalize to warmer climatic conditions in snow-influenced catchments compared to the purely data-driven LSTM model, which does not explicitly represent snow. To address this question, we compare the model simulations of the three model types for snow-dominated and rainfall-dominated catchments in terms of general model accuracy described by KGE (Fig. a–b). During the cold period, the accuracy of all three model types does not differ between the two groups of catchments. That is, model accuracy is as good for snow-dominated catchments as it is for the rainfall-dominated ones. Across both types of catchments, the LSTM performs best out of all models, followed by the hybrid and the HBV model. Notable is the larger spread in the interquartile range of the hybrid model across snow-dominated vs. rainfall-dominated catchments. For snow-dominated catchments, the pattern persists for the two extrapolation periods, i.e. the LSTM keeps showing significantly higher accuracy than the hybrid and regional HBV models (p value <0.05 for the one-sided sign test). For rainfall-dominated catchments, the hybrid model is on par with the LSTM similarly to the previous observations for all catchments combined (p value <0.05 for the cold period and p value >0.05 for the warm period and 5 warmest years) (Fig. ). The LSTM is still significantly more accurate than the HBV for all periods in rainfall-dominated catchments (p value <0.05).

In contrast to absolute accuracy, the per catchment change in accuracy between the cold period and the two extrapolation periods shows clear differences between snow- and rainfall-dominated catchments. For snow-dominated catchments, all models exhibit a similar reduction in accuracy when extrapolating to warmer climate conditions (Fig. c, p value >0.05 for all combinations in the two-sided sign test). In contrast, the accuracy reduction during extrapolation depends on the model for rainfall-dominated catchments. There, the hybrid and the HBV model have a smaller and similar (p value <0.05) reduction in accuracy, while the reduction in KGE of the LSTM is larger and significantly different (p value <0.05) from that of the two other models.

To assess the seasonality of the observed and modelled streamflow regimes in snow-dominated catchments, we consider the discharge averaged by day of the year. We find that the model bias has a distinct seasonal pattern with both the HBV and hybrid model underestimating streamflow during the snowmelt season (April–July) in all periods (Fig. ). The LSTM accurately models the annual streamflow curves during the cold and warm periods, but it underestimates summer flow during the 5 warmest years (Fig. c). This is also reflected in the bias metrics. The hybrid and HBV models underestimate mean streamflow by 14 % and 24 % in snow-dominated catchments during the warm period, while the LSTM only shows an underestimation of 7 % (Table ), with similar bias values for the warmest periods.

To further assess the issue of the hybrid and HBV model in terms of flow estimation during the melt season, we look at the internal snowpack variable of these models. Specifically, we compare the SWE simulated with these two models to catchment averages derived from gridded SWE benchmark datasets (OSHD-CLQM in Switzerland and SNOWGRID in Austria). Both models simulate similar SWE estimates and underestimate snowpack throughout the year, both in Swiss and Austrian catchments (Fig. ). To quantify this SWE underestimation effect, we compute the same metrics also used to evaluate streamflow for simulated SWE (Table ) and find that the hybrid and HBV models underestimate the mean SWE by 24 % and 22 % respectively in the cold period. The degree of underestimation increases during extrapolation from the cold to the warm periods, with the hybrid model underestimating the mean by 26 % and 33 % for the warm and warmest periods and the HBV model underestimating it by 27 % and 37 %. Both regional models have lower bias than the in-sample HBV model in all periods.

Our results show that the LSTM is clearly the most accurate model in terms of modelling streamflow in ungauged basins, i.e. basins lacking streamflow observations for model training (Fig. ). However, it suffers from poor generalizability under changes in climate (Fig. ). This finding is consistent with findings of prior studies which have also found larger accuracy reductions for LSTMs than for conceptual models when evaluating them under climate conditions different from those used during training . In contrast, the hybrid model shows better generalizability, which comes at the cost of slightly reduced accuracy when evaluating it under climate conditions more similar to those of the training period. These results align with previous research showing that hybrid models are better at generalizing to climatic zones not used in training compared to a stand-alone LSTM . The regionalized conceptual model is significantly less accurate than the other two models (Fig. a). However, the conceptual model shows a small reduction in accuracy similar to the one of the hybrid model, which indicates a similar generalizability to warmer climate conditions as the one of the hybrid model (Fig. b). The hybrid model thus effectively combines the strong performance of the LSTM when predicting in ungauged basins with the generalizability of the conceptual HBV model. In years that are +1.4 °C warmer than the average temperature of the cold period, the hybrid model has an accuracy that is indistinguishable from the one of the LSTM. The LSTM, while not generalizing as well as the hybrid and HBV model, still has a significantly higher accuracy than the HBV model even in the warmest period (+2 °C). The lower generalizability of the LSTM compared to the hybrid and HBV model also holds when comparing observed and simulated streamflow signals. The LSTM simulations show a higher sensitivity to changes in meteorological forcings than observations (Fig. ), that is stronger declines in mean and high flows. This might indicate that when moving to even warmer climate conditions, such as future climate projections, the LSTM might project different streamflow changes differently than the other models. A higher streamflow sensitivity to temperature changes by the LSTM compared to conceptual models has previously also been shown by .

The finding that the LSTM is the most accurate model during the training period, while the hybrid model generalizes best to warmer conditions than those used in training, also holds for extreme events. Our results show that the LSTM most accurately models floods and absolute annual low flows during the training period (Fig. a–b). In contrast, all models have similar but lower accuracy for drought deficits, with the hybrid model being the most accurate of the regional models (Fig. c). Differences in model accuracies for floods and droughts are common in hydrological modelling irrespective of the objective function used for training and model performance often deteriorates under drought conditions . The finding that the LSTM is more accurate in ungauged basins than the hybrid model when considering extreme events is in line with previous research . When generalizing to warmer climate conditions, however, the hybrid model most accurately models floods and drought volumes, while the regional HBV model most accurately models annual low flows. Under these warmer conditions, the hybrid model not only equals the absolute accuracy of the LSTM in extrapolation but even surpasses it. The LSTM in turn models floods more accurately than both the in-sample and the regional HBV model and has lower drought volume bias than the regional HBV. The regional HBV model does not just show lower accuracy than the other two models in terms of most metrics, it also shows a large spread in the distribution of the extreme event metrics, indicating large accuracy differences between different catchments. This is in line with findings from , who show that hybrid models have a smaller percentage bias when simulating peak flows in climatic regions withheld from the training set.

While model generalizability varies across model types when considering all catchments (Fig. ), the models show similar generalizability in snow-dominated catchments (Fig. c). In these catchments, the LSTM has the highest absolute accuracy during the cold period and is also the most accurate model in all periods. Similarly, have found LSTMs to outperform hybrid models in cold and polar climates. This contrasts with rainfall-dominated catchments, where the HBV and hybrid model generalize significantly better than the LSTM to periods with warmer climate conditions (Fig. ). These findings suggest that an explicit parametrization of snow and other processes does not always improve generalizability and that process-based approaches can have their own problems under distribution shifts of input variables. Our investigation of simulated SWE for the hybrid and HBV models revealed that both types of models systematically underestimate the water content of the snowpack (Fig. ). This indicates that there are biases in the forcing data and/or that the models are parametrising snow in a suboptimal way. Precipitation undercatch is one form of bias which is particularly prevalent in snow-dominated catchments . While the LSTM can correct for such biases (Fig. , ), the hybrid and HBV models cannot (Figs. , ). This could be an explanation for the underestimation of SWE and summer streamflow in snow-dominated catchments observed in the hybrid and HBV models (Figs. , ). HBV models calibrated for a specific site often have a snowfall correction factor to account for this type of bias, but such a factor is difficult to regionalize and was therefore not included in the conceptual models used in this study. To still account for undercatch, one could directly correct observational precipitation data before feeding it into the hydrological model .

Our findings show that both types of deep learning models (purely data-driven LSTM and hybrid model) achieve the same accuracy in ungauged basins as the in-sample HBV model, even when extrapolating to warmer climatic conditions. Together with their strong ability to extrapolate to ungauged regions, these findings highlight the potential of data-driven and hybrid models for climate change impact assessments in data-scarce regions. The promise of deep learning streamflow models extends to gauged basins because LSTMs and hybrid models have even higher absolute accuracy when evaluated in the catchments they were calibrated in . The strengths of LSTMs and hybrid models extend from general streamflow simulations to extreme events, i.e. high flows and annual low flows (Fig. ). This is crucial for climate change impact assessments since hydrological extreme events such as floods are among the most prevalent natural disasters , particularly affect low- and middle-income countries , and are already changing in response to increasing temperatures . Our results thus suggest that deep learning based models, in particular hybrid models, should be considered as an alternative to conceptual models for climate change impact assessments. In snow-dominated catchments, the stand-alone LSTM can also be considered due to its higher absolute accuracy (Fig. ). It is important to note, that in contrast to conceptual and hybrid models, the LSTM does not directly provide insights into the processes generating its predictions. This would be needed though to attribute changes in streamflow simulations to changes in underlying processes. For this purpose, hybrid models have an advantage because they directly provide access to variables the model has not been trained on such as snowpack, with comparable quality as conceptual models (Fig. , ).

Improving the generalizability of data-driven models is an area of active research, and different techniques have been proposed to achieve this . Hybrid modelling is one possible approach , which is confirmed by our results. Further, model generalizability could be improved by including catchments from regions whose climate is warmer than that of the region of interest as additional training catchments for deep learning based models . This allows these models to learn from a more diverse set of climates, which could help improve generalizability. Studies testing this approach on LSTMs have found that such geographically extended models can provide more realistic projections under climate sensitivity analyses than regional LSTMs . Testing this approach with realistic climate scenarios in the DSST framework could be an interesting avenue of further research. Other studies have also found improved generalizability when adding conservation of mass requirements to LSTMs . Further ideas include modifying the training procedure to optimize for worst-case accuracy across different domains and modifying the loss function to optimize for time invariant residuals, a concept derived from causal inference . It has been demonstrated that the use of such a loss function can improve the generalizability of a conceptual hydrological model under certain circumstances, but comes at the cost of reduced accuracy .

As other model comparison studies, our study faced some challenges and limitations including limited observational data availability and the need to make some modelling choices. Modelling choices e.g. included the choice of an objective function for model training and evaluation. We chose NSE for model training, meaning that we put more emphasis on high flow rather than low flow accuracy . To improve the model simulations for low flows, streamflow transformations or modified objective functions could be used. Another important model choice was the choice of the conceptual model used both for stand-alone modelling and integration into the hybrid framework. We used a model with a snow module relying on a relatively simple degree-day factor, which may not be the optimal choice for simulating streamflow under climate change in snow-dominated catchments . have also shown that using temperature-based PET methods in conceptual models can result in unrealistic water losses under temperature perturbations of +4 °C, compared to when using a more realistic energy-budget-based PET method. This suggests that the parametrizations of hydrological processes in conceptual models can lead to inaccurate extrapolation when these models are included in a hybrid modelling framework. How exactly the choice of hydrological model and the complexity of the snow module affects the generalizability of the hybrid model remains to be investigated in future studies. For example, one could investigate whether the snow module modifications proposed by , switching to an energy-based snow module or calibration against snow observations can further improve the generalizability of hybrid and conceptual models in snow-dominated catchments.

Testing the ability of hydrological models to predict streamflow under changing climatic conditions is challenging due to the lack of ground truth observations and benchmark models. To address this challenge, we used the differential split sample test (DSST; ). While providing insights into model generalizability, this approach limits the assessment of model generalizability to historically observed climate change. To overcome this limitation, other approaches to assess the suitability of hydrological models for climate change impact studies have been proposed. The robustness assessment test can be used to check whether biases in the model depend on the properties of the climate input data used for the simulations. Metamorphic testing; ) defines the expected qualitative response of a system to changes in the input data and tests the model's ability to replicate theses responses. This allows for the assessment of the responses of hydrological models to arbitrary changes in the input data, but is limited by current scientific understanding of the hydrological response to changes in climatic variables and can only make qualitative statements about the predictions . Both methods – the robustness assessment test and metamorphic testing – could be used to further explore the generalizability of data-driven and hybrid models to warmer climate conditions.