We used a subset of a recently published, machine-learning-ready long-term groundwater level dataset for Germany . The subset comprises 667 groundwater wells, each with a total record length of 32 years of weekly data from 1991–2022. Wells were selected from the full dataset based on their Nash–Sutcliffe Efficiency (NSE) in an initial single-well benchmark model. Only wells with an NSE>0.7 were included, in order to exclude those that are clearly influenced by non-meteorological factors such as pumping. As a result, it can be assumed that the groundwater dynamics of the selected wells are primarily driven by meteorological forcing. The wells are spatially well distributed across Germany (Fig. ) and represent a wide range of hydrogeological and climatic conditions see provided time series plots in. To provide an illustrative overview of the variability in groundwater dynamics across wells, Fig.  shows example time series from representative monitoring sites. The selected examples highlight differences in seasonal amplitude, trend behavior, and long- and short-term variability, illustrating the heterogeneity that global models must capture.

The meteorological forcings were also obtained from the dataset mentioned above. They include weekly aggregated variables such as mean, maximum, and minimum air temperature, precipitation sum, and relative humidity, all derived from the HYRAS dataset provided by the German Meteorological Service (DWD) . Additional variables – also from DWD – include actual, potential, and reference evapotranspiration (FAO), as well as soil moisture and soil temperature at 5 cm depth . Further inputs, such as snow water equivalent, snowfall, snowmelt, and surface and subsurface runoff, were sourced from the ERA5-Land reanalysis dataset .

The selection of static environmental features was based on the dataset by . These features include hydrogeological and soil characteristics (e.g., aquifer type, hydraulic conductivity, soil type, recharge), topographic attributes (elevation, slope, aspect, flow direction, etc.), and land use information.

From the full set of static features provided in the dataset , variables related to well depth, screen characteristics, pumping, and pressure state were excluded, as these were sparsely available for the majority of monitoring wells. All categorical static features were label encoded for use in the machine learning models.

Time series features are quantitative metrics derived from groundwater level time series that describe specific aspects of their temporal dynamics. In this study, we adopt the feature definitions introduced by and also applied in . The resulting set consists of a redundancy-reduced selection of nine time series features, which has previously been used successfully to cluster large sets of groundwater wells based on their dynamic behavior .

All time series features were recomputed in this study using only information available within the respective training period. No data from the validation or test periods was used for feature derivation, thereby preventing information leakage. A complete list and detailed descriptions of the time series features are provided in Appendix A.

In all model setups, we used Long Short-Term Memory (LSTM) networks to process the dynamic time series data. LSTMs are the most commonly applied machine learning architecture in water science, and as such, we refrain from providing a detailed description here.

We implemented four different approaches to incorporate static input data into the models (Fig. ). The first two represent the most widely used methods in water science to date, while the latter two were selected based on their promising performance in comparative studies from other disciplines.

Repetition of static data. The static features are replicated at each time step and fed into the Long Short-Term Memory (LSTM) network alongside the dynamic inputs, as first proposed in . In the following, we refer to this architecture as the repetition model (rep model).

In the in-sample (IS) setting using static environmental features, the repetition model achieves the best performance across all error metrics, closely followed by the conditional and concatenation models (Table , Fig. a). The median NSE is 0.81 for the repetition model, 0.80 for the conditional model, and 0.79 for the concatenation model. The attention model lags slightly behind, with a median NSE of 0.77. As expected, all models incorporating static features outperform the dynamic-only baseline, which yields a median NSE of 0.73.

A closer look at the results reveals that model performance often depends on the individual well. For the majority of wells, the results align with the overall trend: the repetition model performs best, followed by the conditional, concatenation, and attention models in that order. However, there are also wells that deviate from this pattern. This becomes evident when counting how many wells were best predicted by each model based on NSE, regardless of the margin.

Out of the 667 wells, 376 were best modeled using the repetition approach, followed by 109 wells for which the conditional model performed best. The concatenation model was optimal for 81 wells, and the attention model for 51 wells. Interestingly, 50 wells were best modeled by the dynamic-only (dynonly) model, suggesting that these wells did not benefit from the inclusion of static features.

We did not analyze this aspect in detail, as it would likely be impractical to deploy different architectures for different wells, even if a relationship to time series dynamics could be established. Nonetheless, this finding is noteworthy, and it would be valuable to investigate in future work whether this variation is merely coincidental or, more likely, as we suspect, related to the underlying temporal behavior of individual wells.

In the out-of-sample (OOS) setting, the performance of all models is significantly lower than in the IS setting, with median NSE values decreasing from 0.73–0.81 (IS) to 0.73–0.74 (OOS), as shown in Table  and Fig. b. The performance differences between the various approaches are minimal, and all models perform approximately on par with the dynamic-only baseline. This indicates that none of the models were able to effectively generalize in space using the available environmental static features.

The lower performance in the OOS setting, compared to the IS setting, can be attributed to three possible factors:

Reduced training data due to cross-validation. In the 10-fold cross-validation setup, the amount of training data is reduced by 10 % in each fold. While this effect could theoretically be minimized using a leave-one-out cross-validation strategy – training the model on all but one well and testing on the remaining one – this would require n separate model runs (where n is the number of wells), making it computationally infeasible. Consequently, 10-fold CV remains a widely accepted and practical alternative.

As shown by , time-series features can lead to improved performance in the OOS setting. Based on this finding, we repeated all analyses using time-series features as static inputs, in order to evaluate whether and to what extent the performance of the different integration approaches changes when informative static features are used. In this context, “informative” refers to features that enable the models to genuinely generalize across different wells based on the information provided.

In the in-sample (IS) setting with time-series features as static inputs, the results are particularly noteworthy. While the performance of the repetition model does not improve (median NSE: 0.81 vs. 0.81 previously), the concatenation and attention models appear to benefit from the inclusion of informative static features (median NSE: 0.81, 0.80 vs. 0.79, 0.77 previously) (Table , Fig. c). The conditional model performs slightly worse (median NSE: 0.79 vs. 0.80 previously).

Although the repetition model remains the top performer for 238 wells, the concatenation model catches up, now performing best for 198 wells. The attention model is best for 125 wells. The conditional model falls further behind, being optimal for only 60 wells, while the dynamic-only model remains best for 46 wells.

In the IS setting, models appear to benefit even from “meaningless” static inputs, likely by using them as a form of unique identifier . However, when informative static features are provided – such as time-series-derived descriptors – the models gain the ability to generalize more effectively based on these inputs. This ability, however, is not equally pronounced across all integration strategies. While the concatenation and attention models show clear improvements with informative static features, the repetition model's performance remains largely unchanged, and the conditional model even shows a slight decline.

In the out-of-sample (OOS) setting, as expected, all models show improved performance compared to when environmental static features were used. All integration approaches now outperform the dynamic-only model across all error metrics. In terms of median NSE, the repetition again performs best (0.82), closely followed by the concatenation model (0.80), conditional model (0.79) and attention model (0.78). The dynamic-only model remains at a lower performance level, with a median NSE of 0.73 (Table , Fig. d). Notably, the OOS results using time-series-based static features are only slightly lower than the in-sample (IS) results, and in case of the repetition model even higher. These results confirm that the models are able to extract relevant, i.e. physically interpretable information from time-series-based static features and use it to generalize across space.

When evaluating which model performs best for the highest number of wells, the repetition model takes the lead with 378 wells. It is followed by the concatenation model with 134 wells. The conditional and attention models lag behind, performing best for only 64 and 58 wells, respectively. The dynamic-only model comes last, showing the best performance in just 33 wells.

In terms of computational effort, the different approaches exhibit noticible differences. While we were unable to consistently track exact runtimes – due to parallel execution across machines with varying computational resources to reduce total runtime – distinct trends emerged. The attention model was the fastest overall, closely followed by the concatenation model. In contrast, both the repetition and conditional models were significantly slower, with the repetition model also demanding considerably more memory.

When comparing our findings to studies from other domains, several consistent patterns emerge regarding the value and integration of static features in deep learning models. However, the effectiveness of particular integration strategies also appears to be domain-specific and strongly dependent on the nature and informativeness of the static features.

In general, studies across domains agree that informative static features can substantially improve predictive performance and spatial generalization – but only when integrated using an appropriate strategy. Our results support this trend: when switching from environmental to time-series-based static features, some of the tested architectures showed clear improvements in both IS and OOS settings, while others remained unchanged or even declined.

In design science, also showed that informed integration methods outperform simpler schemes. Among these, mid-level fusion performed best, allowing the model to first encode temporal dependencies before incorporating static context. This strategy corresponds most closely to our conditional architecture, which also achieved strong performance, albeit not the highest in our experiments.

In medicine, found that parallel encoding and late fusion of static and dynamic features improved generalization across patients. This setup is conceptually similar to our concatenation architecture. They reported that this approach significantly improved performance, particularly for unseen patients, emphasizing that parallel processing followed by concatenation was a key to successful generalization. In our case, while the concatenation strategy also performed well under out-of-sample conditions – especially when static features carried meaningful information – the repetition model still outperformed it overall.

found that both concatenation and conditional initialization strategies achieved the best trade-off between predictive accuracy and computational efficiency. These findings are at least partially consistent with our results, in which both the conditional and concatenation models also achieved strong performance. In particular, the concatenation model represents a favorable compromise between predictive accuracy and computational efficiency.

In agricultural modeling, compared models with and without static inputs and found that early fusion via concatenation of static and dynamic features yielded the best performance in both cross-validation and out-of-sample prediction. Their findings support our conclusion that concatenation is effective when static inputs are informative.

found that their best performance came from concatenating static and dynamic representations, with some benefit from temporal attention – which aligns with our finding that attention-based models can benefit from informative static features, though in our case, attention still slightly trailed behind concatenation and repetition.

Taken together, these studies confirm that the best-performing integration method depends not only on model architecture, but also on the quality and informativeness of the static features. Our work adds to this understanding by systematically evaluating several approaches in a large-scale groundwater modeling context and confirming that when static features are informative, concatenation offers an effective and efficient compromise, though with slightly lower predictive performance compared to the repetition model.