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Abstract Finding similarities between model parameters across different catchments has proved to be challenging. Existing approaches struggle due to catchment heterogeneity and non‐linear dynamics. In particular, attempts to correlate catchment attributes with hydrological responses have failed due to interdependencies among variables and consequent equifinality. Machine Learning (ML), particularly the Long Short‐Term Memory (LSTM) approach, has demonstrated strong predictive and spatial regionalization performance. However, understanding the nature of the regionalization relationships remains difficult. This study proposes a novel approach to partially decouple learning the representation of (a) catchment dynamics by using theHydroLSTMarchitecture and (b) spatial regionalization relationships by using aRandom Forest(RF) clustering approach to learn the relationships between the catchment attributes and dynamics. This coupled approach, calledRegional HydroLSTM, learns a representation of “potential streamflow” using a single cell‐state, while the output gate corrects it to correspond to the temporal context of the current hydrologic regime. RF clusters mediate the relationship between catchment attributes and dynamics, allowing identification of spatially consistent hydrological regions, thereby providing insight into the factors driving spatial and temporal hydrological variability. Results suggest that by combining complementary architectures, we can enhance the interpretability of regional machine learning models in hydrology, offering a new perspective on the “catchment classification” problem. We conclude that an improved understanding of the underlying nature of hydrologic systems can be achieved by careful design of ML architectures to target the specific things we are seeking to learn from the data. 

Several studies have demonstrated the ability of long short-term memory (LSTM) machine-learning-based modeling to outperform traditional spatially lumped process-based modeling approaches for streamflow prediction. However, due mainly to the structural complexity of the LSTM network (which includes gating operations and sequential processing of the data), difficulties can arise when interpreting the internal processes and weights in the model. Here, we propose and test a modification of LSTM architecture that is calibrated in a manner that is analogous to a hydrological system. Our architecture, called “HydroLSTM”, simulates the sequential updating of the Markovian storage while the gating operation has access to historical information. Specifically, we modify how data are fed to the new representation to facilitate simultaneous access to past lagged inputs and consolidated information, which explicitly acknowledges the importance of trends and patterns in the data. We compare the performance of the HydroLSTM and LSTM architectures using data from 10 hydro-climatically varied catchments. We further examine how the new architecture exploits the information in lagged inputs, for 588 catchments across the USA. The HydroLSTM-based models require fewer cell states to obtain similar performance to their LSTM-based counterparts. Further, the weight patterns associated with lagged input variables are interpretable and consistent with regional hydroclimatic characteristics (snowmelt-dominated, recent rainfall-dominated, and historical rainfall-dominated). These findings illustrate how the hydrological interpretability of LSTM-based models can be enhanced by appropriate architectural modifications that are physically and conceptually consistent with our understanding of the system. 

Process-based modelling offers interpretability and physical consistency in many domains of geosciences but struggles to leverage large datasets efficiently. Machine-learning methods, especially deep networks, have strong predictive skills yet are unable to answer specific scientific questions. In this Perspective, we explore differentiable modelling as a pathway to dissolve the perceived barrier between process-based modelling and machine learning in the geosciences and demonstrate its potential with examples from hydrological modelling. ‘Differentiable’ refers to accurately and efficiently calculating gradients with respect to model variables or parameters, enabling the discovery of high-dimensional unknown relationships. Differentiable modelling involves connecting (flexible amounts of) prior physical knowledge to neural networks, pushing the boundary of physics-informed machine learning. It offers better interpretability, generalizability, and extrapolation capabilities than purely data-driven machine learning, achieving a similar level of accuracy while requiring less training data. Additionally, the performance and efficiency of differentiable models scale well with increasing data volumes. Under data-scarce scenarios, differentiable models have outperformed machine-learning models in producing short-term dynamics and decadal-scale trends owing to the imposed physical constraints. Differentiable modelling approaches are primed to enable geoscientists to ask questions, test hypotheses, and discover unrecognized physical relationships. Future work should address computational challenges, reduce uncertainty, and verify the physical significance of outputs. 

We confirm that energy dissipation weighting provides the most accurate approach to determining the effective hydraulic conductivity (Keff) of a binary K grid. A deep learning algorithm (UNET) can infer Keff with extremely high accuracy (R2 > 0.99). The UNET architecture could be trained to infer the energy dissipation weighting pattern from an image of the K distribution, although it was less accurate for cases with highly localized structures that controlled flow. Furthermore, the UNET architecture learned to infer the energy dissipation weighting even if it was not trained directly on this information. However, the weights were represented within the UNET in a way that was not immediately interpretable by a human user. This reiterates the idea that even if ML/DL algorithms are trained to make some hydrologic predictions accurately, they must be designed and trained to provide each user-required output if their results are to be used to improve our understanding of hydrologic systems. 

We develop a simple Quantile Spacing (QS) method for accurate probabilistic estimation of one-dimensional entropy from equiprobable random samples, and compare it with the popular Bin-Counting (BC) and Kernel Density (KD) methods. In contrast to BC, which uses equal-width bins with varying probability mass, the QS method uses estimates of the quantiles that divide the support of the data generating probability density function (pdf) into equal-probability-mass intervals. And, whereas BC and KD each require optimal tuning of a hyper-parameter whose value varies with sample size and shape of the pdf, QS only requires specification of the number of quantiles to be used. Results indicate, for the class of distributions tested, that the optimal number of quantiles is a fixed fraction of the sample size (empirically determined to be ~0.25–0.35), and that this value is relatively insensitive to distributional form or sample size. This provides a clear advantage over BC and KD since hyper-parameter tuning is not required. Further, unlike KD, there is no need to select an appropriate kernel-type, and so QS is applicable to pdfs of arbitrary shape, including those with discontinuous slope and/or magnitude. Bootstrapping is used to approximate the sampling variability distribution of the resulting entropy estimate, and is shown to accurately reflect the true uncertainty. For the four distributional forms studied (Gaussian, Log-Normal, Exponential and Bimodal Gaussian Mixture), expected estimation bias is less than 1% and uncertainty is low even for samples of as few as 100 data points; in contrast, for KD the small sample bias can be as large as −10% and for BC as large as −50%. We speculate that estimating quantile locations, rather than bin-probabilities, results in more efficient use of the information in the data to approximate the underlying shape of an unknown data generating pdf.