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The most accurate rainfall–runoff predictions are currently based on deep learning. There is a concern among hydrologists that the predictive accuracy of data-driven models based on deep learning may not be reliable in extrapolation or for predicting extreme events. This study tests that hypothesis using long short-term memory (LSTM) networks and an LSTM variant that is architecturally constrained to conserve mass. The LSTM network (and the mass-conserving LSTM variant) remained relatively accurate in predicting extreme (high-return-period) events compared with both a conceptual model (the Sacramento Model) and a process-based model (the US National Water Model), even when extreme events were not included in the training period. Adding mass balance constraints to the data-driven model (LSTM) reduced model skill during extreme events.

Although decades of effort have been devoted to building Physical‐Conceptual (PC) models for predicting the time‐series evolution of geoscientific systems, recent work shows that Machine Learning (ML) based Gated Recurrent Neural Network technology can be used to develop models that are much more accurate. However, the difficulty of extracting physical understanding from ML‐based models complicates their utility for enhancing scientific knowledge regarding system structure and function. Here, we propose a physically interpretable Mass‐Conserving‐Perceptron (MCP) as a way to bridge the gap between PC‐based and ML‐based modeling approaches. The MCP exploits the inherent isomorphism between the directed graph structures underlying both PC models and GRNNs to explicitly represent the mass‐conserving nature of physical processes while enabling the functional nature of such processes to be directly learned (in an interpretable manner) from available data using off‐the‐shelf ML technology. As a proof of concept, we investigate the functional expressivity (capacity) of the MCP, explore its ability to parsimoniously represent the rainfall‐runoff (RR) dynamics of the Leaf River Basin, and demonstrate its utility for scientific hypothesis testing. To conclude, we discuss extensions of the concept to enable ML‐based physical‐conceptual representation of the coupled nature of mass‐energy‐information flows through geoscientific systems. Plain Language Summary We develop a physically interpretable computational unit, referred to as the Mass‐Conserving‐Perceptron (MCP). Networks of such units can be used to model the conservative nature of the input‐state‐output dynamics of mass flows in geoscientific systems, while Machine Learning (ML) technology can be used to learn the functional nature of the physical processes governing such system behaviors. Testing using data from the Leaf River Basin demonstrates the considerable functional expressivity (capacity) and interpretability of even a single‐MCP‐node‐based model, while providing excellent predictive performance and the ability to conduct scientific hypothesis testing. The concept can easily be extended to facilitate ML‐based physical‐conceptual representation of the coupled nature of mass‐energy‐information flows through geoscientific systems, thereby facilitating the development of synergistic physics‐AI modeling approaches. Key Points We develop a physically interpretable unit (Mass‐Conserving‐Perceptron) that can be used as a basic component of geoscientific models Off‐the‐shelf Machine Learning technology can be used to learn the functional nature of the physical processes governing system behaviors The concept can be extended to facilitate ML‐based representation of coupled mass‐energy‐information flows in geoscientific systems

Miscalculating the volumes of water withdrawn for irrigation, the largest consumer of freshwater in the world, jeopardizes sustainable water management. Hydrological models quantify water withdrawals, but their estimates are unduly precise. Model imperfections need to be appreciated to avoid policy misjudgements.

The past decade has seen significant progress in characterizing uncertainty in environmental systems models, through statistical treatment of incomplete knowledge regarding parameters, model structure, and observational data. Attention has now turned to the issue of model structural adequacy (MSA, a term we prefer over model structure “error”). In reviewing philosophical perspectives from the groundwater, unsaturated zone, terrestrial hydrometeorology, and surface water communities about how to model the terrestrial hydrosphere, we identify several areas where different subcommunities can learn from each other. In this paper, we (a) propose a consistent and systematic “unifying conceptual framework” consisting of five formal steps for comprehensive assessment of MSA; (b) discuss the need for a pluralistic definition of adequacy; (c) investigate how MSA has been addressed in the literature; and (d) identify four important issues that require detailed attention—structured model evaluation, diagnosis of epistemic cause, attention to appropriate model complexity, and a multihypothesis approach to inference. We believe that there exists tremendous scope to collectively improve the scientific fidelity of our models and that the proposed framework can help to overcome barriers to communication. By doing so, we can make better progress toward addressing the question “How can we use data to detect, characterize, and resolve model structural inadequacies?” Key Points Model building comprises five important formal steps These remain poorly understood, and methods for dealing with them remain ad‐hoc Progress requires a common perspective on epistemic problems of model adequacy

We investigate the applicability of machine learning technologies to the development of parsimonious, interpretable, catchment‐scale hydrologic models using directed‐graph architectures based on the mass‐conserving perceptron (MCP) as the fundamental computational unit. Here, we focus on architectural complexity (depth) at a single location, rather than universal applicability (breadth) across large samples of catchments. The goal is to discover a minimal representation (numbers of cell‐states and flow paths) that represents the dominant processes that can explain the input‐state‐output behaviors of a given catchment, with particular emphasis given to simulating the full range (high, medium, and low) of flow dynamics. We find that a “HyMod Like” architecture with three cell‐states and two major flow pathways achieves such a representation at our study location, but that the additional incorporation of an input‐bypass mechanism significantly improves the timing and shape of the hydrograph, while the inclusion of bi‐directional groundwater mass exchanges significantly enhances the simulation of baseflow. Overall, our results demonstrate the importance of using multiple diagnostic metrics for model evaluation, while highlighting the need for properly selecting and designing the training metrics based on information‐theoretic foundations that are better suited to extracting information across the full range of flow dynamics. This study sets the stage for interpretable regional‐scale MCP‐based hydrological modeling (using large sample data) by using neural architecture search to determine appropriate minimal representations for catchments in different hydroclimatic regimes. Plain Language Summary We show that conventional machine learning technologies can be used to develop parsimonious, interpretable, catchment‐scale hydrologic models using the mass‐conserving perceptron (MCP) as a fundamental computational unit. Using data from the Leaf River Basin, we test a variety of minimal, dominant process, representations that can explain the input‐state‐output dynamics of the catchment. Our results demonstrate the importance of using multiple diagnostic metrics for evaluation and comparison of different model architectures, and highlight the importance of choosing (or designing) objective functions for model training that are properly suited to the task of extracting information across the full range of flow dynamics. This depth‐focus study sets the stage for interpretable regional‐scale MCP‐based hydrological modeling (using large sample data) by using neural architecture search to determine appropriate minimal representations for catchments in different hydroclimatic regimes. Key Points We utilize mass‐conserving perceptron (MCP) directed‐graph architectures to develop concise, interpretable catchment‐scale hydrologic models We focus on model complexity (depth) at a single location, rather than universal applicability (breadth) across large samples of catchments This study set the stage for interpretable MCP‐based modeling to find minimal representations in different hydroclimatic regimes

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 the HydroLSTM architecture and (b) spatial regionalization relationships by using a Random Forest (RF) clustering approach to learn the relationships between the catchment attributes and dynamics. This coupled approach, called Regional 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.

Using information-theoretic quantities in practical applications with continuous data is often hindered by the fact that probability density functions need to be estimated in higher dimensions, which can become unreliable or even computationally unfeasible. To make these useful quantities more accessible, alternative approaches such as binned frequencies using histograms and k-nearest neighbors (k-NN) have been proposed. However, a systematic comparison of the applicability of these methods has been lacking. We wish to fill this gap by comparing kernel-density-based estimation (KDE) with these two alternatives in carefully designed synthetic test cases. Specifically, we wish to estimate the information-theoretic quantities: entropy, Kullback–Leibler divergence, and mutual information, from sample data. As a reference, the results are compared to closed-form solutions or numerical integrals. We generate samples from distributions of various shapes in dimensions ranging from one to ten. We evaluate the estimators’ performance as a function of sample size, distribution characteristics, and chosen hyperparameters. We further compare the required computation time and specific implementation challenges. Notably, k-NN estimation tends to outperform other methods, considering algorithmic implementation, computational efficiency, and estimation accuracy, especially with sufficient data. This study provides valuable insights into the strengths and limitations of the different estimation methods for information-theoretic quantities. It also highlights the significance of considering the characteristics of the data, as well as the targeted information-theoretic quantity when selecting an appropriate estimation technique. These findings will assist scientists and practitioners in choosing the most suitable method, considering their specific application and available data. We have collected the compared estimation methods in a ready-to-use open-source Python 3 toolbox and, thereby, hope to promote the use of information-theoretic quantities by researchers and practitioners to evaluate the information in data and models in various disciplines.

Hydrologic models require the specification of unknown model parameters via calibration to historical input‐output data. For spatially distributed models, the large number of unknowns makes the calibration problem poorly conditioned. Spatial regularization can help to stabilize the problem by facilitating inclusion of additional information. While a common regularization approach is to apply a scalar multiplier to the prior estimate of each parameter field, this can cause problems by simultaneously changing both the mean and the variance of the distribution. This paper explores a multiple‐criteria regularization approach that facilitates adjustment of the mean, variance, and shape of the parameter distribution, using prior information to constrain the problem while providing sufficient degrees of freedom to enable model performance improvements. We also test simple squashing functions to help in maintaining conceptually reasonable parameter values throughout the spatial domain. We apply the method to three basins in the context of the Distributed Model Intercomparison Project (DMIP2), obtaining considerable performance improvements at the basin outlet. However, the prior parameter estimates are found to give much better performance at the interior points (treated as ungauged), suggesting that the spatial information has not been properly exploited. The results also suggest that basin outlet hydrographs may not be particularly sensitive to spatial parameter variability and that an overall basin mean value may be sufficient for flow forecasting at the outlet, although not at the interior points. We discuss weaknesses in our study approach and suggest diagnostically more powerful strategies to be pursued.

Spatially distributed hydrologic models are useful for understanding the water balance dynamics of catchments under changing conditions, thereby providing important information for water resource management and decision making. However, in poorly gauged basins, the absence of reliable and overlapping in situ hydro-meteorological data makes the calibration and evaluation of such models quite challenging. Here, we explored the potential of using streamflow signatures extracted from historical (not current) streamflow data, along with current remote sensing-based evapotranspiration data, to constrain the parameters of a spatially distributed Soil and Water Assessment Tool (SWAT) model of the Mara River Basin (Kenya/Tanzania) that is forced by satellite-based rainfall. The result is a reduced bias of the simulated estimates of streamflow and evapotranspiration. In addition, the simulated water balance dynamics better reflect underlying governing factors such as soil type, land cover and climate at both annual and seasonal time scales, indicating the structural and behavioral consistency of the calibrated model. This study demonstrates that the judicious use of available information can help to facilitate meaningful calibration and evaluation of hydrologic models to support decision making in poorly gauged river basins around the world.

Due largely to challenges associated with physical interpretability of machine learning (ML) methods, and because model interpretability is key to credibility in management applications, many scientists and practitioners are hesitant to discard traditional physical‐conceptual modeling approaches despite their poorer predictive performance. Here, we examine how to develop parsimonious minimally‐optimal representations that can facilitate better insight regarding system functioning. The term “minimally‐optimal” indicates that the desired outcome can be achieved with the smallest possible effort and resources, while “parsimony” is widely held to support understanding. Accordingly, we suggest that ML‐based modeling should use computational units that are inherently physically‐interpretable, and explore how generic network architectures comprised of Mass‐Conserving‐Perceptron can be used to model dynamical systems in a physically‐interpretable manner. In the context of spatially‐lumped catchment‐scale modeling, we find that both physical interpretability and good predictive performance can be achieved using a “distributed‐state” network with context‐dependent gating and “information‐sharing” across nodes. The distributed‐state mechanism ensures a sufficient number of temporally‐evolving properties of system storage while information‐sharing ensures proper synchronization of such properties. The results indicate that MCP‐based ML models with only a few layers (up to two) and relativity few physical flow pathways (up to three) can play a significant role in ML‐based streamflow modeling.