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Accurate hydrological modeling is vital to characterizing how the terrestrial water cycle responds to climate change. Pure deep learning (DL) models have shown to outperform process-based ones while remaining difficult to interpret. More recently, differentiable, physics-informed machine learning models with a physical backbone can systematically integrate physical equations and DL, predicting untrained variables and processes with high performance. However, it was unclear if such models are competitive for global-scale applications with a simple backbone. Therefore, we use – for the first time at this scale – differentiable hydrologic models (fullname δHBV-globe1.0-hydroDL and shorthanded δHBV) to simulate the rainfall-runoff processes for 3753 basins around the world. Moreover, we compare the δHBV models to a purely data-driven long short-term memory (LSTM) model to examine their strengths and limitations. Both LSTM and the δHBV models provide competent daily hydrologic simulation capabilities in global basins, with median Kling-Gupta efficiency values close to or higher than 0.7 (and 0.78 with LSTM for a subset of 1675 basins with long-term records), significantly outperforming traditional models. Moreover, regionalized differentiable models demonstrated stronger spatial generalization ability (median KGE 0.64) than a traditional parameter regionalization approach (median KGE 0.46) and even LSTM for ungauged region tests in Europe and South America. Nevertheless, relative to LSTM, the differentiable model was hampered by structural deficiencies for cold or polar regions, and highly arid regions, and basins with significant human impacts. This study also sets the benchmark for hydrologic estimates around the world and builds foundations for improving global hydrologic simulations. 

Recent advances in differentiable modeling, a genre of physics-informed machine learning that trains neural networks (NNs) together with process-based equations, have shown promise in enhancing hydrological models' accuracy, interpretability, and knowledge-discovery potential. Current differentiable models are efficient for NN-based parameter regionalization, but the simple explicit numerical schemes paired with sequential calculations (operator splitting) can incur numerical errors whose impacts on models' representation power and learned parameters are not clear. Implicit schemes, however, cannot rely on automatic differentiation to calculate gradients due to potential issues of gradient vanishing and memory demand. Here we propose a “discretize-then-optimize” adjoint method to enable differentiable implicit numerical schemes for the first time for large-scale hydrological modeling. The adjoint model demonstrates comprehensively improved performance, with Kling–Gupta efficiency coefficients, peak-flow and low-flow metrics, and evapotranspiration that moderately surpass the already-competitive explicit model. Therefore, the previous sequential-calculation approach had a detrimental impact on the model's ability to represent hydrological dynamics. Furthermore, with a structural update that describes capillary rise, the adjoint model can better describe baseflow in arid regions and also produce low flows that outperform even pure machine learning methods such as long short-term memory networks. The adjoint model rectified some parameter distortions but did not alter spatial parameter distributions, demonstrating the robustness of regionalized parameterization. Despite higher computational expenses and modest improvements, the adjoint model's success removes the barrier for complex implicit schemes to enrich differentiable modeling in hydrology. 

Abstract. As a genre of physics-informed machine learning, differentiable process-based hydrologic models (abbreviated as δ or delta models) with regionalized deep-network-based parameterization pipelines were recently shown to provide daily streamflow prediction performance closely approaching that of state-of-the-art long short-term memory (LSTM) deep networks. Meanwhile, δ models provide a full suite of diagnostic physical variables and guaranteed mass conservation. Here, we ran experiments to test (1) their ability to extrapolate to regions far from streamflow gauges and (2) their ability to make credible predictions of long-term (decadal-scale) change trends. We evaluated the models based on daily hydrograph metrics (Nash–Sutcliffe model efficiency coefficient, etc.) and predicted decadal streamflow trends. For prediction in ungauged basins (PUB; randomly sampled ungauged basins representing spatial interpolation), δ models either approached or surpassed the performance of LSTM in daily hydrograph metrics, depending on the meteorological forcing data used. They presented a comparable trend performance to LSTM for annual mean flow and high flow but worse trends for low flow. For prediction in ungauged regions (PUR; regional holdout test representing spatial extrapolation in a highly data-sparse scenario), δ models surpassed LSTM in daily hydrograph metrics, and their advantages in mean and high flow trends became prominent. In addition, an untrained variable, evapotranspiration, retained good seasonality even for extrapolated cases. The δ models' deep-network-based parameterization pipeline produced parameter fields that maintain remarkably stable spatial patterns even in highly data-scarce scenarios, which explains their robustness. Combined with their interpretability and ability to assimilate multi-source observations, the δ models are strong candidates for regional and global-scale hydrologic simulations and climate change impact assessment. 

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. 

Abstract The behaviors and skills of models in many geosciences (e.g., hydrology and ecosystem sciences) strongly depend on spatially-varying parameters that need calibration. A well-calibrated model can reasonably propagate information from observations to unobserved variables via model physics, but traditional calibration is highly inefficient and results in non-unique solutions. Here we propose a novel differentiable parameter learning (dPL) framework that efficiently learns a global mapping between inputs (and optionally responses) and parameters. Crucially, dPL exhibits beneficial scaling curves not previously demonstrated to geoscientists: as training data increases, dPL achieves better performance, more physical coherence, and better generalizability (across space and uncalibrated variables), all with orders-of-magnitude lower computational cost. We demonstrate examples that learned from soil moisture and streamflow, where dPL drastically outperformed existing evolutionary and regionalization methods, or required only ~12.5% of the training data to achieve similar performance. The generic scheme promotes the integration of deep learning and process-based models, without mandating reimplementation. 

Abstract Predictions of hydrologic variables across the entire water cycle have significant value for water resources management as well as downstream applications such as ecosystem and water quality modeling. Recently, purely data‐driven deep learning models like long short‐term memory (LSTM) showed seemingly insurmountable performance in modeling rainfall runoff and other geoscientific variables, yet they cannot predict untrained physical variables and remain challenging to interpret. Here, we show that differentiable, learnable, process‐based models (calledδmodels here) can approach the performance level of LSTM for the intensively observed variable (streamflow) with regionalized parameterization. We use a simple hydrologic model HBV as the backbone and use embedded neural networks, which can only be trained in a differentiable programming framework, to parameterize, enhance, or replace the process‐based model's modules. Without using an ensemble or post‐processor,δmodels can obtain a median Nash‐Sutcliffe efficiency of 0.732 for 671 basins across the USA for the Daymet forcing data set, compared to 0.748 from a state‐of‐the‐art LSTM model with the same setup. For another forcing data set, the difference is even smaller: 0.715 versus 0.722. Meanwhile, the resulting learnable process‐based models can output a full set of untrained variables, for example, soil and groundwater storage, snowpack, evapotranspiration, and baseflow, and can later be constrained by their observations. Both simulated evapotranspiration and fraction of discharge from baseflow agreed decently with alternative estimates. The general framework can work with models with various process complexity and opens up the path for learning physics from big data. 

Abstract Predicting discharge in contiguously data‐scarce or ungauged regions is needed for quantifying the global hydrologic cycle. We show that prediction in ungauged regions (PUR) has major, underrecognized uncertainty and is drastically more difficult than previous problems where basins can be represented by neighboring or similar basins (known as prediction in ungauged basins). While deep neural networks demonstrated stellar performance for streamflow predictions, performance nonetheless declined for PUR, benchmarked here with a new stringent region‐based holdout test on a US data set with 671 basins. We tested approaches to reduce such errors, leveraging deep network's flexibility to integrate “soft” data, such as satellite‐based soil moisture product, or daily flow distributions which improved low flow simulations. A novel input‐selection ensemble improved average performance and greatly reduced catastrophic failures. Despite challenges, deep networks showed stronger performance metrics for PUR than traditional hydrologic models. They appear competitive for geoscientific modeling even in data‐scarce settings. 

Abstract Recent observations with varied schedules and types (moving average, snapshot, or regularly spaced) can help to improve streamflow forecasts, but it is challenging to integrate them effectively. Based on a long short‐term memory (LSTM) streamflow model, we tested multiple versions of a flexible procedure we call data integration (DI) to leverage recent discharge measurements to improve forecasts. DI accepts lagged inputs either directly or through a convolutional neural network unit. DI ubiquitously elevated streamflow forecast performance to unseen levels, reaching a record continental‐scale median Nash‐Sutcliffe Efficiency coefficient value of 0.86. Integrating moving‐average discharge, discharge from the last few days, or even average discharge from the previous calendar month could all improve daily forecasts. Directly using lagged observations as inputs was comparable in performance to using the convolutional neural network unit. Importantly, we obtained valuable insights regarding hydrologic processes impacting LSTM and DI performance. Before applying DI, the base LSTM model worked well in mountainous or snow‐dominated regions, but less well in regions with low discharge volumes (due to either low precipitation or high precipitation‐energy synchronicity) and large interannual storage variability. DI was most beneficial in regions with high flow autocorrelation: it greatly reduced baseflow bias in groundwater‐dominated western basins and also improved peak prediction for basins with dynamical surface water storage, such as the Prairie Potholes or Great Lakes regions. However, even DI cannot elevate performance in high‐aridity basins with 1‐day flash peaks. Despite this limitation, there is much promise for a deep‐learning‐based forecast paradigm due to its performance, automation, efficiency, and flexibility.