Data assimilation (DA) enables hydrologic models to update their internal states using near-real-time observations for more accurate forecasts. With deep neural networks like long short-term memory (LSTM), using either lagged observations as inputs (called "data integration") or variational DA has shown success in improving forecasts. However, it is unclear which methods are performant or optimal for physics-informed machine learning ("differentiable") models, which represent only a small amount of physically-meaningful states while using deep networks to supply parameters or missing processes. Here we developed variational DA methods for differentiable models, including optimizing adjusters for just precipitation data, just model internal hydrological states, or both. Our results demonstrated that differentiable streamflow models using the CAMELS dataset can benefit strongly and equivalently from variational DA as LSTM, with one-day lead time median Nash-Sutcliffe efficiency (NSE) elevated from 0.75 to 0.82. The resulting forecast matched or outperformed LSTM with DA in the eastern, northwestern, and central Great Plains regions of the conterminous United States. Both precipitation and state adjusters were needed to achieve these results, with the latter being substantially more effective on its own, and the former adding moderate benefits for high flows. Our DA framework does not need systematic training data and could serve as a practical DA scheme for whole river networks.

For a number of years since its introduction to hydrology, recurrent neural networks like long short-term memory (LSTM) have proven remarkably difficult to surpass in terms of daily hydrograph metrics on known, comparable benchmarks. Outside of hydrology, Transformers have now become the model of choice for sequential prediction tasks, making it a curious architecture to investigate. Here, we first show that a vanilla Transformer architecture is not competitive against LSTM on the widely benchmarked CAMELS dataset, and lagged especially for the high-flow metrics due to short-term processes. However, a recurrence-free variant of Transformer can obtain mixed comparisons with LSTM, producing the same Kling-Gupta efficiency coefficient (KGE), along with other metrics. The lack of advantages for the Transformer is linked to the Markovian nature of the hydrologic prediction problem. Similar to LSTM, the Transformer can also merge multiple forcing dataset to improve model performance. While the Transformer results are not higher than current state-of-the-art, we still learned some valuable lessons: (1) the vanilla Transformer architecture is not suitable for hydrologic modeling; (2) the proposed recurrence-free modification can improve Transformer performance so future work can continue to test more of such modifications; and (3) the prediction limits on the dataset should be close to the current state-of-the-art model. As a non-recurrent model, the Transformer may bear scale advantages for learning from bigger datasets and storing knowledge. This work serves as a reference point for future modifications of the model.

Predictions of hydrologic variables across the entire water cycle have significant value for water resource 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 (called {\delta} models 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 modules. Without using an ensemble or post-processor, {\delta} models can obtain a median Nash Sutcliffe efficiency of 0.732 for 671 basins across the USA for the Daymet forcing dataset, compared to 0.748 from a state-of-the-art LSTM model with the same setup. For another forcing dataset, the difference is even smaller: 0.715 vs. 0.722. Meanwhile, the resulting learnable process-based models can output a full set of untrained variables, e.g., soil and groundwater storage, snowpack, evapotranspiration, and baseflow, and 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.