Identifying Structural Priors in a Hybrid Differentiable Model for Stream Water Temperature Modeling

Although deep learning models for stream temperature (Ts) have recently shown exceptional accuracy, they have limited interpretability and cannot output untrained variables. With hybrid differentiable models, neural networks (NNs) can be connected to physically based equations (called structural priors) to output intermediate variables such as water source fractions (specifying what portion of water is groundwater, subsurface, and surface flow). However, it is unclear if such outputs are physically meaningful when only limited physics is imposed, and if structural priors have enough impacts to be identifiable from data. Here, we tested four alternative structural priors describing basin-scale water temperature memory and instream heat processes in a differentiable stream temperature model where NNs freely estimate the water source fractions. We evaluated models' abilities to predict Ts and baseflow ratio. The four priors exhibited noticeably different behaviors in these two metrics and their tradeoffs, with some dominating others. Therefore, the better structural priors can be identified. Moreover, testing different priors yielded valuable insights: having a separate shallow subsurface flow component better matches observations, and a recency-weighted averaging of past air temperature for calculating source water temperature resulted in better Ts and baseflow prediction than traditionally employed simple averaging. However, we also highlight the limitations when insufficient physical constraints are implemented: the internal variables (water source fractions) may not be adequately constrained by a single target variable (stream temperature) alone. To ensure the physical significance of the internal fluxes, one can either employ multivariate data for model selection, or include more physical processes in the priors. A new framework called differentiable modeling combines the benefits from neural networks (NNs) and process-based models. This framework can learn from big data while the process-based model components (called prior knowledge, or priors) are intended to output intermediate physical variables. However, do such priors matter, can we tell if one set of priors is better than another, and do the intermediate outputs represent the intended physical concepts? We explore these questions with a differentiable stream temperature model where the NN replaces the hydrologic component and estimates parameters pertaining to the stream temperature module. The strong optimizing capability of NNs allows us to avoid some complexities and attribute the differences in model outcomes to the assumed priors. Testing different priors thus yielded many important lessons, for example, the need for having a separate shallow groundwater "bucket," the benefit of placing more importance on recent air temperature when estimating groundwater temperature, and the importance of describing in-stream temperature. The results show lots of untapped potential with differentiable modeling and the data we have available. We can identify better structural priors (process equations) using differentiable models, circumventing intertwined parameter issuesConsidering a separate bucket for shallow subsurface water improves both stream temperature and baseflow simulationThe models selected by the multivariate evaluation produce physically meaningful estimates of water source fractions