Identifying structural priors in a hybrid differentiable model for stream water temperature modeling

Although deep learning models for stream temperature (T_s) 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 T_s 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 T_s 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.