A Differentiable Physics-Informed Neural Network (DPINN) for National-Scale River Temperature Modelling

River temperature (T_w) is of fundamental importance to freshwater ecosystem health and the services this provides to society. Yet anthropogenically-induced T_w transformations from pressures like flow regulation, deforestation and climate change induce various thermal impacts globally. As such, evidence-led management approaches are needed to mitigate T_w modifications, but these are often hindered by a paucity of reliable data across river networks. Modelling T_w regimes across unmonitored rivers and forecasting future change is critical for helping safeguard freshwater ecosystems. Hybrid T_w models offer a promising scientific avenue to embed process-based insights within spatially transferrable statistical frameworks, but few studies have applied this across national-scales. However, current hybrid architectures often rely on simplified equations as rigid structural priors which may constrain their flexibility in capturing complex thermal dynamics. In light of this, we have developed a novel hybrid T_w method based on Differentiable Physics-Informed Neural Networks (DPINNs) and applied this to multi-decadal data (1980-2020) from 78 sites across the conterminous United States. This approach integrates a zero-dimensional (0D) heat advection-dispersion equation within a neural network (NN) framework and utilizes the neural network to estimate river heat exchange processes. This capability allows the method to be applied to data from a single site, where establishing a physical process-based model is typically difficult. We observed a Mean Absolute Error (MAE) of 0.68 °C when comparing DPINN model predictions versus observed T_w values. Our results indicated this approach outperformed the established air2stream T_w model and traditional neural networks approaches like MLP (MAE = 0.79 °C) and LSTM (MAE = 0.93 °C). Our results thus highlight that by embedding physical priors to incorporate explicit heat transfer mechanisms, it enhances T_w modelling performance while also reducing the need for large environmental datasets. The strong performance of this innovative DPINN T_w model on a national-scale highlights its potential transferability across a broad range of river environments, and thus could be a vital tool to help predict large-scale T_w dynamics to help underpin effective, ‘climate proofed’ management interventions.