“…For the physical mechanism-constrained deep learning method proposed in this study, the sources of uncertainty are mainly the physical mechanism, the raw data itself and the model structure (Karniadakis et al, 2021). The uncertainty brought by the physical mechanisms is the parameter uncertainty, and for the three physical mechanisms of PIDL, the Richard equation involves some parameters of soil-and water-related properties, which are fitted by the VG formulation, and this fitting method can reduce the parameter uncertainty as low as possible when using a large amount of in situ monitoring data (Gao et al, 2019;Li et al, 2023;Ma et al, 2019Ma et al, , 2023Zhang et al, 2016;Zhao et al, 2020). Uncertainty from raw data, on the other hand, is due to the unavoidable noise in the data collection process and objective errors (truncation error, rounding error) during processing, which belong to chance uncertainty, and this part of uncertainty can usually be reduced by maximizing the accuracy of observations and quantity of data (Huang et al, 2022;Hüllermeier & Waegeman, 2021).…”