Estimation of unfrozen water content in frozen soils based on data interpolation and constrained monotonic neural network

Accurately predicting the unfrozen water content (UWC) in frozen soils is crucial for modeling various soil processes in cold regions. However, existing empirical and theoretical models suffer limitations due to oversimplified assumptions or limited applicable conditions. Meanwhile, data-driven approaches are typically challenged by insufficient high-quality data and poor alignment with the underlying mechanisms in question. In the present study, a monotonic neural network (MNN) model for generalized UWC prediction was developed by leveraging data interpolation and monotonicity constraints. Experimental UWC data of various soils were collected from the literature and a raw dataset was formed. By first best-fitting each UWC curve in the raw dataset and then data interpolation, a new large dataset was generated and used for model development. A constrained MNN architecture for estimating UWC was constructed by constraining the weights of the neural network. The performance of the MNN was then compared with a standard deep neural network (DNN). The results demonstrated that the statistical characteristics of the generated dataset were comparable to that of the raw dataset. Both the MNN and DNN achieved good performance on the generated dataset. However, compared to DNN, the MNN yielded much more accurate prediction when tested on three new soils. In addition, the MNN was able to consistently give monotonic prediction on the UWC-Temperature relationship, even though both the monotonic and non-monotonic data were used for training. Overall, the monotonicity-constrained MNN can provide a robust and physical mechanisms aligned solution for estimating UWC in frozen soils.