Graph-based automated discovery of concise soil hydraulic functions from data: beyond the Mualem - van Genuchten model

Soil hydraulic functions are fundamental to modelling water flow and transport in vadose-zone hydrology and are central to a wide range of hydrological and geoscientific applications. Yet in practice, these functions are still predominantly specified through expert-designed empirical formulations, such as the Mualem-van Genuchten (MvG) model. Although such models have proved highly influential, their derivation relies on predefined functional assumptions that make it difficult to simultaneously achieve accuracy, compactness, and robustness across diverse soil textures. Here we present a graph-based automated model discovery framework for discovering explicit soil hydraulic functions directly from experimental data. Applied to the original datasets used in the development of the MvG model, the method identifies a concise soil water retention function and its associated unsaturated hydraulic conductivity function whose mathematical structure differs fundamentally from classical empirical forms. Across 249 real soil samples spanning diverse textural classes, the discovered functions achieve more accurate predictions of unsaturated hydraulic conductivity than the MvG model. The fitted parameters also exhibit correlations with soil physical properties. This work demonstrates that data-driven model discovery can move beyond traditional empirical derivation and provide a promising route for developing accurate and explicit constitutive models.