Richards Equation Modeling Using Scientific Machine Learning Methodology
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Scientific Machine Learning (SciML) has emerged as a transformative paradigm, combining data-driven techniques with domain-specific knowledge to solve complex physical problems. This work applies the SciML framework to the Richards Equation, a nonlinear partial differential equation that describes unsaturated flow in porous media: $$ \frac{\partial \theta(h)}{\partial t} = \nabla \cdot \left[ K(h) \nabla h \right] + S(h), $$ where $\theta(h)$ is the volumetric water content, $K(h)$ the hydraulic conductivity, $h$ the pressure head, and $S(h)$ a source/sink term. The presentation focuses on the concept of explainable SciML, addressing key aspects such as the development of interpretable learning models, the integration of hydrological domain knowledge, and the enhancement of computational reliability. Strategies for constructing transparent neural architectures are discussed, highlighting the synergy between theoretical understanding and practical modeling needs. The approach aims to increase confidence in data-driven solvers for subsurface flow by bridging physical modeling and deep learning in an interpretable and scientifically grounded manner.
