A Physics-based, Data-driven Numerical Framework for Anomalous Diffusion of Water in Soil

Precision modeling and forecasting of soil moisture are essential for implementing smart irrigation systems and mitigating agricultural drought. Most agro-hydrological models are based on the standard Richards equation, a highly nonlinear, degenerate elliptic-parabolic partial differential equation (PDE) with first order time derivative. However, research has shown that standard Richards equation is unable to model preferential flow in soil with fractal structure. In such a scenario, the soil exhibits anomalous non-Boltzmann scaling behavior. Incorporating the anomalous non-Boltzmann scaling behavior into the Richards equation leads to a generalized, time-fractional Richards equation based on fractional time derivatives. As expected, solving the time-fractional Richards equation for accurate modeling of water flow dynamics in soil faces extensive computational challenges. To target these challenges, we propose a novel numerical method that integrates finite volume method (FVM), adaptive fixed point iteration scheme, and neural network to solve the time-fractional Richards equation. Specifically, we develop an adaptive fixed point iteration scheme to solve the FVM-discretized equation iteratively, which avoids the stability issues when directly solving a stiff and sparse matrix equation. To improve the solution quality which is influenced by numerical errors and computational constraints during actual implementation, we propose to use neural networks that resemble an encoder-decoder architecture to map soil moisture profiles into a latent space and reconstruct them back. Through 1-D examples, we illustrate the accuracy and computational efficiency of our proposed physics-based, data-driven numerical method. Finally, we present a Markov chain Monte Carlo (MCMC) approach to solve the inverse problem to obtain soil-specific parameters given soil moisture solutions.