MP-FVM: Enhancing Finite Volume Method for Water Infiltration Modeling in Unsaturated Soils via Message-passing Encoder-decoder Network
This work addresses the problem of efficiently modeling water flow in unsaturated soils for applications in hydrology and environmental engineering, representing a novel method for a known bottleneck.
The authors tackled the computational challenges of solving the highly nonlinear Richards equation for water infiltration in unsaturated soils by developing the MP-FVM algorithm, which integrates adaptive fixed-point iteration, an encoder-decoder neural network, Sobolev training, and message passing in a finite volume framework, achieving superior accuracy and better preservation of physical laws compared to state-of-the-art methods and the commercial HYDRUS solver.
The spatiotemporal water flow dynamics in unsaturated soils can generally be modeled by the Richards equation. To overcome the computational challenges associated with solving this highly nonlinear partial differential equation (PDE), we present a novel solution algorithm, which we name as the MP-FVM (Message Passing-Finite Volume Method), to holistically integrate adaptive fixed-point iteration scheme, encoder-decoder neural network architecture, Sobolev training, and message passing mechanism in a finite volume discretization framework. We thoroughly discuss the need and benefits of introducing these components to achieve synergistic improvements in accuracy and stability of the solution. We also show that our MP-FVM algorithm can accurately solve the mixed-form $n$-dimensional Richards equation with guaranteed convergence under reasonable assumptions. Through several illustrative examples, we demonstrate that our MP-FVM algorithm not only achieves superior accuracy, but also better preserves the underlying physical laws and mass conservation of the Richards equation compared to state-of-the-art solution algorithms and the commercial HYDRUS solver.