MP-FVM: Enhancing Finite Volume Method for Water Infiltration Modeling in Unsaturated Soils via Message-passing Encoder-decoder Network

TL;DR

This paper introduces MP-FVM, a novel neural network-enhanced finite volume method that improves accuracy and stability in modeling water infiltration in unsaturated soils governed by the Richards equation.

Contribution

The paper presents a new algorithm combining neural networks, message passing, and finite volume methods to solve the Richards equation more accurately and efficiently.

Findings

Achieves superior accuracy over existing methods.

Better preserves physical laws and mass conservation.

Guaranteed convergence under reasonable assumptions.

Abstract

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.