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

TL;DR

This paper tackles the numerical solution of the nonlinear Richards equation for unsaturated soil flow by introducing MP-FVM, a framework that fuses an adaptive fixed-point finite-volume discretization with an encoder–decoder neural network and a message passing mechanism. The method employs Sobolev training and a coarse-to-fine data strategy to boost convergence, accuracy, and mass conservation, while providing theoretical convergence guarantees. Across 1D–3D benchmark problems and a realistic center-pivot scenario, MP-FVM demonstrates superior accuracy and mass-balance preservation compared to state-of-the-art solvers and PINNs, with viable computational efficiency when leveraging pre-trained models. This work offers a scalable, physics-informed, data-driven solver that integrates traditional numerical methods with modern neural architectures for geotechnical applications and potentially other PDEs in porous media.

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.

Figures (14)

• Figure 1: Flowchart of our proposed algorithm to solve the FVM-discretized Richards equation using a message passing mechanism.
• Figure 2: Comparison of pressure head solution profiles at $t=T=360$ seconds under (a) $S = 500$ iterations and (b) $\mathrm{tol}=3.2\times 10^{-5}$ for the 1-D benchmark problem celia1990general using standard and adaptive fixed-point iteration schemes (Equation \ref{['eqn_lscheme2']}). The solutions obtained from celia1990general based on very fine space and time steps are marked as the ground truth solutions.
• Figure 3: The relationships between $1640$ pressure head solutions $\psi$ and $\mu$, which are obtained by two distinct approaches. The resulting nonlinearity present in these reference solutions highlights need for data-driven approach.
• Figure 4: Persistence diagrams edelsbrunner2013persistent for pressure head solutions $\psi$ (left) and $\mu$ (right). The marked differences in topological features illustrate the need for an encoder to map $\psi$ into the topological space of $\mu$. Here, $\infty$ refers to infinite lifespan and $H_0$ are connected components.
• Figure 5: Comparison of pressure head solution profiles at $t=T=360$ seconds produced from adaptive fixed-point iteration scheme only (Equation \ref{['eqn_lscheme2']}) and from MP-FVM algorithm (Equation \ref{['eqn_DRWlscheme']}) with and without implementing Sobolev training.

Theorems & Definitions (8)

• Definition 1.1
• Theorem 4.1
• proof
• Theorem 5.1
• proof
• Lemma A.1: Theorem 6 of fontaine2021
• Lemma A.2: Equation 35 of Berner2019
• proof