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.

翻译：非饱和土壤中的时空水流动力学通常可由Richards方程建模。为克服求解这一高度非线性偏微分方程（PDE）带来的计算挑战，本文提出一种新颖的求解算法，命名为MP-FVM（消息传递-有限体积法），该算法在有限体积离散化框架中整体集成了自适应定点迭代方案、编码器-解码器神经网络架构、Sobolev训练和消息传递机制。我们深入探讨了引入这些组件以实现解的精度与稳定性协同提升的必要性与优势。同时证明，在合理假设下，MP-FVM算法能够精确求解混合形式的$n$维Richards方程，并确保收敛性。通过多个示例验证，相较于现有先进求解算法及商业HYDRUS求解器，MP-FVM算法不仅实现了更高的精度，还能更好地保持Richards方程的基本物理定律与质量守恒特性。