Modeling subsurface flow and transport phenomena is essential for addressing a wide range of challenges in engineering, hydrology, and ecology. The Richards equation is a cornerstone for simulating infiltration, and when coupled with advection–dispersion equations, it provides insights into solute transport. However, the complexity of this coupled model increases significantly when dealing with multiple solute transport. Physics-informed neural networks (PINNs) offer a flexible technique that merges data-driven approaches with the underlying physics principles, enabling the direct incorporation of physical laws or constraints into the neural network training process. Nevertheless, employing PINNs for solving multi-physics problems can present challenges during training, particularly in achieving convergence to realistic concentration profiles. Our study introduces a transfer learning technique to tackle the challenge of modeling multiple species transport in unsaturated soils. This approach aims to improve the accuracy of the PINN framework by decoupling the training process and solving the governing partial differential equations (PDEs) sequentially. We incorporate various strategies to optimize and accelerate the training process. Specifically, we begin by solving the Richards equation and then transfer the acquired knowledge to subsequent solute PINN solvers. This strategy leverages the fact that these PDEs have some similarities in their structure as advection–diffusion equations. To rigorously validate our approach, we conduct 1D numerical experiments and extend our analysis to encompass 2D problems, and inverse problems for homogeneous soils, as well as numerical tests using layered soils. Our findings indicate that transferring learned features is more advantageous than utilizing random features, highlighting the effectiveness of the proposed strategy.

中文翻译：

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一种基于迁移学习物理的深度学习框架，用于模拟非饱和土壤中的多种溶质动力学


模拟地下流动和传输现象对于解决工程、水文学和生态学领域的各种挑战至关重要。理查兹方程是模拟渗透的基石，当与平流-弥散方程结合使用时，它可以提供对溶质输运的深入了解。然而，在处理多种溶质迁移时，这种耦合模型的复杂性显着增加。物理信息神经网络 (PINN) 提供了一种灵活的技术，将数据驱动的方法与基本物理原理相结合，从而能够将物理定律或约束直接纳入神经网络训练过程中。然而，使用 PINN 来解决多物理问题可能会在训练过程中带来挑战，特别是在实现与实际浓度分布的收敛方面。我们的研究引入了迁移学习技术来应对非饱和土壤中多种物种迁移建模的挑战。该方法旨在通过解耦训练过程并依次求解控制偏微分方程 (PDE) 来提高 PINN 框架的准确性。我们采用各种策略来优化和加速培训过程。具体来说，我们首先求解 Richards 方程，然后将获得的知识转移到后续的溶质 PINN 求解器中。该策略利用了这样一个事实：这些偏微分方程在结构上与平流扩散方程有一些相似之处。为了严格验证我们的方法，我们进行了一维数值实验，并将我们的分析扩展到包括二维问题和均质土壤的反问题，以及使用分层土壤的数值测试。 我们的研究结果表明，转移学习的特征比利用随机特征更有优势，凸显了所提出策略的有效性。