Physics-informed neural networks for modeling infiltration and solute transport in porous media

Abstract

This thesis addresses forward and inverse modeling of water infiltration and solute transport in porous media by developing efficient physics-constrained computational numerical models. For the mathematical description of water infiltration, two formulations are considered: the standard Richards equation, which predicts stable and monotonic wetting fronts, and the extended BCJ–R equation, which accounts for unstable infiltration phenomena such as saturation overshoot and gravity-driven fingered flow. To model solute transport, the infiltration models are coupled with advection– dispersion–reaction equations that capture the dynamics of single- and multispecies solute transport. The transport framework incorporates biogeochemical processes including first-order decay of nitrogen compounds through nitrification and denitrification, reversible adsorption of solutes onto soil particles, and transformation pathways such as volatilization, mineralization, and microbial degradation. In the first part of this thesis, a numerical model based on a semi-implicit secondorder mixed finite element method is developed to simulate coupled water flow and solute transport in unsaturated porous media. Water infiltration is considered under a stable front regime using the Richards equation, which is coupled with an advection– dispersion–reaction system to represent fertilizer transport. The developed numerical model avoids nonlinear iterative solvers for the Richards equation, thereby mitigating convergence issues associated with highly nonlinear soil capillary models. Numerical experiments are conducted to solve the coupled system under scenarios including single- and multi-species nitrogen transport, pore-water electrical conductivity, and nitrate migration. Validation against analytical solutions, established numerical codes, and experimental datasets confirms the model’s accuracy in reproducing water infiltration dynamics and solute transport processes. This high-fidelity solver serves both as a benchmark and as a data source for the physics-informed machine learning models developed in subsequent chapters. In the second part, a physics-informed neural network (PINN) is developed for forward modeling of coupled water flow and solute transport. The model requires only the initial and boundary conditions, while the governing equations are embedded directly into the training process. The PINN’s performance is evaluated against the high-fidelity finite element solver from the first part, showing good agreement and confirming the potential of PINNs for forward modeling of water–solute transport in porous media. In the third part, the objective is to identify the most effective PINN strategy for modeling water infiltration and single- and multispecies solute transport in porous media. A comprehensive comparative study of different PINN formulations is conducted, evaluating their performance in forward simulations of coupled flow–transport processes as well as in inverse problems for estimating unknown model parameters. Among the approaches examined, the sequential PINN strategy, where the governing equations for water flow and solute transport are trained separately, emerges as the most effective. Its superiority is demonstrated through extensive validation against synthetic datasets and experimental benchmarks, establishing its robustness for predictive simulations and parameter identification in unsaturated porous media. In the fourth part, the sequential PINN framework is extended with transfer learning to address the complexity of multispecies solute transport in unsaturated soils. The parameters learned during the training of the Richards equation are used to initialize the solute transport PINNs, exploiting structural similarities between the governing PDEs, both of which involve advection–diffusion-type operators. This strategy accelerates convergence, enhances training stability, and improves predictive accuracy compared to randomly initialized models, as demonstrated in forward and inverse simulations across one- and two-dimensional soil configurations. In the fifth part, the sequential PINN paradigm is generalized to arbitrary one-way coupled systems, where each equation depends on the outputs of preceding ones in a fixed causal order. New training schemes are introduced to exploit these directional dependencies, mitigating convergence issues that arise when equations are trained simultaneously. The framework shows clear advantages over standard PINNs, achieving good accuracy and efficiency in modeling the coupled dynamics of water flow and multispecies solute transport in porous media. In the sixth part, the challenging problem of unstable infiltration in unsaturated soils is addressed by numerically solving the fourth-order BCJ–R equation. It was found that the PINN model for the Richards equation is not suitable for this case; therefore, a new solver is developed with tailored training strategies and modified neural architectures to manage the multiscale behavior and stiffness inherent to the BCJ–R equation. The resulting PINN model accurately reproduces key features such as saturation overshoot, shows good agreement with experimental data, and remains consistent with thermodynamic principles, as verified through entropy dissipation analysis in closed systems. Finally, the last part of this thesis investigates multitask learning for infiltration in unsaturated flow through the development of a fast and accurate surrogate model. A deep neural operator is introduced and trained in a meshless, data-free, and physicsinformed setting to learn the mapping from varying initial and boundary conditions to the corresponding soil moisture distributions. To enhance extrapolation capability, a fine-tuning stage is incorporated using either physical constraints or available soil moisture data. The model is validated against the high-fidelity finite element solver, achieving comparable accuracy at a fraction of the computational cost during inference. This framework establishes the neural operator as an efficient and reliable real-time surrogate for optimizing irrigation strategies.