RESOLVING STEEP GRADIENTS FOR PHYSICS-INFORMED NEURAL NETWORKS: RICHARDS’ EQUATION AND CONVECTION-DIFFUSION EQUATION

Abstract

Solving partial differential equations (PDEs) is fundamental to modeling physical, biological, and engineering systems. Traditional numerical methods, while accurate and robust, often struggle with high-dimensional problems, complex geometries, discontinuous boundary conditions, and steep solution gradients. Deep learning, and in particular Physics-Informed Neural Networks (PINNs), offers a promising mesh-free alternative by embedding physical laws directly into the training process of neural networks.

This dissertation investigates and improves the performance of PINNs for PDEs with sharp transitions and nonlinearities. We focus on two key equations: Richards’ Equation, which models unsaturated flow in porous media, and the Convection-Diffusion Equation, which governs advective-diffusive transport processes. For Richards' Equation, we introduce a PINN framework that includes surface flux as an input and discrete residuals to enforce causality. For convection-diffusion problems, we address both known and unknown gradient layer locations. When the location is known, we analyze PINN limitations and propose input transformations that focus model capacity on high-gradient regions. When the location is unknown or complex, we introduce the Auxiliary-Input PINN (AI-PINN), a novel architecture that adapts spatial transformations based on an auxiliary input. A successive training strategy is also used to learn solutions without prior knowledge of gradient positions.

The methods developed in this work offer general strategies for improving the accuracy and robustness of PINNs when applied to challenging PDE problems with steep gradients. Through a combination of architecture design, loss reformulation, and training strategies, this dissertation contributes toward making physics-informed learning a viable tool for complex real-world simulations.