Impact of collocation point sampling techniques on PINN performance in groundwater flow predictions

Physics-Informed Neural Networks (PINNs) represent a promising methodology for addressing partial differential equations in scientific computing. This study examines optimization strategies for PINNs in groundwater flow modeling, concentrating on two main aspects: the distribution of collocation points and training strategies. By examining both point spatial distribution and dynamic resampling approaches, we show how collocation points arrangements can improve solution accuracy, particularly for problems with localized features such as source/sink terms represented by Dirac delta functions.

We introduce and analyze the Chebyshev-exponential (ChebEx) distribution for collocation points, as well as non-adaptive resampling strategies used during training.

In an explainable AI (XAI) setting our findings show that a ChebEx distribution of points improves accuracy over uniform sampling, especially near source terms. We also demonstrate that periodic resampling of collocation points improves training stability. These findings contribute to a better understanding of PINNs optimization and help to broaden our understanding of how spatial point selection influences PINN training dynamics and solution quality, but more research is needed for heterogeneous media and complex boundary conditions.

While our implementation focuses on one-dimensional groundwater flow with homogeneous boundary conditions, the methodologies presented here could be applied to a variety of physical systems governed by partial differential equations (PDEs), including heat transfer, fluid dynamics, and electromagnetic fields.