Treatment of non-physical solutions of the oxygen diffusion in soil by physics-informed neural network

Abstract

We investigate oxygen diffusion in the soil in one dimension by finite differences and the physics-informed neural network. Solving the diffusion equation by either method determines the oxygen concentration profiles inside the soil column at various times. However, while respecting specified Dirichlet and Neumann boundary conditions, the concentration profiles at certain times become negative, which is non-physical per se. We can resolve this situation in finite differences by proclaiming these negative concentration values as zero during the time-stepping scheme. In the case of PINN, we propose an innovative solution with a custom loss function, tailored to avoid such non-physical behavior. Two types of Dirichlet boundary conditions are investigated. The first is constant, and the second one periodically changes, with a period of 24 hours. We demonstrate that the PINN with a customized loss is effective and accurate. The proposed approach to circumvent non-physical solution areas demonstrates promise for application to various analogous problems.