Solving an Advection-Diffusion Equation Using Physics-Informed Neural Networks: Verification against Finite Volume method

Abstract

This thesis presents a concise comparative study between the classical Finite Volume Method (FVM) and the modern, mesh-free Physics-Informed Neural Networks (PINNs). Both approaches are evaluated by solving the advection-diffusion equation, which models bacterial chemotaxis spaces over time. The FVM relies on traditional grid discretization and an implicit Euler scheme, guaranteeing strict physical mass conservation. Conversely, PINNs embed the governing equations directly into a deep learning loss function, utilizing automatic differentiation to bypass classical mesh constraints. Numerical simulations show that while FVM ensures absolute conservative stability, PINNs offer a flexible, mesh-free alternative for complex mathematical domains.