Mass-Conserving Physics-Informed Neural Networks For The One-Dimensional Advection-Diffusion Equation
The advection-diffusion equation is a fundamental model of transport phenomena in which mass conservation is an essential physical constraint. While classical schemes such as Crank-Nicolson preserve this property by construction, Physics-Informed Neural Networks (PINNs) enforce only the local residual of the governing PDE and are therefore not guaranteed to conserve global quantities such as mass over long integration horizons. In this work, we examine the extent of this limitation for the periodic one-dimensional advection-diffusion equation and evaluate a Mass-Penalty PINN that augments the standard PINN loss with a soft mass-conservation constraint. We compare the performance of Vanilla PINN, Mass-Penalty PINN, and the Crank-Nicolson scheme across a range of Peclet numbers spanning diffusion-dominated to advection-dominated regimes, and over two simulation horizons representing short-term and long-term dynamics. The results show that, for short-term simulations, the Mass-Penalty PINN does not always provide a consistent improvement in accuracy. However, for long-term simulations, the Mass-Penalty PINN reduces the relative L2 error and mass conservation error by factors of approximately 9-67 and 15-215, respectively, compared with the Vanilla PINN, across the tested Peclet numbers. Further analysis reveals that the accuracy degradation observed in Vanilla PINN is predominantly caused by the accumulation of mass drift over time. These results demonstrate that incorporating a soft mass-conservation constraint substantially improves the long-term reliability of PINN for conservative transport problems, particularly in mitigating mass drift over extended simulation horizons.