I am currently working on solving a problem to determine the velocity field from the time-varying concentration distribution within a rectangular prism with inflow and outflow sections.

The constraints include the advection-diffusion equation, continuity equation, and the residual of the Navier-Stokes equation to be zero, zero velocity at the walls, zero concentration gradient at the walls, zero pressure at the outflow section, and time-varying concentration data as data constraints.

However, I am facing an issue where the velocity at the outflow section becomes almost zero, causing the continuity equation to be violated across the entire domain. To resolve this, I would like to impose the continuity of flow rate as a condition, but since this is an inverse problem, it is challenging to provide the flow rate accurately.

I would appreciate any ideas on how to improve this prediction. For reference, I have attached screenshots of the current predicted velocity field, the correct velocity field, the concentration distribution at 0.1 seconds, and the concentration distribution at 1.2 seconds.

Thank you for your help.