  Numerical solutions to the equation for advection are determined using
different finite-difference approximations and physics-informed neural networks
(PINNs) under conditions that allow an analytical solution. Their accuracy is
examined by comparing them to the analytical solution. We used a machine
learning framework like PyTorch to implement PINNs. PINNs approach allows
training neural networks while respecting the PDEs as a strong constraint in
the optimization as apposed to making them part of the loss function. In
standard small-scale circulation simulations, it is shown that the conventional
approach incorporates a pseudo diffusive effect that is almost as large as the
effect of the turbulent diffusion model; hence the numerical solution is
rendered inconsistent with the PDEs. This oscillation causes inaccuracy and
computational uncertainty. Of all the schemes tested, only the PINNs
approximation accurately predicted the outcome. We assume that the PINNs
approach can transform the physics simulation area by allowing real-time
physics simulation and geometry optimization without costly and time-consuming
simulations on large supercomputers.
