  In a preliminary attempt to address the problem of data scarcity in
physics-based machine learning, we introduce a novel methodology for data
generation in physics-based simulations. Our motivation is to overcome the
limitations posed by the limited availability of numerical data. To achieve
this, we leverage a diffusion model that allows us to generate synthetic data
samples and test them for two canonical cases: (a) the steady 2-D Poisson
equation, and (b) the forced unsteady 2-D Navier-Stokes (NS)
{vorticity-transport} equation in a confined box. By comparing the generated
data samples against outputs from classical solvers, we assess their accuracy
and examine their adherence to the underlying physics laws. In this way, we
emphasize the importance of not only satisfying visual and statistical
comparisons with solver data but also ensuring the generated data's conformity
to physics laws, thus enabling their effective utilization in downstream tasks.

Fluid Dynamics;Machine Learning