Learning solutions of parametric Navier-Stokes with physics-informed
  neural networks
M. Naderibeni M. J. T. Reinders L. Wu D. M. J. Tax
We leverage Physics-Informed Neural Networks (PINNs) to learn solution
functions of parametric Navier-Stokes Equations (NSE). Our proposed approach
results in a feasible optimization problem setup that bypasses PINNs'
limitations in converging to solutions of highly nonlinear parametric-PDEs like
NSE. We consider the parameter(s) of interest as inputs of PINNs along with
spatio-temporal coordinates, and train PINNs on generated numerical solutions
of parametric-PDES for instances of the parameters. We perform experiments on
the classical 2D flow past cylinder problem aiming to learn velocities and
pressure functions over a range of Reynolds numbers as parameter of interest.
Provision of training data from generated numerical simulations allows for
interpolation of the solution functions for a range of parameters. Therefore,
we compare PINNs with unconstrained conventional Neural Networks (NN) on this
problem setup to investigate the effectiveness of considering the PDEs
regularization in the loss function. We show that our proposed approach results
in optimizing PINN models that learn the solution functions while making sure
that flow predictions are in line with conservational laws of mass and
momentum. Our results show that PINN results in accurate prediction of
gradients compared to NN model, this is clearly visible in predicted vorticity
fields given that none of these models were trained on vorticity labels.