  Physics Informed Neural Networks (PINNs) are shown to be a promising method
for the approximation of Partial Differential Equations (PDEs). PINNs
approximate the PDE solution by minimizing physics-based loss functions over a
given domain. Despite substantial progress in the application of PINNs to a
range of problem classes, investigation of error estimation and convergence
properties of PINNs, which is important for establishing the rationale behind
their good empirical performance, has been lacking. This paper presents
convergence analysis and error estimates of PINNs for a multi-physics problem
of thermally coupled incompressible Navier-Stokes equations. Through a model
problem of Beltrami flow it is shown that a small training error implies a
small generalization error. Posteriori convergence rates of total error with
respect to the training residual and collocation points are presented. This is
of practical significance in determining appropriate number of training
parameters and training residual thresholds to get good PINNs prediction of
thermally coupled steady state laminar flows. These convergence rates are then
generalized to different spatial geometries as well as to different flow
parameters that lie in the laminar regime. A pressure stabilization term in the
form of pressure Poisson equation is added to the PDE residuals for PINNs. This
physics informed augmentation is shown to improve accuracy of the pressure
field by an order of magnitude as compared to the case without augmentation.
Results from PINNs are compared to the ones obtained from stabilized finite
element method and good properties of PINNs are highlighted.
