Error Estimates and Physics Informed Augmentation of Neural Networks for Thermally Coupled Incompressible Navier Stokes Equations

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. \textit{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.