  We propose a novel method for fast and accurate training of physics-informed
neural networks (PINNs) to find solutions to boundary value problems (BVPs) and
initial boundary value problems (IBVPs). By combining the methods of training
deep neural networks (DNNs) and Extreme Learning Machines (ELMs), we develop a
model which has the expressivity of DNNs with the fine-tuning ability of ELMs.
We showcase the superiority of our proposed method by solving several BVPs and
IBVPs which include linear and non-linear ordinary differential equations
(ODEs), partial differential equations (PDEs) and coupled PDEs. The examples we
consider include a stiff coupled ODE system where traditional numerical methods
fail, a 3+1D non-linear PDE, Kovasznay flow and Taylor-Green vortex solutions
to incompressible Navier-Stokes equations and pure advection solution of 1+1 D
compressible Euler equation.
  The Theory of Functional Connections (TFC) is used to exactly impose initial
and boundary conditions (IBCs) of (I)BVPs on PINNs. We propose a modification
to the TFC framework named Reduced TFC and show a significant improvement in
the training and inference time of PINNs compared to IBCs imposed using TFC.
Furthermore, Reduced TFC is shown to be able to generalize to more complex
boundary geometries which is not possible with TFC. We also introduce a method
of applying boundary conditions at infinity for BVPs and numerically solve the
pure advection in 1+1 D Euler equations using these boundary conditions.
