  Physics informed neural networks (PINNs) are nowadays used as efficient
machine learning methods for solving differential equations. However,
vanilla-PINNs fail to learn complex problems as ones involving stiff ordinary
differential equations (ODEs). This is the case of some initial value problems
(IVPs) when the amount of training data is too small and/or the integration
interval (for the variable like the time) is too large. We propose very simple
recipes to improve the training process in cases where only prior knowledge at
initial time of training data is known for IVPs. For example, more physics can
be easily embedded in the loss function in problems for which the total energy
is conserved. A better definition of the training data loss taking into account
all the initial conditions can be done. In a progressive learning approach, it
is also possible to use a growing time interval with a moving grid (of
collocation points) where the differential equation residual is minimized.
These improvements are also shown to be efficient in PINNs modeling for solving
boundary value problems (BVPs) as for the high Reynolds steady-state solution
of advection-diffusion equation.
