Investigating Guiding Information for Adaptive Collocation Point
  Sampling in PINNs
Jose Florido He Wang Amirul Khan Peter K. Jimack
Physics-informed neural networks (PINNs) provide a means of obtaining
approximate solutions of partial differential equations and systems through the
minimisation of an objective function which includes the evaluation of a
residual function at a set of collocation points within the domain. The quality
of a PINNs solution depends upon numerous parameters, including the number and
distribution of these collocation points. In this paper we consider a number of
strategies for selecting these points and investigate their impact on the
overall accuracy of the method. In particular, we suggest that no single
approach is likely to be ``optimal'' but we show how a number of important
metrics can have an impact in improving the quality of the results obtained
when using a fixed number of residual evaluations. We illustrate these
approaches through the use of two benchmark test problems: Burgers' equation
and the Allen-Cahn equation.