  This is the second part of our series works on failure-informed adaptive
sampling for physic-informed neural networks (FI-PINNs). In our previous work
\cite{gao2022failure}, we have presented an adaptive sampling framework by
using the failure probability as the posterior error indicator, where the
truncated Gaussian model has been adopted for estimating the indicator. In this
work, we present two novel extensions to FI-PINNs. The first extension consist
in combining with a re-sampling technique, so that the new algorithm can
maintain a constant training size. This is achieved through a cosine-annealing,
which gradually transforms the sampling of collocation points from uniform to
adaptive via training progress. The second extension is to present the subset
simulation algorithm as the posterior model (instead of the truncated Gaussian
model) for estimating the error indicator, which can more effectively estimate
the failure probability and generate new effective training points in the
failure region. We investigate the performance of the new approach using
several challenging problems, and numerical experiments demonstrate a
significant improvement over the original algorithm.
