Abstract: This is the second part of our series works on failure-informed adaptive sampling for physic-informed neural networks (PINNs). In our previous work (SIAM J. Sci. Comput. 45: A1971–A1994), 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. Here, we present two extensions of that work. The first extension consists 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 the training progress. The second extension is to present the subset simulation (SS) 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.

Key words: Physic-informed neural networks (PINNs), Adaptive sampling, Failure probability