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.

翻译：这是我们对物理信息神经网络（FI-PINNs）进行失效信息自适应采样系列研究的第二部分。在前序工作\cite{gao2022failure}中，我们提出了一种以失效概率作为后验误差指标的自适应采样框架，其中采用截断高斯模型来估计该指标。本文提出了FI-PINNs的两种新颖扩展。第一种扩展在于结合重采样技术，使新算法能够保持恒定的训练规模。该目标通过余弦退火策略实现，该策略在训练过程中逐步将配点采样从均匀分布过渡到自适应分布。第二种扩展是引入子集模拟算法作为后验模型（替代截断高斯模型）来估计误差指标，该方法能更有效地估计失效概率，并在失效区域生成新的有效训练点。我们通过多个具有挑战性的算例验证了新方法的性能，数值实验表明该算法较原始方法有显著改进。