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：本文是我们关于故障导向自适应采样在物理信息神经网络（FI-PINNs）领域系列研究的第二部分。在前期工作《gao2022failure》中，我们提出了基于故障概率作为后验误差指标的自适应采样框架，采用截断高斯模型对该指标进行估计。本研究提出两种FI-PINNs的新型扩展方案：第一种扩展结合了重采样技术，使新算法能够维持恒定训练集规模。通过余弦退火机制实现，该机制随训练进程逐步将配置点的采样方式从均匀采样过渡至自适应采样；第二种扩展采用子集模拟算法替代截断高斯模型作为后验估计模型，该算法不仅能更有效地评估故障概率，还可直接在故障区域生成新的有效训练点。我们通过多个具有挑战性的问题验证了新方法的性能，数值实验表明该算法相比原始方法具有显著优势。