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:这是我们系列工作系列的第二部分,用于对物理知情神经网络(FI-PINNs)进行不知情的适应性适应性抽样(FI-PINNs)。在我们以前的工作 \ cite{gao2022failure} 中,我们提出了一个适应性抽样框架,将失败概率作为事后误差指标,在评估指标时采用了短员高斯模型。在这项工作中,我们向FI-PINNs提供了两个新的扩展。第一个扩展包括与再抽样技术相结合,以便新的算法能够保持一个不变的培训规模。这是通过合作-销毁实现的,通过培训进展,将同地点的抽样从统一转变为适应性。第二个扩展是将子集模拟算法作为估计误差指标的模型(而不是短员高斯模型),这样可以更有效地估计失败概率并产生新的有效培训点。我们利用若干棘手问题来调查新方法的绩效,并用数字实验来证明原算法的显著改进。</s>