论文标题
针对PINN的故障信息自适应抽样
Failure-informed adaptive sampling for PINNs
论文作者
论文摘要
物理知识的神经网络（PINN）已成为解决广泛域中PDE的有效技术。但是，注意到，PINN的性能可以通过不同的采样程序变化。例如，一组固定的（事先选择的）训练点可能无法捕获有效的解决方案区域（尤其是对于奇异性问题）。为了克服这个问题，我们在这项工作中提出了一种自适应策略，称为已发生故障的Pinns（FI-PINNS），该策略的灵感来自可靠性分析的观点。关键思想是定义基于残差的有效故障概率，然后为了将更多样品放置在故障区域中，Fi-Pinns采用了一种失败的富集技术来适应训练集，以使数值准确度适应新的套在一起点。简而言之，与自适应有限元方法相似，所提出的FI-Pinns采用故障概率作为后误差指标来生成新的训练点。我们证明了Fi-Pinns的严格错误范围，并通过几个问题说明了其性能。
Physics-informed neural networks (PINNs) have emerged as an effective technique for solving PDEs in a wide range of domains. It is noticed, however, the performance of PINNs can vary dramatically with different sampling procedures. For instance, a fixed set of (prior chosen) training points may fail to capture the effective solution region (especially for problems with singularities). To overcome this issue, we present in this work an adaptive strategy, termed the failure-informed PINNs (FI-PINNs), which is inspired by the viewpoint of reliability analysis. The key idea is to define an effective failure probability based on the residual, and then, with the aim of placing more samples in the failure region, the FI-PINNs employs a failure-informed enrichment technique to adaptively add new collocation points to the training set, such that the numerical accuracy is dramatically improved. In short, similar as adaptive finite element methods, the proposed FI-PINNs adopts the failure probability as the posterior error indicator to generate new training points. We prove rigorous error bounds of FI-PINNs and illustrate its performance through several problems.
