论文标题
使用神经网络对本地非线性动力学系统的不确定性定量
Uncertainty Quantification of Locally Nonlinear Dynamical Systems using Neural Networks
论文作者
论文摘要
通常以微分方程为代表物理系统的模型给出。在存在不确定性的情况下，使用模型对这些系统的行为进行准确的预测需要了解响应中不确定性的影响。在不确定性量化中，寻求这些物理系统响应的均值和差异之类的统计数据。为了估算这些基于统计的方法，例如蒙特卡洛（Monte Carlo）通常需要对模型的管理方程进行多次评估，以实现不确定性的多个实现。但是，对于大型复杂工程系统，这些方法在计算上变得繁重。在结构工程中，通常有线性结构包含在它们中存在不确定性的空间局部非线性。他们的标准非线性求解器采用基于抽样的方法进行不确定性量化的方法会导致估计响应统计数据的大量计算成本。为了减轻大规模局部非线性动力系统不确定性量化的计算负担，本文提出了一种方法，该方法将响应分解为两个部分 - 标称线性系统的响应和纠正术语。这个纠正术语是包含非线性和不确定性信息的伪造的响应。在本文中，由于计算能力的提高，神经网络是科学机器学习社区中通用功能近似的一种最近流行的工具，以及使用Pytorch和Tensorflow（TensorFlow）的可用性来估计假冒产品。由于仅使用神经网络对非线性和不确定的伪造进行建模，因此可以使用相同的网络来预测系统的不同响应，因此，如果寻求不同响应的统计数据，则不需要训练。
Models are often given in terms of differential equations to represent physical systems. In the presence of uncertainty, accurate prediction of the behavior of these systems using the models requires understanding the effect of uncertainty in the response. In uncertainty quantification, statistics such as mean and variance of the response of these physical systems are sought. To estimate these statistics sampling-based methods like Monte Carlo often require many evaluations of the models' governing equations for multiple realizations of the uncertainty. However, for large complex engineering systems, these methods become computationally burdensome. In structural engineering, often an otherwise linear structure contains spatially local nonlinearities with uncertainty present in them. A standard nonlinear solver for them with sampling-based methods for uncertainty quantification incurs significant computational cost for estimating the statistics of the response. To ease this computational burden of uncertainty quantification of large-scale locally nonlinear dynamical systems, a method is proposed herein, which decomposes the response into two parts -- response of a nominal linear system and a corrective term. This corrective term is the response from a pseudoforce that contains the nonlinearity and uncertainty information. In this paper, neural network, a recently popular tool for universal function approximation in the scientific machine learning community due to the advancement of computational capability as well as the availability of open-sourced packages like PyTorch and TensorFlow is used to estimate the pseudoforce. Since only the nonlinear and uncertain pseudoforce is modeled using the neural networks the same network can be used to predict a different response of the system and hence no new network is required to train if the statistic of a different response is sought.
