In this paper, we propose a general framework for projection-based model order reduction assisted by deep neural networks. The proposed methodology, called ROM-net, consists in using deep learning techniques to adapt the reduced-order model to a stochastic input tensor whose nonparametrized variabilities strongly influence the quantities of interest for a given physics problem. In particular, we introduce the concept of dictionary-based ROM-nets, where deep neural networks recommend a suitable local reduced-order model from a dictionary. The dictionary of local reduced-order models is constructed from a clustering of simplified simulations enabling the identification of the subspaces in which the solutions evolve for different input tensors. The training examples are represented by points on a Grassmann manifold, on which distances are computed for clustering. This methodology is applied to an anisothermal elastoplastic problem in structural mechanics, where the damage field depends on a random temperature field. When using deep neural networks, the selection of the best reduced-order model for a given thermal loading is 60 times faster than when following the clustering procedure used in the training phase.

中文翻译：

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深度神经网络（ROM-net）辅助的模型降阶

在本文中，我们提出了一个基于深度神经网络的基于投影的模型降阶的通用框架。所提出的被称为ROM-net的方法论，包括使用深度学习技术使降阶模型适应于随机输入张量，其随机性的非参数变化强烈影响给定物理问题的关注量。特别是，我们介绍了基于字典的ROM网络的概念，其中深度神经网络从字典中推荐了合适的局部降阶模型。局部降阶模型的字典由简化模拟的聚类构造而成，这些简化模拟可以识别子空间，在该子空间中，解决方案针对不同的输入张量而发展。训练示例由Grassmann流形上的点表示，在哪些距离上进行聚类计算。此方法适用于结构力学中的等温弹塑性问题，其中损伤场取决于随机温度场。当使用深度神经网络时，对于给定的热负荷，选择最佳降阶模型的速度比遵循训练阶段中使用的聚类过程的速度快60倍。