Projection-based model order reduction on nonlinear manifolds has been recently proposed for problems with slowly decaying Kolmogorov n-width such as advection-dominated ones. These methods often use neural networks for manifold learning and showcase improved accuracy over traditional linear subspace-reduced order models. A disadvantage of the previously proposed methods is the potential high computational costs of training the networks on high-fidelity solution snapshots. In this work, we propose and analyze a novel method that overcomes this disadvantage by training a neural network only on subsampled versions of the high-fidelity solution snapshots. This method coupled with collocation-based hyper-reduction and Gappy-POD allows for efficient and accurate surrogate models. We demonstrate the validity of our approach on a 2d Burgers problem.

翻译：基于投影的非线性流形模型降阶方法近期被提出，用于处理科尔莫戈罗夫n宽度缓慢衰减的问题（如对流主导问题）。这类方法通常利用神经网络进行流形学习，相较于传统线性子空间降阶模型展现出更优精度。先前方法的缺陷在于，对高保真解快照进行网络训练可能需要高昂的计算成本。本文提出并分析了一种新型方法，通过仅对高保真解快照的降采样版本训练神经网络，克服了这一缺陷。该方法结合配置点超降阶与Gappy-POD技术，可构建高效精确的代理模型。我们通过二维Burgers问题验证了该方法的有效性。