自编码器与数值分析——面向Navier-Stokes流的流形学习
Autoencoders vs. Numerical Analysis--Informed Manifold Learning for Navier--Stokes Flows
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中文总结 AI 辅助
本文用简约扩散映射替代自编码器进行非线性降阶建模，在旋转圆柱Navier-Stokes流中实现更优或相当的重建预测精度，且计算成本低几个数量级。
中文摘要 AI 辅助
自编码器（AEs）已成为数据驱动降阶建模（ROM）中非线性潜空间构建的主导方法，其解码器将潜在表示提升回环境状态空间。然而，其主导地位掩盖了一种既有的替代方案：基于经典数值分析的流形学习方法。我们使用简约扩散映射（PDMs）重新审视这一替代方案，并将其与基于本征正交分解（POD）的ROM以及多种卷积自编码器架构进行基准比较，研究对象为旋转圆柱后的二维不可压缩流动——这是一个由余维2的Bogdanov-Takens点及其相关的Hopf、鞍结和同宿分岔组织的分岔Navier-Stokes（NS）系统。我们的方法使用PDMs识别一组简约且可解释的固有潜在坐标，并直接从数据中估计其维度。然后，高斯过程回归学习潜在动力学，而PDMs空间中的凸K近邻（K-NN）插值构建原像映射，我们为其建立了逐点一致性。由此产生的非线性ROM显著优于基于POD的ROM，并实现了与基于AE的ROM相当的重建和预测精度——在某些分岔区域甚至更好。同时，使用PDMs进行潜在变量学习所需的计算时间比AE训练少几个数量级。
英文摘要
Autoencoders (AEs) have become a dominant approach to nonlinear latent-space construction in data-driven reduced-order modelling (ROM), with their decoders lifting latent representations back to the ambient state space. Their prominence, however, has overshadowed an established alternative: manifold-learning methods grounded in classical numerical analysis. We revisit this alternative using Parsimonious Diffusion Maps (PDMs), benchmarking them against Proper Orthogonal Decomposition (POD)-based ROMs and several convolutional AE architectures for the two-dimensional incompressible flow past a rotating cylinder ---a bifurcating Navier-Stokes (NS) system organized by a codimension-2 Bogdanov-Takens point and its associated Hopf, saddle-node, and homoclinic bifurcations. Our approach uses PDMs to identify a parsimonious and interpretable set of intrinsic latent coordinates and to estimate their dimension directly from data. Gaussian process regression then learns the latent dynamics, while convex K-nearest-neighbor (K-NN) interpolation in PDMs space constructs the pre-image map, for which we establish pointwise consistency. The resulting nonlinear ROM substantially outperforms POD-based ROMs and achieves reconstruction and prediction accuracy comparable to ---and, in some bifurcating regimes, better than--- that of AE-based ROMs. At the same time, latent-variable learning with PDMs requires orders of magnitude less computational time than AE training.
发表机构
- Scuola Superiore Meridionale, School for Advanced Studies（南方高等研究院）
- SISSA International School for Advanced Studies Mathematics Area, mathLab（国际高等研究学校数学区数学实验室）
- Johns Hopkins University（约翰斯·霍普金斯大学）
- Institute of Science Technology for Energy Sustainable Mobility (STEMS) National Research Council (CNR)（国家研究委员会能源可持续交通科学与技术研究所）
- University of Naples “Federico II
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