提出流形上的非线性降维方法，可捕捉复杂流动的物理结构。
Proper Latent Decomposition
- 用自编码器在流形上学习低维隐空间，实现高维数据压缩。
- 通过微分几何工具构建度量约束的Eikonal求解器，支持隐空间计算。
- 在层流和湍流中识别出具有物理意义的主导模式，优于传统POD。
本文提出一种在流形上推广的恰当潜变量分解（PLD），作为对传统恰当正交分解（POD）的非线性扩展。PLD是一种非线性降阶建模技术，可将高维数据压缩为非线性坐标。首先，通过自编码器推断一个低维内在坐标空间（隐空间），该空间几何上为流形，能以比数值离散更少自由度准确描述流动。其次，利用微分几何工具开发直接作用于隐空间的数值方法，包括度量约束的Eikonal求解器，用于距离计算。基于此框架，提出在流形上执行PLD的算法。最后，分别在层流与湍流柯尔莫戈洛夫（Kolmogorov）流动案例中验证：层流情况下可导出纳维-斯托克斯方程的半解析解；在柯尔莫戈洛夫流动中识别出具有物理结构的主导模式，并与POD结果对比。本工作为分析自编码器、隐空间及高维数据的非线性降阶建模提供了新路径，推动科学洞察。
原文摘要 · Abstract (English)
In this paper, we introduce the proper latent decomposition (PLD) as a generalization of the proper orthogonal decomposition (POD) on manifolds. PLD is a nonlinear reduced-order modeling technique for compressing high-dimensional data into nonlinear coordinates. First, we compute a reduced set of intrinsic coordinates (latent space) to accurately describe a flow with fewer degrees of freedom than the numerical discretization. The latent space, which is geometrically a manifold, is inferred by an autoencoder. Second, we leverage tools from differential geometry to develop numerical methods for operating directly on the latent space; namely, a metric-constrained Eikonal solver for distance computations. With this proposed numerical framework, we propose an algorithm to perform PLD on the manifold. Third, we demonstrate results for a laminar flow case and the turbulent Kolmogorov flow. For the laminar flow case, we are able to identify a semi-analytical expression for the solution of Navier-Stokes; in the Kolmogorov flow case, we are able to identify a dominant mode that exhibits physical structures, which are compared with POD. This work opens opportunities for analyzing autoencoders and latent spaces, nonlinear reduced-order modeling and scientific insights into the structure of high-dimensional data.
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