Linear and Nonlinear Dimensionality Reduction from Fluid Mechanics to Machine Learning

Abstract

Dimensionality reduction is the essence of many data processing problems, including filtering, data compression, reduced-order modeling and pattern analysis. While traditionally tackled using linear tools in the fluid dynamics community, nonlinear tools from machine learning are becoming increasingly popular. This article, halfway between a review and a tutorial, introduces a general framework for linear and nonlinear dimensionality reduction techniques. Differences and links between autoencoders and manifold learning methods are highlighted, and popular nonlinear techniques such as kernel Principal Component Analysis (kPCA), isometric feature learning (ISOMAPs) and Locally Linear Embedding (LLE) are placed in this framework. These algorithms are benchmarked in three classic problems: 1) filtering, 2) identification of oscillatory patterns, and 3) data compression. Their performances are compared against the traditional Proper Orthogonal Decomposition (POD) to provide a perspective on their diffusion in fluid dynamics.

submitted to Measurement Science and Technology

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Cited by in corpus (3)

• Gaussian process regression + deep neural network autoencoder for probabilistic surrogate modeling in nonlinear mechanics of solids
• A meshless method to compute the proper orthogonal decomposition and its variants from scattered data
• Model-based time super-sampling of turbulent flow field sequences