Autoencoders for discovering manifold dimension and coordinates in data from complex dynamical systems

TL;DR

This paper introduces an autoencoder-based method that estimates the intrinsic dimension and coordinates of data from complex dynamical systems, enabling better understanding and modeling of high-dimensional phenomena on lower-dimensional manifolds.

Contribution

The work presents a novel autoencoder framework combining implicit regularization and linear layers to automatically determine manifold dimension and coordinates, with applications to dynamical systems.

Findings

Successfully estimates manifold dimension across various dynamical systems.

Reveals how implicit regularization and weight decay facilitate low-rank learning.

Extends to state-space modeling and forecasting of chaotic PDEs.

Abstract

While many phenomena in physics and engineering are formally high-dimensional, their long-time dynamics often live on a lower-dimensional manifold. The present work introduces an autoencoder framework that combines implicit regularization with internal linear layers and $L_2$ regularization (weight decay) to automatically estimate the underlying dimensionality of a data set, produce an orthogonal manifold coordinate system, and provide the mapping functions between the ambient space and manifold space, allowing for out-of-sample projections. We validate our framework's ability to estimate the manifold dimension for a series of datasets from dynamical systems of varying complexities and compare to other state-of-the-art estimators. We analyze the training dynamics of the network to glean insight into the mechanism of low-rank learning and find that collectively each of the implicit regularizing layers compound the low-rank representation and even self-correct during training. Analysis of gradient descent dynamics for this architecture in the linear case reveals the role of the internal linear layers in leading to faster decay of a "collective weight variable" incorporating all layers, and the role of weight decay in breaking degeneracies and thus driving convergence along directions in which no decay would occur in its absence. We show that this framework can be naturally extended for applications of state-space modeling and forecasting by generating a data-driven dynamic model of a spatiotemporally chaotic partial differential equation using only the manifold coordinates. Finally, we demonstrate that our framework is robust to hyperparameter choices.