Discovery of slow variables in a class of multiscale stochastic systems via neural networks

TL;DR

This paper introduces a neural network-based method to identify slow variables in multiscale stochastic systems, enabling automatic low-dimensional reduction and improved understanding of complex dynamics.

Contribution

It proposes a supervised neural network architecture with an encoder-decoder to extract slow representations, including an error measure and pruning technique for essential coordinate identification.

Findings

Successfully discovers correct slow representations in tested examples

Provides an error metric to evaluate embedding quality

Pruning identifies system's essential coordinates

Abstract

Finding a reduction of complex, high-dimensional dynamics to its essential, low-dimensional "heart" remains a challenging yet necessary prerequisite for designing efficient numerical approaches. Machine learning methods have the potential to provide a general framework to automatically discover such representations. In this paper, we consider multiscale stochastic systems with local slow-fast time scale separation and propose a new method to encode in an artificial neural network a map that extracts the slow representation from the system. The architecture of the network consists of an encoder-decoder pair that we train in a supervised manner to learn the appropriate low-dimensional embedding in the bottleneck layer. We test the method on a number of examples that illustrate the ability to discover a correct slow representation. Moreover, we provide an error measure to assess the quality of the embedding and demonstrate that pruning the network can pinpoint an essential coordinates of the system to build the slow representation.