Discovery of slow variables in a class of multiscale stochastic systems via neural networks
This work addresses the challenge of dimensionality reduction in complex systems for researchers in computational science, but it appears incremental as it builds on existing machine learning frameworks for slow variable discovery.
The paper tackles the problem of reducing high-dimensional multiscale stochastic systems to low-dimensional slow variables by proposing a neural network method with an encoder-decoder architecture, which successfully discovers correct slow representations in tested examples and includes an error measure for quality assessment.
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.