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Learning a reduced basis of dynamical systems using an autoencoder.

David Sondak¹, Pavlos Protopapas¹

¹Institute for Applied Computational Science, Harvard University, Cambridge, Massachusetts 02138, USA.

Physical Review. E

|October 16, 2021

Summary

This summary is machine-generated.

Machine learning, specifically autoencoders, helps discover simplified representations of complex physics equations. This method reveals the underlying dimensions of partial differential equations like Kuramoto-Sivashinsky and Korteweg-de Vries.

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Area of Science:

Computational Physics
Applied Mathematics
Machine Learning

Background:

Machine learning models are increasingly utilized in physics and engineering for complex problem-solving.
Discovering reduced-order models for partial differential equations (PDEs) is crucial for efficient simulation and analysis.

Purpose of the Study:

To apply an autoencoder with latent space penalization to identify approximate finite-dimensional manifolds for canonical PDEs.
To analyze the dimensionality and structure of these manifolds for the Kuramoto-Sivashinsky (K-S), Korteweg-de Vries (KdV), and damped KdV equations.

Main Methods:

Utilized an autoencoder with latent space penalization technique.
Tested the methodology on the Kuramoto-Sivashinsky (K-S) equation, the Korteweg-de Vries (KdV) equation, and the damped KdV equation.
Analyzed the resulting latent space to determine its dimensionality and uncover nonlinear bases.

Main Results:

The optimal latent space for the K-S equation aligns with the dimension of its inertial manifold.
A nonlinear basis representing the K-S equation's latent space manifold was successfully identified.
Reduced latent space recovery for the KdV equation proved challenging, consistent with its infinite-dimensional dynamics.
For the damped KdV equation, the number of active dimensions decreased as the damping coefficient increased.

Conclusions:

The autoencoder method effectively identifies approximate finite-dimensional manifolds for certain PDEs.
The dimensionality of the latent space provides insights into the intrinsic dynamics of the studied PDEs.
The findings suggest potential for using machine learning to simplify and analyze complex dynamical systems in physics and engineering.