# Estimating of the inertial manifold dimension for a chaotic attractor of complex Ginzburg-Landau equation using a neural network

arXiv: https://arxiv.org/abs/1811.11851 · HTML: https://mlsanity.com/paper/1811.11851/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Nov 2018.

## Built on by (most cited first, 1 of 1)

- [Reconstruction of neuromorphic dynamics from a single scalar time series using variational autoencoder and neural network map](https://mlsanity.com/paper/2411.07055/) (2024)

## Abstract (arXiv)

Dimension of an inertial manifold for a chaotic attractor of spatially distributed system is estimated using autoencoder neural network. The inertial manifold is a low dimensional manifold where the chaotic attractor is embedded. The autoencoder maps system state vectors onto themselves letting them pass through an inner state with a reduced dimension. The training processes of the autoencoder is shown to depend dramatically on the reduced dimension: a learning curve saturates when the dimension is too small and decays if it is sufficient for a lossless information transfer. The smallest sufficient value is considered as a dimension of the inertial manifold, and the autoencoder implements a mapping onto the inertial manifold and back. The correctness of the computed dimension is confirmed by its remarkable coincidence with the one obtained as a number of covariant Lyapunov vectors with vanishing pairwise angles. These vectors are called physical modes. Unlike never having zero angles residual ones they are known to span a tangent subspace for the inertial manifold.