Estimating of the inertial manifold dimension for a chaotic attractor of complex Ginzburg-Landau equation using a neural network
This provides a method for estimating inertial manifold dimensions in spatially distributed chaotic systems, which is incremental as it applies an existing neural network technique to a specific problem.
The authors tackled the problem of estimating the inertial manifold dimension for a chaotic attractor in the complex Ginzburg-Landau equation by using an autoencoder neural network, achieving a dimension estimate that remarkably coincided with the number of physical modes from covariant Lyapunov vectors.
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.