Deep learning to discover and predict dynamics on an inertial manifold
This addresses the challenge of modeling chaotic systems for researchers in computational physics and dynamical systems, though it appears incremental as a hybrid extension of existing dimension reduction techniques.
The authors tackled the problem of representing chaotic dynamics on an inertial manifold for the Kuramoto-Sivashinsky equation, developing a hybrid linear-neural network dimension reduction method that substantially outperforms linear reduction alone in reproducing attractor features.
A data-driven framework is developed to represent chaotic dynamics on an inertial manifold (IM), and applied to solutions of the Kuramoto-Sivashinsky equation. A hybrid method combining linear and nonlinear (neural-network) dimension reduction transforms between coordinates in the full state space and on the IM. Additional neural networks predict time-evolution on the IM. The formalism accounts for translation invariance and energy conservation, and substantially outperforms linear dimension reduction, reproducing very well key dynamic and statistical features of the attractor.