Linear versus nonlinear dimensionality reduction of high-dimensional dynamical systems. (English) Zbl 1058.35098
The author uses combinations of the K-L decomposition and neural networks to obtain the intrinsic or true dimension of two PDEs, namely, the 1-d K-S equation and the 2-d N-S equations. For the 1-d K-S equation, he investigates one particular dynamical behavior which, in phase space, is represented by a heteroclinic connection. As for the 2-d N-S equation, a quasi-periodic behavior is examined. In both studies, the powers of neural networks in extracting the intrinsic dimension of both dynamics are presented.
Reviewer: Qin Mengzhao (Beijing)
MSC:
35K55 | Nonlinear parabolic equations |
37N10 | Dynamical systems in fluid mechanics, oceanography and meteorology |
65P20 | Numerical chaos |
65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |