Digital image processing (2nd ed.)
Digital image processing (2nd ed.)
Kuramoto-Sivashinsky dynamics on the center-unstable manifold
SIAM Journal on Applied Mathematics
Adaptive pattern recognition and neural networks
Adaptive pattern recognition and neural networks
Back in the saddle again: a computer assisted study of the Kuramoto-Sivashinsky equation
SIAM Journal on Applied Mathematics
Neural networks: algorithms, applications, and programming techniques
Neural networks: algorithms, applications, and programming techniques
Phase-space analysis of bursting behavior in Kolmogorov flow
Conference proceedings on Interpretation of time series from nonlinear mechanical systems
Neural networks: a tutorial
Preserving symmetries in the proper orthogonal decomposition
SIAM Journal on Scientific Computing
Symmetry of attractors and the Karhunen-Loe`ve decomposition
Trends and perspectives in applied mathematics
Timely Communication: Symmetry and the Karhunen--Loève Analysis
SIAM Journal on Scientific Computing
Simulating Neural Networks with Mathematica
Simulating Neural Networks with Mathematica
A Model for the Unstable Manifold of the Bursting Behavior in the 2D Navier--Stokes Flow
SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
The Whitney Reduction Network: A Method for Computing Autoassociative Graphs
Neural Computation
Hybrid Karhunen-Loeve/neural modelling for a class of distributed parameter systems
International Journal of Intelligent Systems Technologies and Applications
Unsupervised adaptive neural-fuzzy inference system for solving differential equations
Applied Soft Computing
Computers & Mathematics with Applications
Comparison of artificial neural network architecture in solving ordinary differential equations
Advances in Artificial Neural Systems
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The dynamics of two nonlinear partial differential equations (PDEs) known as the Kuramoto-Sivashinsky (K-S) equation and the two-dimensional Navier-Stokes (N-S) equations are analyzed using Karhunen-Loéve (K-L) decomposition and artificial neural networks (ANN). For the K-S equation, numerical simulations using a pseudospectral Galerkin method is presented at a bifurcation parameter α = 17.75, where a dynamical behavior represented by a heteroclinic connection is obtained. We apply K-L decomposition on the numerical simulation data with the task of reducing the data into a set of data coefficients. Then we use ANN to model, and predict the data coefficients at a future time. It is found that training the neural networks with only the first data coefficient is enough to capture the underlying dynamics, and to predict for the other remaining data coefficients. As for the two-dimensional N-S equation, a quasiperiodic behavior represented in phase space by a torus is analyzed at Re= 14.0. Applying the symmetry observed in the two-dimensional N-S equations on the quasiperiodic behavior, eight different tori were obtained. We show that by exploiting the symmetries of the equation and using K-L decomposition in conjunction with neural networks, a smart neural model is obtained.