Learning High-Dimensional Chaos Based on an Echo State Network with Homotopy Transformation

Shikun Wang, Fengjie Geng, Yuting Li () and Hongjie Liu
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Shikun Wang: School of Science, China University of Geosciences (Beijing), Beijing 100083, China
Fengjie Geng: School of Science, China University of Geosciences (Beijing), Beijing 100083, China
Yuting Li: School of Science, China University of Geosciences (Beijing), Beijing 100083, China
Hongjie Liu: School of Urban Construction, Beijing City University, Beijing 101309, China

Mathematics, 2025, vol. 13, issue 6, 1-16

Abstract: Learning high-dimensional chaos is a complex and challenging problem because of its initial value-sensitive dependence. Based on an echo state network (ESN), we introduce homotopy transformation in topological theory to learn high-dimensional chaos. On the premise of maintaining the basic topological properties, our model can obtain the key features of chaos for learning through the continuous transformation between different activation functions, achieving an optimal balance between nonlinearity and linearity to enhance the generalization capability of the model. In the experimental part, we choose the Lorenz system, Mackey–Glass (MG) system, and Kuramoto–Sivashinsky (KS) system as examples, and we verify the superiority of our model by comparing it with other models. For some systems, the prediction error can be reduced by two orders of magnitude. The results show that the addition of homotopy transformation can improve the modeling ability of complex spatiotemporal chaotic systems, and this demonstrates the potential application of the model in dynamic time series analysis.

Keywords: echo state network; homotopy theory; activation function; chaotic systems (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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