Learning High-Dimensional Chaos Based on an Echo State Network with Homotopy Transformation
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 topologica...
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| Format: | Article |
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MDPI AG
2025-03-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/13/6/894 |
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| author | Shikun Wang Fengjie Geng Yuting Li Hongjie Liu |
| author_facet | Shikun Wang Fengjie Geng Yuting Li Hongjie Liu |
| author_sort | Shikun Wang |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-37bb350d7c9d425fbc378f44901e418a |
| institution | DOAJ |
| issn | 2227-7390 |
| language | English |
| publishDate | 2025-03-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj-art-37bb350d7c9d425fbc378f44901e418a2025-08-20T02:42:22ZengMDPI AGMathematics2227-73902025-03-0113689410.3390/math13060894Learning High-Dimensional Chaos Based on an Echo State Network with Homotopy TransformationShikun Wang0Fengjie Geng1Yuting Li2Hongjie Liu3School of Science, China University of Geosciences (Beijing), Beijing 100083, ChinaSchool of Science, China University of Geosciences (Beijing), Beijing 100083, ChinaSchool of Science, China University of Geosciences (Beijing), Beijing 100083, ChinaSchool of Urban Construction, Beijing City University, Beijing 101309, ChinaLearning 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.https://www.mdpi.com/2227-7390/13/6/894echo state networkhomotopy theoryactivation functionchaotic systems |
| spellingShingle | Shikun Wang Fengjie Geng Yuting Li Hongjie Liu Learning High-Dimensional Chaos Based on an Echo State Network with Homotopy Transformation Mathematics echo state network homotopy theory activation function chaotic systems |
| title | Learning High-Dimensional Chaos Based on an Echo State Network with Homotopy Transformation |
| title_full | Learning High-Dimensional Chaos Based on an Echo State Network with Homotopy Transformation |
| title_fullStr | Learning High-Dimensional Chaos Based on an Echo State Network with Homotopy Transformation |
| title_full_unstemmed | Learning High-Dimensional Chaos Based on an Echo State Network with Homotopy Transformation |
| title_short | Learning High-Dimensional Chaos Based on an Echo State Network with Homotopy Transformation |
| title_sort | learning high dimensional chaos based on an echo state network with homotopy transformation |
| topic | echo state network homotopy theory activation function chaotic systems |
| url | https://www.mdpi.com/2227-7390/13/6/894 |
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