Derivative estimation by RKHS regularization for learning dynamics from time-series data
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912305500913664 |
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| author | Guo, Hailong Li, Haibo |
| author_facet | Guo, Hailong Li, Haibo |
| contents | Learning the governing equations from time-series data has gained increasing attention due to its potential to extract useful dynamics from real-world data. Despite significant progress, it becomes challenging in the presence of noise, especially when derivatives need to be calculated. To reduce the effect of noise, we propose a method that simultaneously fits both the derivative and trajectory from noisy time-series data. Our approach formulates derivative estimation as an inverse problem involving integral operators within the forward model, and estimates the derivative function by solving a regularization problem in a vector-valued reproducing kernel Hilbert space (vRKHS). We derive an integral-form representer theorem, which enables the computation of the regularized solution by solving a finite-dimensional problem and facilitates efficiently estimating the optimal regularization parameter. By embedding the dynamics within a vRKHS and utilizing the fitted derivative and trajectory, we can recover the underlying dynamics from noisy data by solving a linear regularization problem. Several numerical experiments are conducted to validate the effectiveness and efficiency of our method. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_01289 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Derivative estimation by RKHS regularization for learning dynamics from time-series data Guo, Hailong Li, Haibo Numerical Analysis 37M10, 65P99, 65R32 Learning the governing equations from time-series data has gained increasing attention due to its potential to extract useful dynamics from real-world data. Despite significant progress, it becomes challenging in the presence of noise, especially when derivatives need to be calculated. To reduce the effect of noise, we propose a method that simultaneously fits both the derivative and trajectory from noisy time-series data. Our approach formulates derivative estimation as an inverse problem involving integral operators within the forward model, and estimates the derivative function by solving a regularization problem in a vector-valued reproducing kernel Hilbert space (vRKHS). We derive an integral-form representer theorem, which enables the computation of the regularized solution by solving a finite-dimensional problem and facilitates efficiently estimating the optimal regularization parameter. By embedding the dynamics within a vRKHS and utilizing the fitted derivative and trajectory, we can recover the underlying dynamics from noisy data by solving a linear regularization problem. Several numerical experiments are conducted to validate the effectiveness and efficiency of our method. |
| title | Derivative estimation by RKHS regularization for learning dynamics from time-series data |
| topic | Numerical Analysis 37M10, 65P99, 65R32 |
| url | https://arxiv.org/abs/2504.01289 |