Backpropagation in hyperbolic chaos via adjoint shadowing
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866911762575523840 |
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| author | Ni, Angxiu |
| author_facet | Ni, Angxiu |
| contents | To generalize the backpropagation method to both discrete-time and continuous-time hyperbolic chaos, we introduce the adjoint shadowing operator $\mathcal{S}$ acting on covector fields. We show that $\mathcal{S}$ can be equivalently defined as:
(a) $\mathcal{S}$ is the adjoint of the linear shadowing operator $S$;
(b) $\mathcal{S}$ is given by a `split then propagate' expansion formula;
(c) $\mathcal{S}(ω)$ is the only bounded inhomogeneous adjoint solution of $ω$.
By (a), $\mathcal{S}$ adjointly expresses the shadowing contribution, a significant part of the linear response, where the linear response is the derivative of the long-time statistics with respect to system parameters. By (b), $\mathcal{S}$ also expresses the other part of the linear response, the unstable contribution. By (c), $\mathcal{S}$ can be efficiently computed by the nonintrusive shadowing algorithm in Ni and Talnikar (2019 J. Comput. Phys. 395 690-709), which is similar to the conventional backpropagation algorithm. For continuous-time cases, we additionally show that the linear response admits a well-defined decomposition into shadowing and unstable contributions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_06648 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Backpropagation in hyperbolic chaos via adjoint shadowing Ni, Angxiu Dynamical Systems Numerical Analysis To generalize the backpropagation method to both discrete-time and continuous-time hyperbolic chaos, we introduce the adjoint shadowing operator $\mathcal{S}$ acting on covector fields. We show that $\mathcal{S}$ can be equivalently defined as: (a) $\mathcal{S}$ is the adjoint of the linear shadowing operator $S$; (b) $\mathcal{S}$ is given by a `split then propagate' expansion formula; (c) $\mathcal{S}(ω)$ is the only bounded inhomogeneous adjoint solution of $ω$. By (a), $\mathcal{S}$ adjointly expresses the shadowing contribution, a significant part of the linear response, where the linear response is the derivative of the long-time statistics with respect to system parameters. By (b), $\mathcal{S}$ also expresses the other part of the linear response, the unstable contribution. By (c), $\mathcal{S}$ can be efficiently computed by the nonintrusive shadowing algorithm in Ni and Talnikar (2019 J. Comput. Phys. 395 690-709), which is similar to the conventional backpropagation algorithm. For continuous-time cases, we additionally show that the linear response admits a well-defined decomposition into shadowing and unstable contributions. |
| title | Backpropagation in hyperbolic chaos via adjoint shadowing |
| topic | Dynamical Systems Numerical Analysis |
| url | https://arxiv.org/abs/2207.06648 |