通过伴随跟踪实现双曲混沌中的反向传播
Backpropagation in hyperbolic chaos via adjoint shadowing
AI总结：
本文提出伴随跟踪算子 S，将反向传播推广到离散与连续时间双曲混沌，并统一刻画线性响应中的跟踪与不稳定贡献。
摘要：
为了将反向传播方法推广到离散时间和连续时间双曲混沌，我们引入作用于余向量场上的伴随跟踪算子 S。我们证明 S 可以等价地定义为：(a) S 是线性跟踪算子 S 的伴随；(b) S 由一个“先分裂后传播”的展开公式给出；(c) S(ω) 是 ω 的唯一有界非齐次伴随解。由 (a)，S 以伴随方式表示跟踪贡献，这是线性响应的重要组成部分，其中线性响应是长时间统计量关于系统参数的导数。由 (b)，S 还表示线性响应的另一部分，即不稳定贡献。由 (c)，S 可通过 Ni 和 Talnikar（2019年，《计算物理学杂志》395卷，690–709页）中的非侵入式跟踪算法高效计算，该算法类似于传统的反向传播算法。对于连续时间情形，我们还证明线性响应具有明确定义的跟踪贡献与不稳定贡献分解。
英文摘要：
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.
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