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) $\mathcal{S}(\omega)$ is the only bounded inhomogeneous adjoint solution of $\omega$. 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.

翻译：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，我们引入了作用于协向量场的伴随阴影算子$\mathcal{S}$。我们证明$\mathcal{S}$可等价定义为：（a）$\mathcal{S}$是线性阴影算子$S$的伴随算子；（b）$\mathcal{S}$由"先分裂后传播"的展开公式给出；（c）$\mathcal{S}(\omega)$是$\omega$唯一的有界非齐次伴随解。根据（a），$\mathcal{S}$伴随地表达了线性响应中的重要部分——阴影贡献，其中线性响应是长时间统计量对系统参数的导数。根据（b），$\mathcal{S}$还表达了线性响应的另一部分——不稳定贡献。根据（c），$\mathcal{S}$可以通过Ni和Talnikar（2019 J. Comput. Phys. 395 690-709）提出的非侵入式阴影算法高效计算，该算法类似于传统的反向传播算法。对于连续时间情况，我们进一步证明线性响应可分解为阴影贡献和不稳定贡献的明确定义形式。