To generalise the backpropagation method to both discrete-time and continuous-time hyperbolic chaos, we introduce the adjoint shadowing operator S acting on covector fields. We show that S can be equivalently defined as: S is the adjoint of the linear shadowing operator S; S is given by a ‘split then propagate’ expansion formula; S(ω) is the only bounded inhomogeneous adjoint solution of ω.By (a), 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), S also expresses the other part of the linear response, the unstable contribution. By (c), 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.

中文翻译：

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通过伴随阴影在双曲混沌中反向传播

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，我们引入了伴随阴影算子 S 作用于协向量场。我们表明 S 可以等效地定义为： S 是线性阴影算子的伴随S; S 由“分裂然后传播”展开公式给出； S（ω） 是唯一有界非齐次伴随解ω。由（一）， S 伴随地表示了阴影贡献，这是线性响应的重要部分，其中线性响应是长期统计数据相对于系统参数的导数。由(b)， S 还表示线性响应的另一部分，即不稳定贡献。由(c)， S 可以通过 Ni 和 Talnikar (2019J. 计算机。物理。 第395章690–709），这与传统的反向传播算法类似。对于连续时间情况，我们还表明线性响应允许明确分解为阴影和不稳定贡献。