论文标题
双曲线混乱中的反向传播通过伴随阴影
Backpropagation in hyperbolic chaos via adjoint shadowing
论文作者
论文摘要
为了将反向传播方法概括为离散的时间和连续的双曲线混乱，我们介绍了在Covector Fields上作用的伴随阴影操作员$ \ MATHCAL {S} $。我们表明$ \ MATHCAL {S} $可以等效地定义为： （a）$ \ MATHCAL {s} $是线性阴影操作员$ s $的伴随； （b）$ \ MATHCAL {S} $由`split tht the stapagate''扩展公式给出； （c）$ \ MATHCAL {S}（ω）$是$ω$的唯一有限的不均匀伴随解决方案。 通过（a），$ \ Mathcal {s} $ auppointly表示阴影贡献，这是线性响应的重要部分，其中线性响应是长期统计信息相对于系统参数的派生。通过（b），$ \ Mathcal {s} $还表示线性响应的另一部分，即不稳定的贡献。通过（c），可以通过Ni和Talnikar中的非感染阴影算法来有效计算$ \ Mathcal {s} $（2019 J.Comput。395690-709），这与常规的返回算法相似。对于连续的时间案例，我们还表明，线性响应将定义明确的分解构成对阴影和不稳定的贡献。
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.
