Adjoint shadowing for backpropagation in hyperbolic chaos

For 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; (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, the most significant part of the linear response, where the linear response is the derivative of the long-time statistics with respect to 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, 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.
Comment: 20 pages