Backpropagation in hyperbolic chaos via adjoint shadowing

To generalise 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:

1. $\mathcal{S}$ is the adjoint of the linear shadowing operator S;

2. $\mathcal{S}$ is given by a 'split then propagate' expansion formula;

3. $\mathcal{S}(\omega)$ 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.