Backpropagation in hyperbolic chaos via adjoint shadowing

Understand

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.