Adjoint shadowing for backpropagation in hyperbolic chaos
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For 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: (a) 𝒮 is the adjoint of the linear shadowing operator; (b) 𝒮 is given by a `split then propagate' expansion formula; (c) 𝒮(ω) is the only bounded inhomogeneous adjoint solution of ω. By (a), 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), 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, 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.
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