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No-propagate algorithm for linear responses of random chaotic systems. (arXiv:2308.07841v1 [math.DS])

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We develop the no-propagate algorithm for sampling the linear response of
random dynamical systems, which are non-uniform hyperbolic deterministic
systems perturbed by noise with smooth density. We first derive a Monte-Carlo
type formula and then the algorithm, which is different from the ensemble
(stochastic gradient) algorithms, finite-element algorithms, and fast-response
algorithms; it does not involve the propagation of vectors or covectors, and
only the density of the noise is differentiated, so the formula is not cursed
by gradient explosion, dimensionality, or non-hyperbolicity. We demonstrate our
algorithm on a tent map perturbed by noise and a chaotic neural network with 51
layers $\times$ 9 neurons.


By itself, this algorithm approximates the linear response of non-hyperbolic
deterministic systems, with an additional error proportional to the noise. We
also discuss the potential of using this algorithm as a part of a bigger
algorithm with smaller error.

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