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We develop the no-propagate algorithm for sampling the linear response ofrandom dynamical systems, which are non-uniform hyperbolic deterministicsystems perturbed by noise with smooth density. We first derive a Monte-Carlotype formula and then the algorithm, which is different from the ensemble(stochastic gradient) algorithms, finite-element algorithms, and fast-responsealgorithms; it does not involve the propagation of vectors or covectors, andonly the density of the noise is differentiated, so the formula is not cursedby gradient explosion, dimensionality, or non-hyperbolicity. We demonstrate ouralgorithm on a tent map perturbed by noise and a chaotic neural network with 51layers $\times$ 9 neurons.  By itself, this algorithm approximates the linear response of non-hyperbolicdeterministic systems, with an additional error proportional to the noise. Wealso discuss the potential of using this algorithm as a part of a biggeralgorithm with smaller error.

No-propagate algorithm for linear responses of random chaotic systems