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.

翻译：针对随机动力系统（即受噪声扰动的非一致双曲确定性系统，且噪声具有光滑密度）的线性响应采样问题，我们提出了无传播算法。首先推导出蒙特卡洛型公式，进而构建该算法。与集成（随机梯度）算法、有限元算法及快速响应算法不同，该算法无需传播向量或余向量，仅需对噪声密度进行微分，因此不会受到梯度爆炸、维数灾难或非双曲性问题的困扰。我们通过受噪声扰动的帐篷映射及包含51层×9个神经元的混沌神经网络验证了该算法的有效性。该算法本身可近似非双曲确定性系统的线性响应，其附加误差与噪声强度成正比。文中还探讨了将该算法作为更大算法组成部分以降低误差的潜在可能性。