computing the derivative of long-time-averaged observables with respect to system parameters is a central problem for many numerical applications. conventionally, there are three straight-forward formulas for this derivative: the pathwise perturbation formula (including the backpropagation method used by the machine learning community), the divergence formula, and the kernel differentiation formula. we shall explain why none works for the general case, which is typically chaotic (also known as the gradient explosion phenomenon), high-dimensional, and small-noise.

we present the fast response formula, which is a 'monte-carlo' type formula for the parameter-derivative of hyperbolic chaos. it is the average of some function of u-many vectors over an orbit, where u is the unstable dimension, and those vectors can be computed recursively. the fast response overcomes all three difficulties under hyperbolicity assumptions. then we discuss how to further incorporate the kernel differentiation trick to overcome non-hyperbolicity.