  Using equilibrium fluctuations to understand the response of a physical
system to an externally imposed perturbation is the basis for linear response
theory, which is widely used to interpret experiments and shed light on
microscopic dynamics. For nonequilibrium systems, perturbations cannot be
interpreted simply by monitoring fluctuations in a conjugate observable --
additional dynamical information is needed. The theory of linear response
around nonequilibrium steady states relies on path ensemble averaging, which
makes this theory inapplicable to perturbations that affect the diffusion
constant or temperature in a stochastic system. Here, we show that a separate,
``effective'' physical process can be used to describe the perturbed dynamics
and that this dynamics in turn allows us to accurately calculate the response
to a change in the diffusion. Interestingly, the effective dynamics contains an
additional drift that is proportional to the ``score'' of the instantaneous
probability density of the system -- this object has also been studied
extensively in recent years in the context of denoising diffusion models in the
machine learning literature. Exploiting recently developed algorithms for
learning the score, we show that we can carry out nonequilibrium response
calculations on systems for which the exact score cannot be obtained.
