  We introduce an approach for analyzing the responses of dynamical systems to
external perturbations that combines score-based generative modeling with the
Generalized Fluctuation-Dissipation Theorem (GFDT). The methodology enables
accurate estimation of system responses, including those with non-Gaussian
statistics. We numerically validate our approach using time-series data from
three different stochastic partial differential equations of increasing
complexity: an Ornstein-Uhlenbeck process with spatially correlated noise, a
modified stochastic Allen-Cahn equation, and the 2D Navier-Stokes equations. We
demonstrate the improved accuracy of the methodology over conventional methods
and discuss its potential as a versatile tool for predicting the statistical
behavior of complex dynamical systems.
