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.

翻译：我们提出了一种结合基于评分的生成式建模与广义涨落耗散定理（GFDT）的方法，用于分析动力系统对外部扰动的响应。该方法能够准确估计系统响应，包括具有非高斯统计特性的响应。我们使用来自三个复杂度递增的随机偏微分方程的时间序列数据，对所提方法进行了数值验证：具有空间相关噪声的Ornstein-Uhlenbeck过程、修正的随机Allen-Cahn方程以及二维Navier-Stokes方程。我们证明了该方法相较于传统方法具有更高的准确性，并讨论了其作为预测复杂动力系统统计行为的通用工具的潜力。