We introduce an approach for analyzing the responses of dynamical systems to
external perturbations that combines score-based generative modeling with the
Fluctuation-Dissipation Theorem (FDT). The methodology enables accurate
estimation of system responses, especially for systems with non-Gaussian
statistics, often encountered in dynamical systems far from equilibrium. Such
cases often present limitations for conventional approximate methods. We
numerically validate our approach using time-series data from a stochastic
partial differential equation where the score function is available
analytically. Furthermore, we demonstrate the improved accuracy of our
methodology over conventional methods and its potential as a versatile tool for
understanding complex dynamical systems. Applications span disciplines from
climate science and finance to neuroscience.

利用基于评分的生成建模与波动耗散定理（FDT）相结合的方法，我们介绍了一种分析动力系统对外部扰动响应的方法。通过该方法，可以准确估计系统响应，特别是对于非高斯统计的动力系统，特别是在远离平衡的情况下经常遇到的系统。我们通过使用时间序列数据从一个具有可用解析评分函数的随机偏微分方程，对我们的方法进行了数值验证。此外，我们展示了我们方法相对于传统方法的改进精度，并表明其作为理解复杂动力系统的多功能工具的潜力，其应用领域涵盖气候科学，金融和神经科学。