This study introduces a training-free conditional diffusion model for learning unknown stochastic differential equations (SDEs) using data. The proposed approach addresses key challenges in computational efficiency and accuracy for modeling SDEs by utilizing a score-based diffusion model to approximate their stochastic flow map. Unlike the existing methods, this technique is based on an analytically derived closed-form exact score function, which can be efficiently estimated by Monte Carlo method using the trajectory data, and eliminates the need for neural network training to learn the score function. By generating labeled data through solving the corresponding reverse ordinary differential equation, the approach enables supervised learning of the flow map. Extensive numerical experiments across various SDE types, including linear, nonlinear, and multi-dimensional systems, demonstrate the versatility and effectiveness of the method. The learned models exhibit significant improvements in predicting both short-term and long-term behaviors of unknown stochastic systems, often surpassing baseline methods like GANs in estimating drift and diffusion coefficients.

本研究提出了一种无训练的条件扩散模型，用于通过数据学习未知的随机微分方程(SDE)。该方法通过利用基于分数的扩散模型来逼近其随机流映射，成功解决了建模SDE的计算效率和准确性问题。数值实验表明，该方法在预测未知随机系统的短期和长期行为方面显著优于传统方法，如GAN。

一种无训练的条件扩散模型用于学习随机动力系统