无需训练即可学习随机微分方程，用数据直接推导系统演化规律。
A Training-Free Conditional Diffusion Model for Learning Stochastic Dynamical Systems
- 基于解析得分函数，用轨迹数据通过蒙特卡洛法估算，免去神经网络训练。
- 在多类随机系统上预测短期与长期行为均显著优于传统方法。
- 适合需要快速建模未知随机动力系统的研究人员使用。
本文提出一种无需训练的条件扩散模型，用于从数据中学习未知的随机微分方程（SDE）。该方法利用基于解析推导的闭合形式精确得分函数，通过蒙特卡洛方法高效估计，无需神经网络学习得分函数。通过求解对应的反向常微分方程生成标注数据，实现对流映射的监督学习。在多种类型的SDE（包括线性、非线性和高维系统）上的大量数值实验表明，该方法具有强泛化能力。所学模型在预测未知随机系统短时与长时行为方面表现优异，通常在漂移项和扩散项估计上超越生成对抗网络（GANs）等基线方法。
原文摘要 · Abstract (English)
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.
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