A TRAINING-FREE CONDITIONAL DIFFUSION MODEL FOR LEARNING STOCHASTIC DYNAMICAL SYSTEMS

This study introduces a training-free conditional diffusion model for learning unknown 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 methods 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 ODE, the approach enables supervised learning of the flow map. Extensive numerical experiments across various SDE types, including linear, nonlinear, and multidimensional 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 such as generative adversarial networks in estimating drift and diffusion coefficients.