A Training-Free Conditional Diffusion Model for Learning Stochastic Dynamical Systems

TL;DR

This paper presents a novel training-free conditional diffusion model that efficiently learns stochastic differential equations from data without neural network training, improving accuracy and computational efficiency.

Contribution

It introduces a closed-form score function for SDEs, enabling direct estimation and supervised learning of the stochastic flow map without training neural networks.

Findings

Outperforms baseline methods like GANs in estimating drift and diffusion.

Effective across linear, nonlinear, and multi-dimensional SDEs.

Enhances short-term and long-term prediction accuracy.

Abstract

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.