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

TL;DR

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, based on an analytically derived closed-form exact score function, which can be efficiently estimated by Monte Carlo method using the trajectory data.

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.

Figures (25)

• Figure 1: One-dimensional OU process: comparison of the mean and the standard deviation of solutions with the initial state being $X_0=1.5$, obtained by the generative model and the ground truth.
• Figure 2: One-dimensional OU process: comparison of effective drift and diffusion functions obtained by the simulated trajectories using the generative model and the exact SDE. Left: drift $a(x)= \mu-x$; Right: diffusion $b(x)=\sigma$.
• Figure 3: One-dimensional OU process: comparison of conditional PDF $p_{X_{t+\Delta t}|X_t}(x_{t+\Delta t}|x_{t}=1.5)$ determined by the generative model $G_\theta$ and the exact flow map $F_{\Delta t}$.
• Figure 4: One-dimensional GBM: comparison of the mean and the standard deviation of solutions with the initial state being $X_0=0.5$, obtained by the generative model and the ground truth.
• Figure 5: One-dimensional GBM: comparison of effective drift and diffusion functions obtained by the simulated trajectories using the generative model and the exact SDE. Left: drift $a(x)= \mu x$; Right: diffusion $b(x)=\sigma x$.

Theorems & Definitions (1)

• Remark 4.1: Reproducibility