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Training-free score-based diffusion for parameter-dependent stochastic dynamical systems

Minglei Yang, Sicheng He

TL;DR

This work addresses the computational bottleneck of simulating parameter-dependent SDEs by introducing a training-free conditional diffusion framework that learns a stochastic flow map across a continuous parameter domain. It combines a closed-form, Monte Carlo–based conditional score estimator with a two-stage learning pipeline (probability-flow data generation followed by supervised regression) to enable trajectory generation for any parameter value within the training range without retraining. The approach is validated on three progressively complex examples, demonstrating accurate conditional distributions, robust interpolation across state and parameter spaces, and reliable long-horizon statistics. The method offers significant speedups for parameter studies, uncertainty quantification, and real-time filtering in stochastic dynamical systems while providing a scalable path toward higher-dimensional applications.

Abstract

Simulating parameter-dependent stochastic differential equations (SDEs) presents significant computational challenges, as separate high-fidelity simulations are typically required for each parameter value of interest. Despite the success of machine learning methods in learning SDE dynamics, existing approaches either require expensive neural network training for score function estimation or lack the ability to handle continuous parameter dependence. We present a training-free conditional diffusion model framework for learning stochastic flow maps of parameter-dependent SDEs, where both drift and diffusion coefficients depend on physical parameters. The key technical innovation is a joint kernel-weighted Monte Carlo estimator that approximates the conditional score function using trajectory data sampled at discrete parameter values, enabling interpolation across both state space and the continuous parameter domain. Once trained, the resulting generative model produces sample trajectories for any parameter value within the training range without retraining, significantly accelerating parameter studies, uncertainty quantification, and real-time filtering applications. The performance of the proposed approach is demonstrated via three numerical examples of increasing complexity, showing accurate approximation of conditional distributions across varying parameter values.

Training-free score-based diffusion for parameter-dependent stochastic dynamical systems

TL;DR

This work addresses the computational bottleneck of simulating parameter-dependent SDEs by introducing a training-free conditional diffusion framework that learns a stochastic flow map across a continuous parameter domain. It combines a closed-form, Monte Carlo–based conditional score estimator with a two-stage learning pipeline (probability-flow data generation followed by supervised regression) to enable trajectory generation for any parameter value within the training range without retraining. The approach is validated on three progressively complex examples, demonstrating accurate conditional distributions, robust interpolation across state and parameter spaces, and reliable long-horizon statistics. The method offers significant speedups for parameter studies, uncertainty quantification, and real-time filtering in stochastic dynamical systems while providing a scalable path toward higher-dimensional applications.

Abstract

Simulating parameter-dependent stochastic differential equations (SDEs) presents significant computational challenges, as separate high-fidelity simulations are typically required for each parameter value of interest. Despite the success of machine learning methods in learning SDE dynamics, existing approaches either require expensive neural network training for score function estimation or lack the ability to handle continuous parameter dependence. We present a training-free conditional diffusion model framework for learning stochastic flow maps of parameter-dependent SDEs, where both drift and diffusion coefficients depend on physical parameters. The key technical innovation is a joint kernel-weighted Monte Carlo estimator that approximates the conditional score function using trajectory data sampled at discrete parameter values, enabling interpolation across both state space and the continuous parameter domain. Once trained, the resulting generative model produces sample trajectories for any parameter value within the training range without retraining, significantly accelerating parameter studies, uncertainty quantification, and real-time filtering applications. The performance of the proposed approach is demonstrated via three numerical examples of increasing complexity, showing accurate approximation of conditional distributions across varying parameter values.
Paper Structure (29 sections, 46 equations, 11 figures, 1 algorithm)

This paper contains 29 sections, 46 equations, 11 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Parameter-dependent stochastic differential equations
  4. The stochastic flow map
  5. Training-free conditional diffusion framework
  6. Generative model for the stochastic flow map
  7. Training data from Monte Carlo simulations
  8. Overview of the diffusion-based approach
  9. Training-free score estimation
  10. Algorithm and implementation
  11. Numerical experiments
  12. Example 1: Brownian motion with parameter-dependent drift
  13. Exact solutions
  14. Training procedure
  15. Results
  16. ...and 14 more sections

Figures (11)

  • Figure 1: Example 1: Conditional distribution $p(X_{n+1} \mid X_n = 2, \mu)$. Left: PDF comparison for $\mu = -0.5$ and $\mu = 0.5$; solid lines show the exact Gaussian density \ref{['eq:ex1_exact_cond']}, markers show learned estimates. Right: Conditional mean $\mathbb{E}[X_{n+1} \mid X_n = 2, \mu]$ vs. $\mu$; solid line is the exact formula $x + \mu \Delta t$, markers are learned estimates.
  • Figure 2: Example 1: Heatmap of conditional distribution $p(X_{n+1} \mid X_n = 0, \mu)$ over the parameter range $\mu \in [-1, 1]$. Left: Exact distribution from \ref{['eq:ex1_exact_cond']}. Right: Learned distribution. The heatmaps are constructed using $5{,}000$ samples per $\mu$ value. The learned model captures the linear shift in mean with $\mu$.
  • Figure 3: Example 1: Terminal distribution $p(X_T)$ at $T = 1.0$ with Gaussian initial distribution $X_0 \sim \mathcal{N}(0, 0.25)$, obtained via $10$ iterative applications of the learned one-step flow map. Left panel: $\mu = -0.5$. Right panel: $\mu = 0.5$. Solid lines: Monte Carlo ground truth ($50{,}000$ samples); dashed lines: analytical formula \ref{['eq:ex1_terminal']}; markers: learned method ($50{,}000$ generated trajectories). Vertical dotted lines indicate the analytical mean $m_0 + \mu T$.
  • Figure 4: Example 2: Conditional distribution $p(X_{n+1} \mid X_n = 1, \mu)$. Left: PDF comparison for $\mu = 0.5$ and $\mu = 2.0$; solid lines show exact Gaussian density \ref{['eq:ex2_exact_cond']}, markers show learned estimates. Right: Conditional mean $\mathbb{E}[X_{n+1} \mid X_n = 1, \mu]$ vs. $\mu$; solid line is the exact formula $x e^{-\mu \Delta t}$, markers are learned estimates.
  • Figure 5: Example 2: Heatmap of conditional distribution $p(X_{n+1} \mid X_n = 0, \mu)$. Left: Exact. Right: Learned. The one-step conditional variance increases monotonically with $\mu$. Constructed using $20{,}000$ samples per $\mu$ value.
  • ...and 6 more figures

Theorems & Definitions (2)

  • Remark 2.2
  • Remark 5.1: Reproducibility