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

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.

💡 Research Summary

This paper addresses the computational bottleneck associated with simulating parameter‑dependent stochastic differential equations (SDEs), where each parameter value traditionally requires a separate high‑fidelity simulation. The authors propose a training‑free conditional diffusion framework that learns the stochastic flow map of such SDEs without the need to train a neural network to approximate the score function. The key technical contribution is a joint kernel‑weighted Monte Carlo estimator that approximates the conditional score function from trajectory data collected at a finite set of discrete parameter values, thereby enabling smooth interpolation across both the state space and the continuous parameter domain.

The methodology consists of four components. First, a generative model Gθ is defined to map a standard Gaussian latent variable z to the one‑step displacement Δx = x_{n+1}−x_n, conditioned on the current state x_n and the physical parameter μ. Second, trajectory data are generated by standard Monte Carlo simulation of the SDE at a collection of parameter values {μ^{(k)}}. Third, the conditional score S(z,τ;x_n,μ)=∇_z log Q(z|x_n,μ) is derived in closed form for the variance‑preserving (VP) diffusion process and expressed as a weighted average over the unknown displacement distribution. The weights involve the Gaussian transition density and two kernel functions that measure proximity in state space and parameter space. Because the exact displacement distribution is unavailable, the authors approximate the score by selecting the N nearest neighbours of (x_n,μ) in the joint (state,parameter) space, computing their displacements, and applying kernel‑weighted Monte Carlo averaging. This yields a training‑free estimate \bar{S} that can be evaluated for any query (x_n,μ).

Fourth, the reverse‑time ordinary differential equation (ODE) associated with the VP diffusion is integrated deterministically using the estimated score. This reverse ODE provides a one‑to‑one mapping from a latent Gaussian sample z∼N(0,I) to a displacement Δx_{n+1}, effectively generating a labeled dataset D_aug = {(x_n, μ, z, Δx_{n+1})}. The generative network Gθ is then trained in a supervised fashion on D_aug using a simple mean‑squared error loss, converting the originally unsupervised generative modeling problem into a standard regression task.

The authors validate the approach on three increasingly complex examples: (i) a one‑dimensional linear SDE, (ii) a two‑dimensional nonlinear multi‑stable system, and (iii) a high‑dimensional diffusion‑reaction model with multiple reaction‑rate parameters. In all cases, the trained model accurately reproduces the conditional transition distributions for any μ within the training range, achieving speed‑ups of one to two orders of magnitude compared with direct Monte Carlo simulation. Quantitative metrics—including KL divergence, score‑estimation error, and computational time—demonstrate that the kernel‑weighted estimator respects the theoretical O(d) error bound in ℓ₂ and O(log d) in ℓ_∞, where d is the state dimension.

By eliminating the need for neural‑network‑based score training and by enabling continuous parameter interpolation via kernel weighting, this work offers a scalable, efficient surrogate for parameter‑dependent stochastic dynamics. Potential extensions include adaptive kernel bandwidth selection, integration with differentiable filtering (e.g., Kalman or particle filters), and application to high‑dimensional physical systems where real‑time inference is essential.