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

arXiv: https://arxiv.org/abs/2410.03108 · HTML: https://mlsanity.com/paper/2410.03108/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 24 of its 39 references that are in the map.
- First posted on arXiv 4 Oct 2024.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Builds on (most cited first, 24 of 24)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Improved Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2102.09672/) (2021)
- [Improved Techniques for Training GANs](https://mlsanity.com/paper/1606.03498/) (2016)
- [Diffusion Models: A Comprehensive Survey of Methods and Applications](https://mlsanity.com/paper/2209.00796/) (2022)
- [Diffusion-LM Improves Controllable Text Generation](https://mlsanity.com/paper/2205.14217/) (2022)
- [DiffuSeq: Sequence to Sequence Text Generation with Diffusion Models](https://mlsanity.com/paper/2210.08933/) (2022)
- [Sliced Score Matching: A Scalable Approach to Density and Score Estimation](https://mlsanity.com/paper/1905.07088/) (2019)
- [Analog Bits: Generating Discrete Data using Diffusion Models with Self-Conditioning](https://mlsanity.com/paper/2208.04202/) (2022)
- [AUC Maximization in the Era of Big Data and AI: A Survey](https://mlsanity.com/paper/2203.15046/) (2022)
- [Data Driven Governing Equations Approximation Using Deep Neural Networks](https://mlsanity.com/paper/1811.05537/) (2018)
- [Latent Diffusion Energy-Based Model for Interpretable Text Modeling](https://mlsanity.com/paper/2206.05895/) (2022)
- [Solving Inverse Stochastic Problems from Discrete Particle Observations Using the Fokker-Planck Equation and Physics-informed Neural Networks](https://mlsanity.com/paper/2008.10653/) (2020)
- [An Ensemble Score Filter for Tracking High-Dimensional Nonlinear Dynamical Systems](https://mlsanity.com/paper/2309.00983/) (2023)
- [Learning Stochastic Dynamical System via Flow Map Operator](https://mlsanity.com/paper/2305.03874/) (2023)
- [A Data-Driven Approach for Discovering Stochastic Dynamical Systems with Non-Gaussian Levy Noise](https://mlsanity.com/paper/2005.03769/) (2020)
- [One-Shot Learning of Stochastic Differential Equations with Data Adapted Kernels](https://mlsanity.com/paper/2209.12086/) (2022)
- [Diffusion-Model-Assisted Supervised Learning of Generative Models for Density Estimation](https://mlsanity.com/paper/2310.14458/) (2023)
- [On the sample complexity of stabilizing linear dynamical systems from data](https://mlsanity.com/paper/2203.00474/) (2022)
- [Bi-fidelity Variational Auto-encoder for Uncertainty Quantification](https://mlsanity.com/paper/2305.16530/) (2023)
- [Diffusion map particle systems for generative modeling](https://mlsanity.com/paper/2304.00200/) (2023)

## Built on by (most cited first, 19 of 19)

- [Bifidelity Parameter Estimation Using Conditional Diffusion Models](https://mlsanity.com/paper/2504.01894/) (2025)
- [Generative AI Models for Learning Flow Maps of Stochastic Dynamical Systems in Bounded Domains](https://mlsanity.com/paper/2507.15990/) (2025)
- [Learning Epidemiological Dynamics via the Finite Expression Method](https://mlsanity.com/paper/2412.21049/) (2024)
- [Learning Stochastic Hamiltonian Systems via Stochastic Generating Function Neural Network](https://mlsanity.com/paper/2507.14467/) (2025)
- [Moment Estimate and Variational Approach for Learning Generalized Diffusion with Non-gradient Structures](https://mlsanity.com/paper/2508.01854/) (2025)
- [A Weak Penalty Neural ODE for Learning Chaotic Dynamics from Noisy Time Series](https://mlsanity.com/paper/2511.06609/) (2025)
- [Training-free score-based diffusion for parameter-dependent stochastic dynamical systems](https://mlsanity.com/paper/2602.02113/) (2026)
- [MVNN: A Measure-Valued Neural Network for Learning McKean-Vlasov Dynamics from Particle Data](https://mlsanity.com/paper/2604.00333/) (2026)
- [A Robust Model-Based Approach for Continuous-Time Policy Evaluation with Unknown L'evy Process Dynamics](https://mlsanity.com/paper/2504.01482/) (2025)
- [Identifying Unknown Stochastic Dynamics via Finite expression methods](https://mlsanity.com/paper/2504.07085/) (2025)
- [Divergence-kernel method for linear responses of densities and generative models](https://mlsanity.com/paper/2509.03992/) (2025)
- [The finite expression method for turbulent dynamics with high-order moment recovery](https://mlsanity.com/paper/2605.10687/) (2026)
- [Weighted Flow Matching and Physics-Informed Nonlinear Filtering for Parameter Estimation in Digital Twins](https://mlsanity.com/paper/2605.17146/) (2026)
- [Diagnosing the conditional-mean barrier in scientific machine-learning surrogates](https://mlsanity.com/paper/2605.28076/) (2026)
- [Error Bounds for a Diffusion Model-Based Drift Estimator](https://mlsanity.com/paper/2606.02115/) (2026)
- [Data-driven Effective Modeling of Stochastic Chemical Reaction Networks](https://mlsanity.com/paper/2608.25421/) (2026)
- [Memory-Conditioned Diffusion Model for Generalized Langevin Dynamics](https://mlsanity.com/paper/2609.28371/) (2026)
- [AECSF: Adaptive Ensemble Conditional Score Filtering for High-Dimensional Nonlinear Data Assimilation](https://mlsanity.com/paper/2609.32411/) (2026)
- [Generative Modeling of Stochastic Dynamics for Long-Time Evolution](https://mlsanity.com/paper/2610.00546/) (2026)

## Abstract (arXiv)

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.