mlsanityData to 7 Oct 2026

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

arXiv 2410.03108 ↗First posted 4 Oct 2024cs.LG math.DScited by 19

On the map: Diffusion models → Diffusion foundations and sampling

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.

Key facts. Cited by 19 papers in the map (data through 7 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.

Builds on

24 of 39 references.

Closest ancestors · 8

Same direction, newest first

  1. 2023Diffusion-Model-Assisted Supervised Learning of Generative Models for Density Estimation22 Oct
  2. 2022Diffusion Models: A Comprehensive Survey of Methods and Applications2 Sep
  3. 2022Analog Bits: Generating Discrete Data using Diffusion Models with Self-Conditioning8 Aug
  4. 2021High-Resolution Image Synthesis with Latent Diffusion Models20 Dec
  5. 2021Diffusion Models Beat GANs on Image Synthesis11 May
  6. 2021Improved Denoising Diffusion Probabilistic Models18 Feb
  7. 2020Score-Based Generative Modeling through Stochastic Differential Equations26 Nov
  8. 2020Denoising Diffusion Probabilistic Models19 Jun

Other references · 16

  1. 2019Generative Modeling by Estimating Gradients of the Data Distributioncited by 4,067
  2. 2016Improved Techniques for Training GANscited by 2,938
  3. 2022Diffusion-LM Improves Controllable Text Generationcited by 1,006
  4. 2022DiffuSeq: Sequence to Sequence Text Generation with Diffusion Modelscited by 439
  5. 2019Sliced Score Matching: A Scalable Approach to Density and Score Estimationcited by 381
  6. 2022AUC Maximization in the Era of Big Data and AI: A Surveycited by 100
  7. 2018Data Driven Governing Equations Approximation Using Deep Neural Networkscited by 94
  8. 2022Latent Diffusion Energy-Based Model for Interpretable Text Modelingcited by 67
  9. 2020Solving Inverse Stochastic Problems from Discrete Particle Observations Using the Fokker-Planck Equation and Physics-informed Neural Networkscited by 52
  10. 2023An Ensemble Score Filter for Tracking High-Dimensional Nonlinear Dynamical Systemscited by 33
  11. 2023Learning Stochastic Dynamical System via Flow Map Operatorcited by 24
  12. 2020A Data-Driven Approach for Discovering Stochastic Dynamical Systems with Non-Gaussian Levy Noisecited by 15
4 more
  1. 2022One-Shot Learning of Stochastic Differential Equations with Data Adapted Kernelscited by 15
  2. 2022On the sample complexity of stabilizing linear dynamical systems from datacited by 8
  3. 2023Bi-fidelity Variational Auto-encoder for Uncertainty Quantificationcited by 8
  4. 2023Diffusion map particle systems for generative modelingcited by 6

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2024: 1; 2025: 8; 2026: 10.

19 papers.

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  1. 2025Bifidelity Parameter Estimation Using Conditional Diffusion Modelscited by 7
  2. 2025Generative AI Models for Learning Flow Maps of Stochastic Dynamical Systems in Bounded Domainscited by 3
  3. 2024Learning Epidemiological Dynamics via the Finite Expression Methodcited by 2
  4. 2025Learning Stochastic Hamiltonian Systems via Stochastic Generating Function Neural Networkcited by 1
  5. 2025Moment Estimate and Variational Approach for Learning Generalized Diffusion with Non-gradient Structurescited by 1
  6. 2025A Weak Penalty Neural ODE for Learning Chaotic Dynamics from Noisy Time Seriescited by 1
  7. 2026Training-free score-based diffusion for parameter-dependent stochastic dynamical systemscited by 1
  8. 2026MVNN: A Measure-Valued Neural Network for Learning McKean-Vlasov Dynamics from Particle Datacited by 1
  9. 2025A Robust Model-Based Approach for Continuous-Time Policy Evaluation with Unknown L'evy Process Dynamicscited by 0
  10. 2025Identifying Unknown Stochastic Dynamics via Finite expression methodscited by 0
  11. 2025Divergence-kernel method for linear responses of densities and generative modelscited by 0
  12. 2026The finite expression method for turbulent dynamics with high-order moment recoverycited by 0
7 more
  1. 2026Weighted Flow Matching and Physics-Informed Nonlinear Filtering for Parameter Estimation in Digital Twinscited by 0
  2. 2026Diagnosing the conditional-mean barrier in scientific machine-learning surrogatescited by 0
  3. 2026Error Bounds for a Diffusion Model-Based Drift Estimatorcited by 0
  4. 2026Data-driven Effective Modeling of Stochastic Chemical Reaction Networkscited by 0
  5. 2026Memory-Conditioned Diffusion Model for Generalized Langevin Dynamicscited by 0
  6. 2026AECSF: Adaptive Ensemble Conditional Score Filtering for High-Dimensional Nonlinear Data Assimilationcited by 0
  7. 2026Generative Modeling of Stochastic Dynamics for Long-Time Evolutioncited by 0

Similar, not cited

Close in wording, no citation either way

  1. 2025A Score-based Diffusion Model Approach for Adaptive Learning of Stochastic Partial Differential Equation Solutions
  2. 2023Closing the ODE-SDE gap in score-based diffusion models through the Fokker-Planck equation
  3. 2023Score-based Generative Modeling Through Backward Stochastic Differential Equations: Inversion and Generation
  4. 2024Score-based Diffusion Models via Stochastic Differential Equations -- a Technical Tutorial
  5. 2025Generating solution paths of Markovian stochastic differential equations using diffusion models
  6. 2024Diffusion models as probabilistic neural operators for recovering unobserved states of dynamical systems
  7. 2022A Flexible Diffusion Model
  8. 2026A Tutorial on Diffusion Theory: From Differential Equations to Diffusion Models