# User-defined Event Sampling and Uncertainty Quantification in Diffusion Models for Physical Dynamical Systems

arXiv: https://arxiv.org/abs/2306.07526 · HTML: https://mlsanity.com/paper/2306.07526/

## Key facts

- Cited by 30 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 22 of its 49 references that are in the map.
- First posted on arXiv 13 Jun 2023.

## Placed in

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

## Builds on (most cited first, 22 of 22)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Photorealistic Text-to-Image Diffusion Models with Deep Language Understanding](https://mlsanity.com/paper/2205.11487/) (2022)
- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [RePaint: Inpainting using Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2201.09865/) (2022)
- [Diffusion Posterior Sampling for General Noisy Inverse Problems](https://mlsanity.com/paper/2209.14687/) (2022)
- [Variational Diffusion Models](https://mlsanity.com/paper/2107.00630/) (2021)
- [FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models](https://mlsanity.com/paper/1810.01367/) (2018)
- [Improving Diffusion Models for Inverse Problems using Manifold Constraints](https://mlsanity.com/paper/2206.00941/) (2022)
- [Score-based diffusion models for accelerated MRI](https://mlsanity.com/paper/2110.05243/) (2021)
- [Come-Closer-Diffuse-Faster: Accelerating Conditional Diffusion Models for Inverse Problems through Stochastic Contraction](https://mlsanity.com/paper/2112.05146/) (2021)
- [Residual Flows for Invertible Generative Modeling](https://mlsanity.com/paper/1906.02735/) (2019)
- [Diffusion models as plug-and-play priors](https://mlsanity.com/paper/2206.09012/) (2022)
- [Simplifying Hamiltonian and Lagrangian Neural Networks via Explicit Constraints](https://mlsanity.com/paper/2010.13581/) (2020)
- [A Theoretical Justification for Image Inpainting using Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2302.01217/) (2023)
- [Flood Prediction Using Machine Learning Models: Literature Review](https://mlsanity.com/paper/1908.02781/) (2019)
- [Model-assisted deep learning of rare extreme events from partial observations](https://mlsanity.com/paper/2111.04857/) (2021)

## Built on by (most cited first, 30 of 30)

- [Learning Diffusion Priors from Observations by Expectation Maximization](https://mlsanity.com/paper/2405.13712/) (2024)
- [On conditional diffusion models for PDE simulations](https://mlsanity.com/paper/2410.16415/) (2024)
- [DEFT: Efficient Fine-Tuning of Diffusion Models by Learning the Generalised h-transform](https://mlsanity.com/paper/2406.01781/) (2024)
- [Variational Diffusion Posterior Sampling with Midpoint Guidance](https://mlsanity.com/paper/2410.09945/) (2024)
- [Divide-and-Conquer Posterior Sampling for Denoising Diffusion Priors](https://mlsanity.com/paper/2403.11407/) (2024)
- [A framework for conditional diffusion modelling with applications in motif scaffolding for protein design](https://mlsanity.com/paper/2312.09236/) (2023)
- [Adding Conditional Control to Diffusion Models with Reinforcement Learning](https://mlsanity.com/paper/2406.12120/) (2024)
- [Free Hunch: Denoiser Covariance Estimation for Diffusion Models Without Extra Costs](https://mlsanity.com/paper/2410.11149/) (2024)
- [Improving Flow Matching by Aligning Flow Divergence](https://mlsanity.com/paper/2602.00869/) (2026)
- [Diffusion Methods for Generating Transition Paths](https://mlsanity.com/paper/2309.10276/) (2023)
- [Equivariant Denoisers Cannot Copy Graphs: Align Your Graph Diffusion Models](https://mlsanity.com/paper/2405.17656/) (2024)
- [Efficient Zero-Shot Inpainting with Decoupled Diffusion Guidance](https://mlsanity.com/paper/2512.18365/) (2025)
- [A Comprehensive Survey on Diffusion Models and Their Applications](https://mlsanity.com/paper/2408.10207/) (2024)
- [Back-Projection Diffusion: Solving the Wideband Inverse Scattering Problem with Diffusion Models](https://mlsanity.com/paper/2408.02866/) (2024)
- [Generative Learning for Slow Manifolds and Bifurcation Diagrams](https://mlsanity.com/paper/2504.20375/) (2025)
- [PriorGuide: Test-Time Prior Adaptation for Simulation-Based Inference](https://mlsanity.com/paper/2510.13763/) (2025)
- [Supervised Guidance Training for Infinite-Dimensional Diffusion Models](https://mlsanity.com/paper/2601.20756/) (2026)
- [RMFlow: Refined Mean Flow by a Noise-Injection Step for Multimodal Generation](https://mlsanity.com/paper/2602.00849/) (2026)
- [Score-Based Turbo Message Passing for Plug-and-Play Compressive Image Recovery](https://mlsanity.com/paper/2503.22140/) (2025)
- [Discrete Spatial Diffusion: Intensity-Preserving Diffusion Modeling](https://mlsanity.com/paper/2505.01917/) (2025)
- [Using Ensemble Diffusion to Estimate Uncertainty for End-to-End Autonomous Driving](https://mlsanity.com/paper/2506.00560/) (2025)
- [Improved probabilistic regression using diffusion models](https://mlsanity.com/paper/2510.04583/) (2025)
- [Latent Refinement via Flow Matching for Training-free Linear Inverse Problem Solving](https://mlsanity.com/paper/2511.06138/) (2025)
- [LatentFlow: A General Framework for Conditioning Stochastic Processes](https://mlsanity.com/paper/2607.12922/) (2026)
- [Diffusing Differentiable Representations](https://mlsanity.com/paper/2412.06981/) (2024)
- [Second Order Drifting Models](https://mlsanity.com/paper/2608.07924/) (2026)
- [Diffusion-based subsurface CO2 multiphysics monitoring and forecasting](https://mlsanity.com/paper/2407.18426/) (2024)
- [Improving the Noise Estimation of Latent Neural Stochastic Differential Equations](https://mlsanity.com/paper/2412.17499/) (2024)
- [Briding Diffusion Posterior Sampling and Monte Carlo methods: a survey](https://mlsanity.com/paper/2510.14114/) (2025)
- [Target-Aware Linear Regression Under Distribution Shift](https://mlsanity.com/paper/2606.22775/) (2026)

## Abstract (arXiv)

Diffusion models are a class of probabilistic generative models that have been widely used as a prior for image processing tasks like text conditional generation and inpainting. We demonstrate that these models can be adapted to make predictions and provide uncertainty quantification for chaotic dynamical systems. In these applications, diffusion models can implicitly represent knowledge about outliers and extreme events; however, querying that knowledge through conditional sampling or measuring probabilities is surprisingly difficult. Existing methods for conditional sampling at inference time seek mainly to enforce the constraints, which is insufficient to match the statistics of the distribution or compute the probability of the chosen events. To achieve these ends, optimally one would use the conditional score function, but its computation is typically intractable. In this work, we develop a probabilistic approximation scheme for the conditional score function which provably converges to the true distribution as the noise level decreases. With this scheme we are able to sample conditionally on nonlinear userdefined events at inference time, and matches data statistics even when sampling from the tails of the distribution.