# A Unified Framework for Bayesian Data Assimilation with Generative Models and Observation Interpolants

arXiv: https://arxiv.org/abs/2610.03396 · HTML: https://mlsanity.com/paper/2610.03396/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 26 of its 35 references that are in the map.
- First posted on arXiv 2 Oct 2026.

## Placed in

- Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/)

## Builds on (most cited first, 26 of 26)

- [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)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Diffusion Posterior Sampling for General Noisy Inverse Problems](https://mlsanity.com/paper/2209.14687/) (2022)
- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [SiT: Exploring Flow and Diffusion-based Generative Models with Scalable Interpolant Transformers](https://mlsanity.com/paper/2401.08740/) (2024)
- [PDE-Refiner: Achieving Accurate Long Rollouts with Neural PDE Solvers](https://mlsanity.com/paper/2308.05732/) (2023)
- [A Survey on Diffusion Models for Inverse Problems](https://mlsanity.com/paper/2410.00083/) (2024)
- [D-Flow: Differentiating through Flows for Controlled Generation](https://mlsanity.com/paper/2402.14017/) (2024)
- [PnP-Flow: Plug-and-Play Image Restoration with Flow Matching](https://mlsanity.com/paper/2410.02423/) (2024)
- [Score-based Data Assimilation](https://mlsanity.com/paper/2306.10574/) (2023)
- [Generator Matching: Generative modeling with arbitrary Markov processes](https://mlsanity.com/paper/2410.20587/) (2024)
- [Training-free Linear Image Inverses via Flows](https://mlsanity.com/paper/2310.04432/) (2023)
- [Benchmarking Autoregressive Conditional Diffusion Models for Turbulent Flow Simulation](https://mlsanity.com/paper/2309.01745/) (2023)
- [Probabilistic Forecasting with Stochastic Interpolants and F"ollmer Processes](https://mlsanity.com/paper/2403.13724/) (2024)
- [An Ensemble Score Filter for Tracking High-Dimensional Nonlinear Dynamical Systems](https://mlsanity.com/paper/2309.00983/) (2023)
- [Deep Generative Data Assimilation in Multimodal Setting](https://mlsanity.com/paper/2404.06665/) (2024)
- [Generalised Latent Assimilation in Heterogeneous Reduced Spaces with Machine Learning Surrogate Models](https://mlsanity.com/paper/2204.03497/) (2022)
- [FlowDAS: A Stochastic Interpolant-based Framework for Data Assimilation](https://mlsanity.com/paper/2501.16642/) (2025)
- [Reduced-Order Autodifferentiable Ensemble Kalman Filters](https://mlsanity.com/paper/2301.11961/) (2023)
- [Physics-aware generative models for turbulent fluid flows through energy-consistent stochastic interpolants](https://mlsanity.com/paper/2504.05852/) (2025)
- [D-Flow SGLD: Source-Space Posterior Sampling for Scientific Inverse Problems with Flow Matching](https://mlsanity.com/paper/2602.21469/) (2026)
- [The Deep Latent Space Particle Filter for Real-Time Data Assimilation with Uncertainty Quantification](https://mlsanity.com/paper/2406.02204/) (2024)
- [SURGE: Approximation and Training Free Particle Filter for Diffusion Surrogate](https://mlsanity.com/paper/2605.18745/) (2026)

## Abstract (arXiv)

Bayesian data assimilation combines model forecasts with noisy observations, but sampling high-dimensional, non-Gaussian posteriors remains challenging. We introduce an observation-interpolant framework that turns pretrained stochastic interpolant, flow matching, and diffusion models into posterior samplers without retraining. Conditioning the interpolant path on observations yields a shared likelihood-score correction to the drift or velocity, unifying stochastic and deterministic posterior sampling. The resulting SDEs and ODEs sample the exact posterior when the intermediate likelihood score is known. For practical computation, we approximate this score using a closed-form Gaussian surrogate with a bias-corrected mean and covariance inflated by the model's source covariance. Jacobian-free and ensemble-shared approximations make the method tractable in high dimensions. We evaluate the framework on linear-Gaussian dynamics, stochastic two-dimensional Navier-Stokes, and urban airflow with up to O(104) degrees of freedom.