# Closed-form conditional diffusion models for data assimilation Source: [Closed-form conditional diffusion models for data assimilation]() Collection index: [llms.txt]() Data updated: 2026-04-03T05:00:04.568Z Authors: Brianna Binder, Agnimitra Dasgupta, Assad Oberai Published: 2026-03-22T15:25:23.000Z ## Abstract We propose closed-form conditional diffusion models for data assimilation. Diffusion models use data to learn the score function (defined as the gradient of the log-probability density of a data distribution), allowing them to generate new samples from the data distribution by reversing a noise injection process. While it is common to train neural networks to approximate the score function, we leverage the analytical tractability of the score function to assimilate the states of a system with measurements. To enable the efficient evaluation of the score function, we use kernel density estimation to model the joint distribution of the states and their corresponding measurements. The proposed approach also inherits the capability of conditional diffusion models of operating in black-box settings, i.e., the proposed data assimilation approach can accommodate systems and measurement processes without their explicit knowledge. The ability to accommodate black-box systems combined with the superior capabilities of diffusion models in approximating complex, non-Gaussian probability distributions means that the proposed approach offers advantages over many widely used filtering methods. We evaluate the proposed method on nonlinear data assimilation problems based on the Lorenz-63 and Lorenz-96 systems of moderate dimensionality and nonlinear measurement models. Results show the proposed approach outperforms the widely used ensemble Kalman and particle filters when small to moderate ensemble sizes are used. ## Recorded benchmark results A research result applies to its stated dataset, split, metric and evaluation configuration. Compare like-for-like settings, preserve the metric's unit and direction, and distinguish paper-reported results from independent reproduction. The best result recorded here is not a claim of exhaustive, current global coverage. - [Filtering on Lorenz–63](): Filtering; Lorenz '63; Average EW2: 5.744. - [Data Assimilation on Lorenz-96 20d (S=10 simulations)](): Data Assimilation; 20-dimensional Lorenz–96 system S = 10 independent simulations; Average ERMSE: 1.439. - [Filtering on Lorenz-96 10-dimensional](): Filtering; Lorenz-96 10-dimensional; RMSE: 0.952. ## Original sources and implementation [Original paper]() [PDF]()