# Probabilistic Spatial Interpolation of Sparse Data using Diffusion Models

arXiv: https://arxiv.org/abs/2506.00033 · HTML: https://mlsanity.com/paper/2506.00033/

## 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 12 of its 26 references that are in the map.
- First posted on arXiv 26 May 2025.

## 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, 12 of 12)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [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)
- [GLIDE: Towards Photorealistic Image Generation and Editing with Text-Guided Diffusion Models](https://mlsanity.com/paper/2112.10741/) (2021)
- [Image Super-Resolution via Iterative Refinement](https://mlsanity.com/paper/2104.07636/) (2021)
- [Demystifying MMD GANs](https://mlsanity.com/paper/1801.01401/) (2018)
- [DiffusionDet: Diffusion Model for Object Detection](https://mlsanity.com/paper/2211.09788/) (2022)
- [ClimODE: Climate and Weather Forecasting with Physics-informed Neural ODEs](https://mlsanity.com/paper/2404.10024/) (2024)
- [DiffDA: a Diffusion Model for Weather-scale Data Assimilation](https://mlsanity.com/paper/2401.05932/) (2024)
- [Latent assimilation with implicit neural representations for unknown dynamics](https://mlsanity.com/paper/2309.09574/) (2023)

## Abstract (arXiv)

The large underlying assumption of climate models today relies on the basis of a "confident" initial condition, a reasonably plausible snapshot of the Earth for which all future predictions depend on. However, given the inherently chaotic nature of our system, this assumption is complicated by sensitive dependence, where small uncertainties in initial conditions can lead to exponentially diverging outcomes over time. This challenge is particularly salient at global spatial scales and over centennial timescales, where data gaps are not just common but expected. The source of uncertainty is two-fold: (1) sparse, noisy observations from satellites and ground stations, and (2) internal variability stemming from the simplifying approximations within the models themselves. In practice, data assimilation methods are used to reconcile this missing information by conditioning model states on partial observations. Our work builds on this idea but operates at the extreme end of sparsity. We propose a conditional data imputation framework that reconstructs full temperature fields from as little as 1% observational coverage. The method leverages a diffusion model guided by a prekriged mask, effectively inferring the full-state fields from minimal data points. We validate our framework over the Southern Great Plains, focusing on afternoon (12:00-6:00 PM) temperature fields during the summer months of 2018-2020. Across varying observational densities--from swath data to isolated in-situ sensors--our model achieves strong reconstruction accuracy, highlighting its potential to fill in critical data gaps in both historical reanalysis and real-time forecasting pipelines.