# Incomplete Data, Complete Dynamics: A Diffusion Approach

arXiv: https://arxiv.org/abs/2509.20098 · HTML: https://mlsanity.com/paper/2509.20098/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 35 of its 58 references that are in the map.
- First posted on arXiv 24 Sep 2025.

## Builds on (most cited first, 30 of 35)

- [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)
- [Masked Autoencoders Are Scalable Vision Learners](https://mlsanity.com/paper/2111.06377/) (2021)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [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)
- [Palette: Image-to-Image Diffusion Models](https://mlsanity.com/paper/2111.05826/) (2021)
- [Masked Autoencoders As Spatiotemporal Learners](https://mlsanity.com/paper/2205.09113/) (2022)
- [Analyzing and Improving the Training Dynamics of Diffusion Models](https://mlsanity.com/paper/2312.02696/) (2023)
- [From data to functa: Your data point is a function and you can treat it like one](https://mlsanity.com/paper/2201.12204/) (2022)
- [Noise2Score: Tweedie's Approach to Self-Supervised Image Denoising without Clean Images](https://mlsanity.com/paper/2106.07009/) (2021)
- [Ambient Diffusion: Learning Clean Distributions from Corrupted Data](https://mlsanity.com/paper/2305.19256/) (2023)
- [DiffusionPDE: Generative PDE-Solving Under Partial Observation](https://mlsanity.com/paper/2406.17763/) (2024)
- [Stochastic interpolants with data-dependent couplings](https://mlsanity.com/paper/2310.03725/) (2023)
- [Restart Sampling for Improving Generative Processes](https://mlsanity.com/paper/2306.14878/) (2023)
- [MisGAN: Learning from Incomplete Data with Generative Adversarial Networks](https://mlsanity.com/paper/1902.09599/) (2019)
- [VAEM: a Deep Generative Model for Heterogeneous Mixed Type Data](https://mlsanity.com/paper/2006.11941/) (2020)
- [When Physics Meets Machine Learning: A Survey of Physics-Informed Machine Learning](https://mlsanity.com/paper/2203.16797/) (2022)
- [not-MIWAE: Deep Generative Modelling with Missing not at Random Data](https://mlsanity.com/paper/2006.12871/) (2020)
- [Improved Techniques for Maximum Likelihood Estimation for Diffusion ODEs](https://mlsanity.com/paper/2305.03935/) (2023)
- [Learning Signal-Agnostic Manifolds of Neural Fields](https://mlsanity.com/paper/2111.06387/) (2021)
- [Image Restoration Through Generalized Ornstein-Uhlenbeck Bridge](https://mlsanity.com/paper/2312.10299/) (2023)
- [Learning from Irregularly-Sampled Time Series: A Missing Data Perspective](https://mlsanity.com/paper/2008.07599/) (2020)
- [MissDiff: Training Diffusion Models on Tabular Data with Missing Values](https://mlsanity.com/paper/2307.00467/) (2023)
- [Diffusion Models for Audio Restoration](https://mlsanity.com/paper/2402.09821/) (2024)
- [Diffusion Probabilistic Fields](https://mlsanity.com/paper/2303.00165/) (2023)
- [Unifying Bayesian Flow Networks and Diffusion Models through Stochastic Differential Equations](https://mlsanity.com/paper/2404.15766/) (2024)
- [Recent Advances on Machine Learning for Computational Fluid Dynamics: A Survey](https://mlsanity.com/paper/2408.12171/) (2024)
- [DiffPuter: Empowering Diffusion Models for Missing Data Imputation](https://mlsanity.com/paper/2405.20690/) (2024)

## Built on by (most cited first, 3 of 3)

- [When Does AI for PDEs Yield Scientific Evidence?](https://mlsanity.com/paper/2608.22504/) (2026)
- [OceanDepths: A Global Dataset of Paired Subsurface and Surface Ocean Observations](https://mlsanity.com/paper/2608.16373/) (2026)
- [Learning Field Reconstruction from Incomplete Data by Globally Correcting Local Estimates](https://mlsanity.com/paper/2610.05375/) (2026)

## Abstract (arXiv)

Learning physical dynamics from data is a fundamental challenge in machine learning and scientific modeling. Real-world observational data are inherently incomplete and irregularly sampled, posing significant challenges for existing data-driven approaches. In this work, we propose a principled diffusion-based framework for learning physical systems from incomplete training samples. To this end, our method strategically partitions each such sample into observed context and unobserved query components through a carefully designed splitting strategy, then trains a conditional diffusion model to reconstruct the missing query portions given available contexts. This formulation enables accurate imputation across arbitrary observation patterns without requiring complete data supervision. Specifically, we provide theoretical analysis demonstrating that our diffusion training paradigm on incomplete data achieves asymptotic convergence to the true complete generative process under mild regularity conditions. Empirically, we show that our method significantly outperforms existing baselines on synthetic and real-world physical dynamics benchmarks, including fluid flows and weather systems, with particularly strong performance in limited and irregular observation regimes. These results demonstrate the effectiveness of our theoretically principled approach for learning and imputing partially observed dynamics.