Incomplete Data, Complete Dynamics: A Diffusion Approach

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. Learning physical dynamics from observational data represents a cornerstone challenge in machine learning and scientific computing, with applications spanning weather forecasting (Conti, 2024; Zhang et al., 2025b), fluid dynamics (Wang et al., 2024; Brunton & Kutz, 2024), biological systems modeling (Qi et al., 2024; Goshisht, 2024), and beyond. Classical physics-based approaches require explicit specification of governing equations and boundary conditions, while data-driven methods offer the promise of discovering hidden dynamics directly from observations (Luo et al., 2025; Meng et al., 2025). However, a fundamental bottleneck persists: real-world observational data are inherently incomplete, irregularly sampled, and subject to various forms of missing information, making it difficult for existing approaches to learn accurate representations of the underlying dynamics.