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.