PhysDEM: Physics-Defined Energy-Matching Diffusion for Spatiotemporal Field Generation under Scarce Measurements

2026-10-01

AI Summary

This study addresses the limitation of conventional diffusion models that rely on full-field datasets, which hinders their ability to generate spatiotemporal physical fields from sparse measurements. To overcome this, we propose a physics-defined energy-matching diffusion framework that eliminates the need for pre-assembling complete datasets by deeply integrating governing equations with sparse observations. Furthermore, we introduce Gibbs target reweighting, conditional mean identities, and physical displacement probability flows to effectively model the distributional characteristics of multi-solution physical fields. This approach enables amortized spatiotemporal field inference, achieving efficient sampling and the recovery of highly coherent physical fields across both synthetic partial differential equation benchmarks and real-world applications while maintaining diagnostic stability.

Abstract

Generating and predicting spatiotemporal physical fields from scarce measurements is challenging, as observations are insufficient to characterize a distribution over complete fields. This limits conventional data-driven diffusion models that rely on full-field datasets. We introduce PhysDEM, a physics-defined diffusion framework that combines governing equations with spatially sparse observations to generate multiple plausible fields. First, we construct a Gibbs target by reweighting a measurement-conditioned Gaussian reference with PDE residual energy. Second, we derive an exact conditional-mean identity that reduces denoising to supervised learning of the standardized energy-induced mean correction. Third, a physics-displacement probability flow cancels Gaussian reference terms and enables amortized sampling with changing measurements through Gaussian conditioning, without retraining. Experiments on synthetic PDE systems and real-world-informed applications demonstrate that PhysDEM supports coherent field recovery and efficient sampling while maintaining stable diagnostics under tested noise levels, illustrating its practical value for field assessment. To our knowledge, PhysDEM is the first physics-defined diffusion model enabling amortized spatiotemporal field inference without preassembled full-field datasets.

Problem

• spatiotemporal field generation
• scarce measurements
• physics-informed diffusion
• partial differential equations
• field inference

Innovation

• Physics-defined diffusion model
• Spatiotemporal field generation
• Gibbs target distribution
• Amortized sampling
• PDE residual energy