PHDME: Physics-Informed Diffusion Models without Explicit Governing Equations

TL;DR

This paper introduces PHDME, a physics-informed diffusion framework that models complex dynamical systems with incomplete physics and sparse data, improving trajectory forecasting accuracy and physical consistency.

Contribution

PHDME leverages port-Hamiltonian structures without requiring explicit governing equations, enabling effective diffusion modeling with limited observations.

Findings

Enhanced accuracy in PDE benchmarks

Improved physical consistency under data scarcity

Effective uncertainty quantification with conformal calibration

Abstract

Diffusion models provide expressive priors for forecasting trajectories of dynamical systems, but are typically unreliable in the sparse data regime. Physics-informed machine learning (PIML) improves reliability in such settings; however, most methods require \emph{explicit governing equations} during training, which are often only partially known due to complex and nonlinear dynamics. We introduce \textbf{PHDME}, a port-Hamiltonian diffusion framework designed for \emph{sparse observations} and \emph{incomplete physics}. PHDME leverages port-Hamiltonian structural prior but does not require full knowledge of the closed-form governing equations. Our approach first trains a Gaussian process distributed Port-Hamiltonian system (GP-dPHS) on limited observations to capture an energy-based representation of the dynamics. The GP-dPHS is then used to generate a physically consistent artificial dataset for diffusion training, and to inform the diffusion model with a structured physics residual loss. After training, the diffusion model acts as an amortized sampler and forecaster for fast trajectory generation. Finally, we apply split conformal calibration to provide uncertainty statements for the generated predictions. Experiments on PDE benchmarks and a real-world spring system show improved accuracy and physical consistency under data scarcity.