Title: PHDME: Physics-Informed Diffusion Models without Explicit Governing Equations
Primary area: neurosymbolic & hybrid AI systems (physics-informed, logic & formal reasoning, etc.)
Keywords: Physics-informed Learning, Diffusion Model, Port-Hamiltonian system, Uncertainty Quantification, Gaussian process

Abstract: Diffusion models are expressive priors for generating and predicting data from high-dimensional dynamical systems. Yet, purely data-driven approaches often lack reliability and trustworthiness, motivating growing interest in physics-informed machine learning (PIML). Most existing PIML methods, however, assume access to exact governing equations during training—an assumption that fails when the dynamics are unknown or too complex to model accurately. To address this gap, we introduce PHDME (Port-Hamiltonian Diffusion Model), a physics-informed diffusion framework that learns system dynamics without requiring exact 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 and diverse dataset for diffusion training. To enforce physics-consistency, we embed the GP-dPHS structure directly into the diffusion training objective through a loss that penalizes deviations from the learned Hamiltonian dynamics, weighted by the GP’s predictive uncertainty. After training, we employ conformal prediction to provide distribution-free uncertainty quantification of the generated trajectories. In this way, PHDME is designed for regimes with scarce data and unknown equations, enabling data-efficient, physically valid trajectory generation with calibrated uncertainty estimates.

TL;DR: PHDM: diffusion guided by GP-learned Port-Hamiltonian energy gradients from sparse data. No exact fuction needed; structure is conserved and uncertainty is calibrated for spatiotemporal prediction.