PHDME: Physics-Informed Diffusion Models without Explicit Governing Equations

2026-01-29

AI Summary

This work proposes PHDME, a novel framework that integrates port-Hamiltonian systems with diffusion models to address the challenge of reliably predicting trajectories of dynamical systems under sparse observational data and incomplete physical equations. By leveraging the energy-based structural priors of port-Hamiltonian systems and Gaussian processes, PHDME constructs physically consistent dynamic representations and generates synthetic data to augment training. The method incorporates incomplete physical knowledge without requiring explicit governing equations, employing a structured physics-informed residual loss and a split conformal calibration strategy to jointly ensure predictive accuracy and reliable uncertainty quantification. Experiments on partial differential equation benchmarks and a real-world spring-mass system demonstrate that PHDME significantly improves both prediction accuracy and physical consistency in data-scarce regimes.

Technology Category

• Humans and AI: Human-Aware Planning and Behavior Prediction
• Reasoning under Uncertainty: Relational Probabilistic Models
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

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.

Problem

• sparse data
• incomplete physics
• governing equations
• dynamical systems
• physics-informed machine learning

Innovation

• Physics-Informed Diffusion
• Port-Hamiltonian Systems
• Sparse Data
• Gaussian Process
• Conformal Calibration