PHDME: Physics-Informed Diffusion Models without Explicit Governing Equations

TL;DR

PHDME tackles the challenge of forecasting dynamical systems when governing equations are unknown and data are scarce by learning a probabilistic energy-based prior (GP-dPHS) and amortizing it into a diffusion model. The two-stage approach first builds a GP-dPHS from limited observations to capture Hamiltonian structure, then trains a diffusion model with a physics-consistency loss weighted by GP uncertainty, producing fast, batched trajectory samples. Split conformal prediction furnishes finite-sample calibrated uncertainty on the generated trajectories, enabling reliable risk-aware forecasting. Experiments on canonical PDE benchmarks and a real-world spring demonstrate improved accuracy and physical fidelity under data scarcity, with robust uncertainty quantification against baselines that lack strong physics priors.

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.

Figures (15)

• Figure 1: The left panel depicts a typical soft robot scenario in which a flexible continuum manipulator exhibits dynamics that are difficult to specify. The middle panel adopts a top-down parameterization with the y-axis as spatial projection along the arm direction, the z-axis (pixel value) as displacement, and the x-axis as temporal evolution. This converts the evolution into an image form, enabling the diffusion model to synthesize the full spatiotemporal field of multiple time steps in a single shot rather than step-by-step rollouts.
• Figure 2: This figure visualize the two-stage training of the PHDME, where we firstly train a rather slow but structured deep prior. Then we leverage this prior to inform the diffusion training for rapid sample generations.
• Figure 3: From left to right, the figure illustrates the process starting with the original video, skeletonization, and final spring movement data over time (overlay).
• Figure 4: On the left side, PHDME beats the baselines with pure-data driven and limited physics access by having the minimum MSE over iterations. On the right side, we further investigate the potential impacts of the physics-loss term percentage regarding the performance.
• Figure 5: Generated trajectories from unseen initial conditions with sparse training data. Columns (left to right) compare NeuralODE, PHDME, and the simulator ground truth; rows show $p(x,t)$ (top) and $q(x,t)$ (bottom). PHDME yields rollouts that closely match the simulator's dominant wave patterns and temporal evolution, while NeuralODE exhibits noticeable distortions, highlighting PHDME's advantage in data-scarce generalization.