Identify Then Project: Contrastive Learning of Latent Dynamics from Partial Observations with Port-Hamiltonian Structure

TL;DR

This paper introduces a two-stage contrastive learning framework for identifying and projecting latent port-Hamiltonian dynamics from partial observations, ensuring physical consistency and improved reliability over single-stage methods.

Contribution

The paper proposes a novel two-stage identify-then-project approach for learning structured latent port-Hamiltonian systems from partial data, combining contrastive learning with affine projection.

Findings

Two-stage approach preserves dynamics while enforcing physical structure.

Method outperforms single-stage learning, especially in dissipative and high-dimensional settings.

Affine projection naturally bridges contrastive identification and port-Hamiltonian systems.

Abstract

Identifying latent state representations and dynamics is essential when direct modeling in observation space is infeasible, particularly under partial and high-dimensional observations. In such settings, representation learning and physics-aware modeling are inherently coupled. We study this problem for latent port-Hamiltonian systems, a structured class encompassing both conservative and dissipative dynamics. We propose a two-stage identify-then-project framework. First, a contrastive teacher learns continuous-time latent dynamics from partial observations. Then, a student projects the identified teacher representation and dynamics onto a port-Hamiltonian submanifold via a learned affine chart, yielding a physically consistent realization. As a conceptual counterfactual, we also consider a single-stage variant that jointly learns latent identification and port-Hamiltonian structure, but find it to be less reliable, motivating the proposed two-stage teacher-student framework. We show theoretically that affine projection is the natural bridge between the affine gauge of contrastive latent identification and the port-Hamiltonian systems. Empirically, we demonstrate that the proposed two-stage approach preserves the teacher's dynamics while enforcing physical structure, and performs more reliably than the single-stage alternative, particularly in dissipative regimes and high-dimensional visual settings.