Mesh Field Theory: Port-Hamiltonian Formulation of Mesh-Based Physics

2026-05-01

AI Summary

This work addresses the challenges of ambiguous coupling between physical structure and learning models, energy non-conservation, and poor extrapolation in mesh-based modeling of continuous physical systems. The authors propose Mesh Field Theory and its neural implementation, MeshFT-Net, which uniquely decouples topological and metric structures within mesh representations. By formulating dynamics in port-Hamiltonian form and learning only the metric-dependent component—while enforcing conservative interconnections dictated solely by topology—the approach embeds strong physically consistent inductive biases. The network architecture integrates principles from port-Hamiltonian systems, locality, permutation equivariance, directional covariance, and energy dissipation constraints. Experiments demonstrate that the method achieves near-zero energy drift, accurate dispersion relations, and exact momentum conservation across diverse physical systems, while exhibiting superior extrapolation capability and high data efficiency.

Abstract

We present Mesh Field Theory (MeshFT) and its neural realization, MeshFT-Net: a structure-preserving framework for mesh-based continuum physics that cleanly separates the physics' topological structure from its metric structure. Imposing minimal physical principles (locality, permutation equivariance, orientation covariance, and energy balance/dissipation inequality), we prove a reduction theorem for mesh-based physics. Under these conditions, the physical dynamics admit a local factorization into a port-Hamiltonian form: the conservative interconnection is fixed uniquely by mesh topology, whereas metric effects enter only through constitutive relations and dissipation. This reduction clarifies what must be fixed and what should be learned, directly informing MeshFT-Net's design. Across evaluations on analytic and realistic datasets, physics-consistency tests, and out-of-distribution validation, MeshFT-Net achieves near-zero energy drift and strong physical fidelity (correct dispersion and momentum conservation) along with robust extrapolation and high data efficiency. By eliminating non-physical degrees of freedom and learning only metric-dependent structure, MeshFT provides a principled inductive bias for stable, faithful, and data-efficient learning-based physical simulation.

Problem

• mesh-based physics
• port-Hamiltonian systems
• structure-preserving learning
• energy conservation
• physical simulation

Innovation

• Mesh Field Theory
• port-Hamiltonian systems
• structure-preserving learning
• physics-informed neural networks
• topology-metric separation