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

TL;DR

Mesh Field Theory (MeshFT) introduces a structure-preserving mesh-based physics framework and its neural implementation, MeshFT-Net, which ensures physical fidelity, stability, and data efficiency in simulations by separating topological and metric effects.

Contribution

The paper develops MeshFT and MeshFT-Net, a novel neural framework that encodes mesh-based physics with a port-Hamiltonian structure, enhancing stability and physical accuracy.

Findings

MeshFT-Net achieves near-zero energy drift in simulations.

It maintains strong physical fidelity, including dispersion and momentum conservation.

The approach demonstrates robust extrapolation 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.