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

arXiv cs.LG / 5/4/2026

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Key Points

The paper introduces Mesh Field Theory (MeshFT) and MeshFT-Net, a structure-preserving neural framework for mesh-based continuum physics that separates topology from metric effects.
By enforcing basic physical constraints (locality, permutation equivariance, orientation covariance, and energy balance/dissipation inequality), the authors prove a reduction theorem that yields a unique port-Hamiltonian interconnection determined by mesh topology.
Metric influences are confined to constitutive relations and dissipation, reducing the need to learn non-physical degrees of freedom and improving physical interpretability.
Experiments on analytic and realistic datasets show strong physics consistency, near-zero energy drift, correct dispersion and momentum conservation, and good out-of-distribution extrapolation with high data efficiency.
The resulting theoretical insights directly guide MeshFT-Net’s design, offering an inductive bias for stable, faithful, and data-efficient learning-based physical simulation.

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.