Title: Mesh Field Theory: Port–Hamiltonian Formulation of Mesh-Based Physics
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)
Keywords: Mesh Field Theory, Mesh-Based Physics, Port–Hamiltonian Dynamics, Structure-Preserving 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 physical simulation.

TL;DR: Mesh Field Theory disentangles mesh-based physics into a fixed topological wiring and learned metric operators, yielding stable long-horizon rollouts, high physical fidelity, strong data efficiency, and robust out-of-distribution performance.