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

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.

Mesh-based simulations are widely used to analyze physical dynamics such as fluids and deformable bodies, but they are often computationally expensive. Machine learning can make them faster, yet overly flexible models may lose the physical structure needed for stable long-term prediction. This paper proposes Mesh Field Theory (MeshFT) and its neural realization, MeshFT-Net. Instead of relying on heuristic design, MeshFT derives the model structure from universal physical principles: locality, symmetry, orientation consistency, and energy balance. Its key idea is to keep the mesh’s direction-aware connection pattern fixed, while learning system-dependent effects such as geometry, material response, and energy loss. Across wave, nonlinear, dissipative, and deformable-body benchmarks, MeshFT-Net achieved more stable rollouts, higher physical fidelity, and better data efficiency than baseline simulators.