Title: Mesh Field Theory: A Port-Hamiltonian Approach to Mesh-Based Physics

Abstract:

This work introduces Mesh Field Theory (MeshFT) alongside its neural implementation, MeshFT-Net, offering a structure-preserving framework for continuum physics based on meshes. This approach effectively decouples the topological architecture of the physics from its metric characteristics. By enforcing a set of fundamental physical principles—including locality, permutation equivariance, orientation covariance, and energy balance/dissipation inequalities—we establish a reduction theorem for mesh-based physics. These constraints demonstrate that physical dynamics can be locally factorized into a port-Hamiltonian structure. Within this formulation, the conservative interconnection is uniquely determined by the mesh's topology, while metric influences are confined to constitutive relations and dissipation mechanisms. This theoretical reduction delineates which aspects of the system must be predefined versus those that should be learned, thereby guiding the architecture of MeshFT-Net. Empirical evaluations across both analytical and realistic datasets, including physics-consistency checks and out-of-distribution tests, reveal that MeshFT-Net maintains near-zero energy drift and high physical accuracy, ensuring correct momentum conservation and dispersion. Furthermore, the model demonstrates robust extrapolation capabilities and significant data efficiency. By discarding non-physical degrees of freedom and focusing learning exclusively on metric-dependent structures, MeshFT offers a rigorous inductive bias that enables stable, accurate, and data-efficient learning-based physical simulations.

Source: arXiv Generated at: 2026-06-02 00:00:00 UTC