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

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.

• The paper introduces a formal reduction theorem proving that any mesh-based physical system admits a local port-Hamiltonian factorization under four minimal physical requirements.
• The methodology cleanly separates fixed mesh topology from learnable metric and dissipation operators, ensuring energy/momentum conservation and eliminating spurious modes.
• Empirical results demonstrate MeshFT-Net's orders-of-magnitude improvements in stability, long-horizon fidelity, and data efficiency compared to existing neural mesh simulators.

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

Overview

This paper introduces Mesh Field Theory (MeshFT) and its neural instantiation, MeshFT-Net, as a unified framework for mesh-based physical simulation grounded strictly in physical principles. The central contribution is a formal reduction theorem: under four minimal requirements—locality, permutation equivariance, orientation covariance, and an energy balance/dissipation inequality—any mesh-based physical system admits a local port-Hamiltonian factorization. The unique aspect is a clean and provable separation between the algebraic-topological structure, entirely fixed by mesh incidences, and the metric-dependent (physical/material) operators, which remain the only learnable degrees of freedom. This topological-metric split is then used as the architectural foundation for MeshFT-Net, offering robust physical consistency, energy/momentum preservation, and strong data efficiency well beyond existing neural mesh simulators.

Theoretical Foundations: Local Port-Hamiltonian Reduction

MeshFT's core is the rigorous reduction theorem for mesh-based dynamics. It is shown that when a model satisfies locality, permutation equivariance, orientation covariance, and energy balance, its Jacobian locally factorizes as

$\frac{\partial F}{\partial z} = (J(z) - R(z)) G(z)$

where $J(z)$ is skew-symmetric and follows a precise sparsity/sign pattern dictated by mesh signed-incidences, $R(z)$ is positive-semidefinite (dissipative), and $G(z)$ is positive-definite (metric/constitutive). Crucially, this structure is not assumed a priori but deduced from physical principles, elucidating which model components must be hard-coded (topological incidences) versus which must be learned (local SPD/PSD constitutive and dissipation).

A key insight is that existing message-passing architectures, such as MeshGraphNets, often only enforce locality and permutation symmetry, lacking orientation covariance and energy balance. Without these, spurious and non-physical modes inevitably emerge. MeshFT systematically eliminates these issues by enforcing orientation covariance (sign-gauge symmetry under flipping mesh elements) and strict energy monotonicity.

Architectural Implementation: MeshFT-Net

Guided directly by the reduction theorem, MeshFT-Net fixes the topology-wired conservative coupling (signed incidence structure—hard-coded, not learned), and parameterizes only the metric and dissipation operators via small, local, orientation-even neural networks (e.g., degreewise MLPs, SPD or PSD constraints). Training proceeds via standard supervised losses on ground-truth pairs or trajectories, without any PDE-residual or explicit enforcement of governing equations. Time integration is carried out using symplectic splitting schemes to maintain long-horizon stability.

In this construction, the learnable functional class is sharply restricted by physical principle—constitutive/dissipation maps must be local, permutation-equivariant, and orientation-gauge-invariant. This principled restriction ensures that learned behaviors are physically admissible and generalize strongly across mesh resolutions, frequencies, and even parameter regimes.

Empirical Results

Comprehensive experiments compare MeshFT-Net with a spectrum of neural mesh simulators:

• Plane-Wave Benchmarks: On analytic benchmarks for 2D waves, MeshFT-Net attains lowest one-step MSE and near-zero energy drift (e.g., $1.3 \times 10^{-4}$), compared with orders-of-magnitude larger drift in MGN, HNN, and others.
• Dissipative Dynamics: MeshFT-Net most faithfully preserves energy dissipation, with competitive prediction error and minimal normalized energy error (NEE).
• Long-Horizon and OOD: MeshFT-Net maintains energy/momentum conservation and correct modal dispersion across long rollouts, frequency/resolution shifts, and generalizes strongly under out-of-distribution physical parameters. Competing models, including MGN, HNN, FNO, and PI-MGN, suffer significant degradation, often diverging entirely or failing to conserve invariants.
• Physical Consistency Diagnostics: Quantitative tests verify superior wave speed accuracy, canonical correspondence, minimal PDE residuals, and near-perfect conservation of momentum and kinetic-potential balance.
• Acoustic Scattering: On The Well acoustic datasets (realistic, discontinuous media), MeshFT-Net matches physical wavefronts and reflection patterns, with only minor smoothing at high frequencies.
• Ablations: Structural ablations reinforce that violating topology/sign structure or SPD-ness of metric blocks results in instability, rapid energy drift, and spurious phenomena—affirming the necessity of all four MeshFT assumptions.

Theoretical and Practical Implications

MeshFT advances the field in several dimensions:

• Separation of Concerns: By enforcing a provable split between topology (hard-wired) and metric/dissipative content (learned), it formalizes which aspects are architecture and which are trainable, avoiding the pitfalls of template selection/mismatch in global port-Hamiltonian or Hamiltonian NN approaches.
• Inductive Bias: It provides an explicit, physically-justified inductive bias for ML-based simulation, confining learning to only the metric/dissipation side.
• Elimination of Spurious Degrees of Freedom: By construction, MeshFT removes all non-physical couplings and enforces exact preservation of topological invariants such as discrete divergence-free constraints.
• Adaptability and Generalization: The principle-driven reduction holds independently of mesh dimension/type (structured/unstructured, manifold/non-manifold) and cleanly supports spatial heterogeneity and varying constitutive behavior.

Future Directions

The framework suggests several natural research avenues:

• Extension to Nonlinear and Non-Local Constitutive Laws: While the paper explores local, orientation-even parameterizations, extension to nonlinear state-dependent SPD/PSD maps and non-local constitutive behavior may further improve expressivity.
• Structure-Preserving Model Reduction: Identification of minimal parameterizations for low-data or multi-fidelity simulation, while maintaining physical structure, is enabled by the MeshFT split.
• Integration with Data-Driven Exterior Calculus: MeshFT’s architecture aligns closely with discrete exterior calculus schemes, opening the door for integration and transfer across mesh-based discretization and learned physics.

Conclusion

Mesh Field Theory and MeshFT-Net formalize and enforce a physically-principled, topology-metric separation for mesh-based neural simulation. The local reduction theorem ensures conservative interconnections strictly follow mesh incidences, while geometry/material and dissipation are learned locally. MeshFT-Net demonstrates orders-of-magnitude improvements in long-horizon fidelity, generalization, and data efficiency, robustly eliminating spurious modes and non-physical drift. The MeshFT framework thus establishes a rigorous foundation for stable, data-efficient, and physically-consistent neural simulators on arbitrary meshes (2605.00394).