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

Conference: ICML 2026
arXiv: 2605.00394
Code: None
Area: 3D Physics Simulation / Structure-Preserving Neural Networks / Mesh Learning
Keywords: Mesh Physics, Port-Hamiltonian, Topology-Metric Separation, Energy Conservation, MeshGraphNet

TL;DR

Starting from four physical principles—locality, permutation equivariance, orientation covariance, and energy conservation/dissipation inequality—this work proves that any mesh-based physical dynamics satisfying these axioms can be locally reduced to a port-Hamiltonian form at the Jacobian level. In this form, the conserved interconnection structure \(J\) is entirely determined by the mesh topology (signed incidence matrix \(D_k\)), while the metric and dissipation are learned through \(G\) and \(R\). The proposed MeshFT-Net achieves near-zero energy drift, correct dispersion and momentum over long rollouts, and significantly outperforms MGN and HNN.

Background & Motivation

Background: Two main approaches dominate mesh-based physics learning (e.g., fluids, elastics, acoustics): (1) GNN/message-passing methods (e.g., MeshGraphNets, SPH-Net, FNO), and (2) explicit structure-preserving networks (e.g., HNN, LNN, port-Hamiltonian NN, GENERIC), which hard-code energy/symplectic structures into the architecture.

Limitations of Prior Work: Pure MGN-like methods suffer from energy drift and non-physical modes during long rollouts. HNN/global port-Hamiltonian NNs require manually chosen global Hamiltonians/templates, which are not robust to model mis-specifications. Neither approach clarifies which degrees of freedom in mesh physics are non-physical and should be structurally eliminated.

Key Challenge: In exterior differential geometry, the exterior derivative \(d\) is topological (metric-independent), while geometric/material properties enter only through metric operators like the Hodge star \(\star\). Existing learned simulators conflate these, allowing metric learning to contaminate topology and amplifying metric errors through topological inaccuracies.

Goal: (1) Establish clean physical principles; (2) formally prove that these principles constrain dynamics to a port-Hamiltonian form at the Jacobian level; (3) design a network that "fixes topology, learns metrics," and validate its long-term stability, dispersion, momentum, and OOD generalization.

Key Insight: MeshGraphNet can be viewed as satisfying locality (L) and permutation equivariance (P) but lacking orientation covariance (O) and energy balance (E). Enforcing (O) and (E) eliminates non-physical degrees of freedom structurally, leaving the classical DEC (Discrete Exterior Calculus) topological skeleton plus local metric operators.

Core Idea: "Physical principles ⇒ Jacobian factorization ⇒ Fixed topology, learnable metrics"—topology is fixed by \(D_k\) (signed incidence matrix), while only the positive-definite metric \(G_\theta\) and semi-positive-definite dissipation \(R_\theta\) are learnable.

Method

Overall Architecture

The input is a fixed directed cell complex \(\mathcal{K}\) and initial state \(z^0 = (z_k^0, z_{k+1}^0)\) (cochain degrees of freedom, e.g., node potentials + edge fluxes). The output is the next state \(z^{n+1}\). The pipeline: (1) Use the reduction theorem to constrain dynamics to the port-Hamiltonian form \(\dot z = (J - R(z)) G(z) z\); (2) \(J = \begin{pmatrix} 0 & -D_k^\top \\ D_k & 0 \end{pmatrix}\) is entirely determined by the mesh incidence matrix and is not trained; (3) Use a Strang splitting integrator to alternate "half-step dissipation + conservative step + half-step dissipation," with all operations as sparse matrix-vector products, achieving \(O(N)\) complexity.

Key Designs

1. Four Axioms + Local Port-Hamiltonian Reduction Theorem:

   • Function: Provides an axiomatic definition of "valid mesh-based physical dynamics" and proves that any \(F\) satisfying these axioms can be written at the Jacobian level as \(\partial F / \partial z = (J(z) - R(z)) G(z)\).
   • Mechanism: The four axioms are (L) locality, (P) permutation equivariance, (O) orientation covariance (reversing cell orientation flips only oriented variable signs, leaving scalar quantities \(H, e^\top \dot z\) invariant), and (E) energy balance—dynamics decompose into a conservative part \(F_\text{con}\) (satisfying \(e^\top F_\text{con} = 0\)) and a dissipative part \(F_\text{diss}\) (satisfying \(e^\top F_\text{diss} \le 0\)). The theorem proves that the Jacobian of the conservative part must be skew-symmetric, the dissipative part must be semi-negative-definite, and the off-diagonal blocks of the conservative interconnection must take the signed-incidence structure \(J_{k,k+1} = -D_k^\top C_k(z)\) and \(J_{k+1,k} = C_k(z) D_k\) (Corollary 3.4 shows \(C_k\) can be absorbed into coordinate scaling when constant).
   • Design Motivation: Unlike directly positing a port-Hamiltonian template, this work deduces "what must be fixed and what can be learned," formalizing the topology-metric separation as a structural theorem rather than an engineering heuristic.

2. MeshFT-Net: Fixed Topology, Learnable Metrics:

   • Function: Implements the theorem as a neural network architecture—\(J\) is fixed as \(\begin{pmatrix} 0 & -D_k^\top \\ D_k & 0 \end{pmatrix}\), and learnable weights are restricted to SPD metrics \(G_\theta\) and PSD dissipation \(R_\theta\).
   • Mechanism: Energy is quadratic \(H_\theta(z) = \tfrac{1}{2} z^\top G_\theta z\), with co-energy \(e = G_\theta z\). \(G_\theta\) is implemented via softplus diagonals or small Cholesky blocks, conditioned on local geometry/material features through permutation-equivariant and orientation-even MLPs. \(R_\theta(z)\) adopts a Rayleigh form \(z \mapsto \gamma(\cdot) G_\theta^{-1} z\) to ensure PSD.
   • Design Motivation: Moves "topology" from the training set to the "mesh itself"—since it is mesh-given and should not be learned—and delegates "metrics/materials/dissipation" to the network. This ensures that even with limited or OOD data, the topological structure remains intact.

3. Strang Splitting Time Integrator + CFL Guard:

   • Function: Maintains symplecticity of the conservative part while precisely integrating dissipation within a single update layer.
   • Mechanism: Algorithm 1 implements a KDK scheme: first a half-step dissipation \(\exp(-\tfrac{\Delta t}{2} R G) z\), then symmetric leapfrog updates for the conservative part \(z_k \leftarrow z_k - \tfrac{\Delta t}{2} D_k^\top G_{k+1} z_{k+1}\) → \(z_{k+1} \leftarrow z_{k+1} + \Delta t D_k G_k z_k^\text{half}\) → another half-step conservation, and finally another half-step dissipation. CFLGuard(Δt) scales the step size based on the local maximum eigenvalue to prevent numerical blow-up.
   • Design Motivation: Conventional Euler integration cannot conserve energy precisely. Strang splitting ensures that the conservative and dissipative subflows do not interfere, enabling an analytically provable \(\dot H = -e^\top R(z) e \le 0\).

Loss & Training

Supervised one-step prediction: \(\sum_k \text{Loss}(\hat z_k^{n+1}, z_k^{n+1})\), without PDE residual terms. The inductive bias comes entirely from the fixed \(J\) and SPD/PSD structures. Multi-step stacking with supervision on the final output is used for rollout tasks.

Key Experimental Results

Main Results

Benchmarked on analytic plane waves (regular grid + Delaunay), Rayleigh damped oscillations, acoustic scattering from The Well, and OOD settings (frequency/wave speed/resolution shifts) against MGN, MGN-HP, HNN, PI-MGN, FNO, and GraphCON.

Task	Model	One-step MSE	TSMSE (rollout)	Energy Drift
Plane Waves (Regular Grid)	MGN	\(1.6{\times}10^{-7}\)	\(1.3{\times}10^{-1}\)	\(25.9\)
Plane Waves	HNN	\(3.5{\times}10^{-8}\)	\(3.0{\times}10^{-3}\)	\(1.0{\times}10^{-2}\)
Plane Waves	MeshFT-Net	\(\mathbf{1.3{\times}10^{-9}}\)	\(\mathbf{9.6{\times}10^{-5}}\)	\(\mathbf{1.3{\times}10^{-4}}\)
Rayleigh Damping	MGN	\(5.2{\times}10^{-8}\)	\(1.7{\times}10^{-1}\)	NEE \(2.2\)
Rayleigh Damping	MeshFT-Net	\(1.2{\times}10^{-7}\)	\(\mathbf{2.1{\times}10^{-2}}\)	NEE \(\mathbf{2.1{\times}10^{-2}}\)

Ablation Study

Configuration	TSMSE	Energy Drift
Fixed \(J\) + Diagonal \(G\)	\(4.52{\times}10^{-5}\)	\(0.115\)
Fixed \(J\) + Full \(G\)	\(3.28{\times}10^{-5}\)	\(0.028\)
\(z\)-Dependent \(J\) + Diagonal \(G\)	\(\mathbf{6.77{\times}10^{-6}}\)	\(0.025\)
\(z\)-Dependent \(J\) + Full \(G\)	\(6.17{\times}10^{-6}\)	\(0.030\)

Physical consistency diagnostics (Table 3a excerpt) show MeshFT-Net ranks first in wave speed error, gauge relations, PDE residuals (short/long-range), kinetic-potential energy equipartition, and momentum conservation, with momentum error as low as \(4.9{\times}10^{-8}\) (MGN: \(0.39\), HNN: \(1.07\)).

Key Findings

• One-step MSE does not strongly correlate with long-term rollout performance: MGN achieves the lowest one-step MSE in damping tasks but has a rollout TSMSE nearly 10× higher than MeshFT-Net, indicating short-term accuracy does not imply long-term physical fidelity.
• Momentum conservation is not explicitly constrained, but MeshFT-Net naturally inherits action-reaction symmetry by enforcing orientation covariance (O). Methods without (O) exhibit orders-of-magnitude higher momentum drift.
• Under OOD conditions, MGN/FNO/PI-MGN diverge (\(>100\)) under resolution or wave speed shifts, while MeshFT-Net maintains energy drift \(<\mathcal{O}(10^{-1})\), demonstrating the generalization power of fixed-topology inductive bias.
• Nonlinear shallow-water toy experiments show that state-dependent \(J\) significantly improves performance when coefficients depend on the state. However, "fixed topology + full \(G\)" compensates to some extent, balancing model capacity and structure.

Highlights & Insights

• "Theorem-driven architecture design" is the paper's key methodological contribution: proving that a set of axioms structurally constrains the solution space to port-Hamiltonian form, then hard-coding these constraints into the architecture rather than reverse-engineering from a global Hamiltonian template.
• "Topology-metric separation" is abstract but highly practical: topology (incidence matrix \(D_k\)) belongs to the mesh and is never learned, while metrics (\(G, R\)) belong to physics and are learnable. This injects a layer of "non-generalizable physics" into GNNs, leaving learnable parts to material/geometry-related properties.
• The interpretation of MGN is clear—MGN is not wrong but overly permissive. Adding (O) and (E) shrinks the state space to a physically reasonable subset, providing a clear "energy + orientation" patching route for future mesh-based simulators.

Limitations & Future Work

• The main experiments use state-independent \(G_\theta\) (quadratic storage). Strongly nonlinear PDEs (e.g., Navier-Stokes, plasticity, phase transitions) require nonlinear constitutive \(G_\theta(z)\)/\(\Psi_\theta(e)\), which are only explored in supplementary toy experiments.
• Axioms (O) and (E) are sufficient conditions but do not guarantee correctness under external sources, boundary conditions, or multi-physics coupling. The paper explicitly notes that external source terms require further extensions.
• The framework relies on the cell complex's incidence structure \(D_k\), making adaptation to fully unstructured/time-varying topologies (e.g., fracture materials, adaptive meshes) non-trivial.
• In some OOD shifts, MeshFT-Net outperforms FNO/GraphCON only relatively, with absolute TSMSE still non-zero, indicating that topological constraints cannot fully replace sufficiently rich data.

Related Work & Insights

• vs MGN: MGN satisfies (L) + (P), while this work enforces (O) + (E), structurally eliminating non-physical degrees of freedom and improving long-term stability by orders of magnitude.
• vs HNN/port-Hamiltonian NN (Desai et al.): Those methods learn on a "global Hamiltonian template," requiring correct templates. This work uses "local Jacobian factorization," making it more robust to template errors.
• vs DEC/data-driven exterior calculus (Trask et al.): Both share the topology-metric separation idea. This work derives it from physical axioms rather than differential geometry templates, making it more general.
• vs PI-MGN/FNO: PDE residual/operator learning is data-driven + weakly physical. This work is structure-driven, requiring no knowledge of PDE forms, with more stable generalization.

Rating

• Novelty: ⭐⭐⭐⭐ Formalizes topology-metric separation from exterior differential geometry into a strict GNN design principle, with one-to-one correspondence between theory and implementation.
• Experimental Thoroughness: ⭐⭐⭐⭐ Covers analytic + real data, physical diagnostics, OOD, and nonlinear ablations comprehensively.
• Writing Quality: ⭐⭐⭐⭐ Theorems are clearly stated, Algorithm 1 is reproducible, and supplementary material is almost unnecessary.
• Value: ⭐⭐⭐⭐ Provides a theorem-driven design paradigm for the learned simulator community, enabling future structure-preserving networks to follow the "axioms ⇒ Jacobian constraints ⇒ fixed topology" workflow.