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

Conference: ICML 2026
arXiv: 2605.00394
Code: None
Area: 3D Physical 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 + energy conservation/dissipation inequality"—it is proven that any mesh-based physical dynamics satisfying these axioms can be locally reduced to a port-Hamiltonian form at the Jacobian level. In this formulation, the conservative interconnection structure \(J\) is completely fixed by the mesh topology (the signed incidence matrix \(D_k\)), while metric and dissipation enter through learnable \(G\) and \(R\). MeshFT-Net, designed based on this theory, achieves near-zero energy drift and correct dispersion/momentum in long-term rollouts, significantly outperforming MGN and HNN.

Background & Motivation

Background: The field is rapidly evolving along two paths: using GNNs/message passing to learn mesh physics (fluids, elastodynamics, acoustics) (e.g., MeshGraphNets, SPH-Net, FNO), and using explicit structure-preserving networks (HNN, LNN, port-Hamiltonian NN, GENERIC) that hard-code energy or symplectic structures into the architecture.

Limitations of Prior Work: Pure MGN-based methods suffer from energy drift and non-physical modes in long-term rollouts. HNN or global port-Hamiltonian NNs require manual selection of a global Hamiltonian/template, showing poor robustness when the model is misspecified. Neither path clearly explains which degrees of freedom in mesh physics are non-physical and should be eliminated by the structure.

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 \(\star\). Unfortunately, existing learned simulators entangle the two, allowing metric learning to pollute the topological structure, which in turn causes topological errors to amplify metric errors.

Goal: (1) Provide a clean set of physical principles; (2) Formally prove that these principles force the dynamics into a port-Hamiltonian form at the Jacobian level; (3) Design a network that "fixes the topology and learns only the metric," and verify its long-term stability, dispersion, momentum, and OOD generalization.

Key Insight: MeshGraphNet is viewed as a superset that already satisfies Locality (L) and Permutation Equivariance (P) but lacks Orientation Covariance (O) and Energy Balance (E). By enforcing O and E, redundant non-physical degrees of freedom are structurally eliminated, leaving exactly the topological skeleton of classical Discrete Exterior Calculus (DEC) plus local metric operators.

Core Idea: "Physical Principles \(\Rightarrow\) Jacobian Factorization \(\Rightarrow\) Fixed Topology, Learnable Metric"—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 oriented cell complex \(\mathcal{K}\) and an initial state \(z^0 = (z_k^0, z_{k+1}^0)\) (cochain degrees of freedom, e.g., node potentials + edge flows). The output is the state \(z^{n+1}\) at the next time step. The pipeline consists of: (1) Using the reduction theorem to constrain dynamics to a port-Hamiltonian form \(\dot z = (J - R(z)) G(z) z\); (2) Setting \(J = \begin{pmatrix} 0 & -D_k^\top \\ D_k & 0 \end{pmatrix}\) fixed by the mesh incidence matrix (untrained); (3) Using a Strang splitting integrator to alternate between "half-step dissipation + conservation step + half-step dissipation," where all operations are sparse matvecs with \(O(N)\) complexity.

Key Designs

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

    • Function: Provides an axiomatic definition for "valid mesh 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 direction flips the sign of oriented variables but keeps scalar quantities like \(H\) and \(e^\top \dot z\) invariant), and (E) Energy Balance—dynamics are split into a conservative part \(F_\text{con}\) (satisfying \(e^\top F_\text{con} = 0\)) and a dissipative part \(F_\text{diss}\) (satisfying \(e^\top F_\text{diss} \le 0\)). It is proven that the Jacobian of the conservative part must be skew-symmetric and the dissipative part must be negative semi-definite. Furthermore, 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 \(J_{k+1,k} = C_k(z) D_k\).
    • Design Motivation: Unlike "positing a port-Hamiltonian template," this work deduces "what must be fixed and what can be learned," framing the division of labor between topology and metric as a structural theorem rather than engineering heuristics.

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

    • Function: Directly implements the theorem as a neural network architecture—fixing \(J\) as \(\begin{pmatrix} 0 & -D_k^\top \\ D_k & 0 \end{pmatrix}\) and restricting learnable weights to the SPD metric \(G_\theta\) and PSD dissipation \(R_\theta\).
    • Mechanism: Energy is defined as a quadratic form \(H_\theta(z) = \tfrac{1}{2} z^\top G_\theta z\), with co-energy \(e = G_\theta z\); \(G_\theta\) is implemented using softplus diagonals or small Cholesky blocks, conditioned on local geometric/material features via permutation-equivariant + orientation-even MLPs. \(R_\theta(z)\) takes 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 given by the mesh and should not be learned from data—while leaving "metric/materials/dissipation" to the network. Consequently, the topological structure remains stable even with limited or out-of-distribution data.

  3. Strang Splitting Time Integrator + CFL Guard:

    • Function: Maintains symplecticity of the conservative part while accurately integrating dissipation within a single update layer.
    • Mechanism: Algorithm 1 provides a KDK pattern: a half-step dissipation \(\exp(-\tfrac{\Delta t}{2} R G) z\), followed by a symmetric leapfrog for the conservative part \(z_k \leftarrow z_k - \tfrac{\Delta t}{2} D_k^\top G_{k+1} z_{k+1} \rightarrow z_{k+1} \leftarrow z_{k+1} + \Delta t D_k G_k z_k^\text{half} \rightarrow\) another half-step conservation, and finally another half-step dissipation. CFLGuard(Δt) scales the step size based on local maximum eigenvalues to prevent numerical explosion.
    • Design Motivation: Standard Euler schemes cannot accurately conserve energy; Strang splitting allows the conservative and dissipative sub-flows to proceed without interference, yielding an analytically provable \(\dot H = -e^\top R(z) e \le 0\) when paired with an exact skew-symmetric \(J\).

Loss & Training

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

Key Experimental Results

Main Results

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

Task Model One-step MSE TSMSE (rollout) Energy Drift
Analytic Plane Wave (Regular) MGN \(1.6{\times}10^{-7}\) \(1.3{\times}10^{-1}\) \(25.9\)
Analytic Plane Wave HNN \(3.5{\times}10^{-8}\) \(3.0{\times}10^{-3}\) \(1.0{\times}10^{-2}\)
Analytic Plane Wave Ours \(\mathbf{1.3{\times}10^{-9}}\) \(\mathbf{9.6{\times}10^{-5}}\) \(\mathbf{1.3{\times}10^{-4}}\)
Rayleigh Damping MGN \(5.2{\times}10^{-8}\) \(1.7{\times}10^{-1}\) NEE \(2.2\)
Rayleigh Damping Ours \(1.2{\times}10^{-7}\) \(\mathbf{2.1{\times}10^{-2}}\) 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 show that 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 \(4.9{\times}10^{-8}\) (vs. \(0.39\) for MGN and \(1.07\) for HNN).

Key Findings

  • One-step MSE is not strongly correlated with long-range rollout performance: MGN achieved the lowest one-step MSE in damping tasks but had nearly 10x higher rollout TSMSE than MeshFT-Net, indicating that short-term local accuracy does not imply long-term physical fidelity.
  • Momentum conservation is not an explicit constraint, but MeshFT-Net naturally inherits action-reaction relations by enforcing orientation covariance (O). Methods without enforced (O) show momentum drift orders of magnitude higher.
  • On OOD tasks, MGN/FNO/PI-MGN diverge (\(>100\)) when resolution or wave speed shifts, whereas MeshFT-Net maintains energy drift \(<\mathcal{O}(10^{-1})\). The inductive bias of the fixed topology provides genuine generalization.
  • Nonlinear shallow-water experiments show that when coefficients are state-dependent, changing \(J\) to be state-dependent improves performance, though "fixed unlearnable topology + full \(G\)" can compensate to some extent, representing a trade-off between model capacity and structure.

Highlights & Insights

  • "Theorem-driven architecture design" is the primary methodological contribution: first proving the solution space is structurally constrained to port-Hamiltonian forms under specific axioms, then treating these constraints as hard architecture rather than working backward from a global Hamiltonian template.
  • The "topology–metric separation" is abstract yet practical: topology (incidence matrix \(D_k\)) belongs to the mesh and is never learned; metric (\(G, R\)) belongs to physics and is learnable. This manually injects a layer of "physically non-generalizable information" into the GNN, leaving the learnable parts for properties truly related to material and geometry.
  • The interpretation of the MGN series is very clear: MGN is not "wrong," but its axioms are too broad. By adding (O) and (E), the state space of the dynamics is refined to a physically plausible subset, providing a clear "energy + orientation" patch for future mesh-based simulators.

Limitations & Future Work

  • The main experiments utilized state-independent \(G_\theta\) (quadratic storage). Strongly nonlinear PDEs (e.g., Navier-Stokes, plasticity, phase transitions) require nonlinear constitutive models \(G_\theta(z)\) / \(\Psi_\theta(e)\), explored only in supplementary toy experiments.
  • Axioms (O) and (E) are sufficient conditions but do not guarantee correct behavior under external sources, complex boundary conditions, or multi-physics coupling; external source terms require further extension.
  • The framework relies on the incidence structure \(D_k\) of the cell complex, making it not directly applicable to completely unstructured or time-varying topologies (e.g., fracturing materials, adaptive meshing).
  • In some OOD shifts, MeshFT-Net is only "relatively" better than FNO/GraphCON; absolute TSMSE remains non-zero, suggesting topological constraints cannot fully replace sufficient data.
  • vs MGN: MGN satisfies only (L) + (P); this work enforces (O) + (E), structurally eliminating non-physical degrees of freedom and improving long-range stability by orders of magnitude.
  • vs HNN / port-Hamiltonian NN: Those methods learn on a "global Hamiltonian template," requiring the template to be correct. This work uses "local Jacobian factorization," making it more robust to template errors.
  • vs DEC / Data-driven Exterior Calculus: Both share the topology–metric separation idea; this work derives it from physical axioms rather than starting from differential geometry templates, making it more general.
  • vs PI-MGN / FNO: PDE residual/operator learning is data-driven + weak physics; this work is structure-driven and does not require knowledge of the PDE form, leading to more stable generalization.

Rating

  • Novelty: ⭐⭐⭐⭐ Strictly translates the topology–metric separation of exterior differential geometry into GNN architecture design principles.
  • Experimental Thoroughness: ⭐⭐⭐⭐ Analytic + real data + physical diagnostics + OOD + nonlinear ablations.
  • Writing Quality: ⭐⭐⭐⭐ Clear theorem statements; Algorithm 1 is reproducible almost without the appendix.
  • Value: ⭐⭐⭐⭐ Provides a theorem-driven design paradigm for the learned simulator community.