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Elucidating the Conformal Structure of the Brinkman Penalisation Method for Geometry-Adapted, Structure-Preserving Operator Learning of Hamiltonian PDEs

Teo Deveney, Baige Xu, Takaharu Yaguchi

arXiv:2609.29847v1math.NAcs.LGcs.NA

TL;DR

Operator learning on complex domains faces persistent deployment obstacles despite its promise for accelerating physical simulation. The paper derives a multi-conformal structure for Brinkman-penalised Hamiltonian PDEs and builds structure-preserving integrators and neural operators around it. Scattering experiments report local energy-budget fidelity and improved operator-learning behavior relative to FNO baselines.

  • Problem

    Operator-learning methods face persistent obstacles to deployment, while SciML methods are pursued partly to accelerate physical simulation.

    Source: §1 · p. 2

  • Method

    The paper derives a compatibility-based multi-conformal structure for Brinkman-penalised PDEs and uses exact dissipative flows with multi-symplectic evolution in integrators and neural operators.

    Source: §1 · p. 3

  • Results

    In scattering experiments, the methods reproduce local energy budgets; the Maxwell conformal symplectic neural operator reduces FNO rollout error by an order of magnitude, while the wave operator stays close to ground-truth discrete energy where FNO energy grows.

    Source: §1 · p. 3

  • Takeaways & Limitations

    The identified structure supports integration and operator learning on arbitrary geometries without specialised meshing or retraining, with conservation properties compared favorably to standard approaches.

    Source: §7 · pp. 18–19

    A rigorous approximation theory for the operator setting is beyond scope, and nonlinear Hamiltonian PDEs in the penalised class remain to be explored.

    Source: §5.2 · p. 13, §7 · p. 19

Abstract

from arXiv · show

The Brinkman penalisation method embeds boundary-value problems on complex domains into a simple computational box by modeling the solid region as a strongly dissipative medium, avoiding body-fitted mesh generation. We show that multi-symplectic Hamiltonian PDEs regularised by Brinkman-type penalisation retain a multi-conformal symplectic structure under a compatibility condition linking the symplectic matrix and the penalisation projection. This yields an exact local conservation law, under which the multi-symplectic two-form is conserved in the fluid region and decays exponentially inside the solid. The linear wave equation with Brinkman friction and Maxwell's equations with artificial Ohmic conductivity satisfy this condition, with explicit modified Hamiltonian densities. Building on this, we propose (i) structure-preserving numerical integrators via Strang splitting that satisfy a discrete conformal conservation law, and (ii) conformal symplectic neural operators that interleave exact dissipative flows with learnable multi-symplectic evolution operators, allowing geometry-dependent operator learning. Numerical experiments on wave and electromagnetic scattering demonstrate that our methods reproduce correct local energy budgets and avoid unphysical energy drift, providing a principled framework for physics-consistent scientific machine learning on complex domains.

1 Introduction

The paper identifies a geometric structure in Brinkman-penalised Hamiltonian PDEs and uses it to build structure-preserving integrators and neural operators for complex-domain problems. Experiments on wave and electromagnetic scattering illustrate their energy and operator-learning behavior.

Motivation

  • The work targets geometry-dependent operators and physically inaccurate training data, amid SciML’s promise of accelerating physical simulation.Brinkman penalisation avoids body-fitted meshing, while structure-preserving integration addresses invariant errors in conventional numerical training data.Source: §1 · p. 2

Geometry of penalised Hamiltonian PDEs

  • Under the compatibility condition, Brinkman-penalised systems have multi-conformal symplectic structure, conserving symplectic density in fluid and decaying exponentially at rate 1/η in solid.The linear wave equation with Brinkman friction and Maxwell’s equations with artificial Ohmic conductivity satisfy the condition.Source: §1 · p. 3

Structure-preserving integrators

  • Strang splitting combines an exactly integrated pointwise dissipative flow with a multi-symplectic integrator for the conservative flow, yielding a discrete conformal conservation law and the correct local energy budget.The exact dissipative map contracts the symplectic density pointwise by e^−χτ/η under the compatibility condition.Source: §1 · p. 3

Conformal symplectic neural operators

  • The proposed conformal symplectic neural operator sandwiches a learnable multi-symplectic evolution operator between exact penalisation flows, with the geometry indicator supplied as input.The indicator input allows a single trained model to cover families of domains.Source: §1 · p. 3

Numerical experiments

  • For Maxwell scattering, the conformal symplectic neural operator reduces FNO rollout error by an order of magnitude; for wave scattering, it stays close to ground-truth discrete energy while FNO energy grows significantly.The wave integrator also demonstrates an exact energy budget on obstacle scattering, where a non-geometric explicit baseline exhibits catastrophic energy growth.Source: §1 · p. 3

Related work

  • The contribution extends prior conformal conservation work from spatially constant damping to partial, spatially inhomogeneous damping, and adapts SympNets to penalised complex-domain PDEs.The algebraic criterion links the symplectic matrix and penalisation projection, with an explicit modification of the Hamiltonian density.Source: §1 · p. 4

2 Preliminaries

The preliminaries introduce Brinkman penalisation on a computational box, multi-symplectic Hamiltonian PDEs, and local conformal conservation laws that accommodate spatially varying damping.

2.1 Brinkman penalisation

  • Brinkman penalisation models the solid as a strongly dissipative medium on the whole computational box; as η tends to zero, penalised fields vanish in the solid and boundary conditions are recovered.Geometry enters through the indicator χ, and the examples damp wave velocity ut or Maxwell’s electric field E while leaving u or H undamped.Source: §2.1 · p. 4

2.2 Multi-symplectic Hamiltonian PDEs

  • A multi-symplectic Hamiltonian PDE is a first-order system with constant skew-symmetric structure matrices and a smooth Hamiltonian density, whose solutions satisfy a multi-symplectic conservation law.A numerical scheme is multi-symplectic when its discrete variational equation satisfies the analogous law.Source: §2.2 · p. 5
  • The wave and Maxwell examples fit the multi-symplectic formulation, with symplectic densities formed from their dynamical fields and spatial fluxes.For the wave equation, the densities include ω = du ∧ dv and κ = c du ∧ dw; Maxwell’s equations are represented through curl operators.Source: §2.2 · p. 5

2.3 Conformal symplectic systems and their PDE analogue

  • Multi-conformal symplectic conservation is local: the symplectic density is transported conservatively where damping vanishes and decays exponentially at the prescribed local damping rate where it is positive.This locality matches Brinkman penalisation’s confinement of dissipation to the solid region.Source: §2.3 · p. 6

3 Multi-conformal symplectic structure of Brinkman-penalised Hamiltonian PDEs

The section identifies a compatibility condition under which Brinkman-penalised Hamiltonian PDEs retain a multi-conformal symplectic structure with a modified Hamiltonian density. The wave equation and conductivity-penalised Maxwell equations satisfy the condition, yielding a local conservation law that distinguishes fluid and solid regions.

3.1 The penalised class and a naive attempt

  • Naively replacing partial Brinkman damping with canonical conformal damping changes the wave equation by adding a stiff restoring force supported in the solid.The discrepancy arises because canonical damping acts on all multi-symplectic variables, whereas Brinkman friction acts only on selected components.Source: §3.1 · p. 7

3.2 Main structural theorem

  • Under the compatibility condition K = PK + KP, the penalised system becomes multi-conformal symplectic after adding an explicit quadratic correction to its Hamiltonian density.The condition means damped variables are K-conjugate to undamped variables; jointly damping a conjugate pair does not satisfy this partial-damping condition.Source: §3.2 · pp. 7–8

3.3 Example I: the wave equation with Brinkman friction

  • The Brinkman-penalised wave equation satisfies the theorem, with a solid-supported position–velocity correction and a local conformal conservation law.The correction may also be interpreted as potential energy stored by the penalisation medium.Source: §3.3 · p. 9

3.4 Example II: Maxwell’s equations with conductivity penalisation

  • Conductivity-penalised Maxwell equations satisfy the theorem, and their Hamiltonian correction is a solid-supported electric–magnetic coupling requiring no change of variables.The correction is the bilinear density χ/(2η) E · H.Source: §3.4 · pp. 9–10

3.5 Physical meaning of the conservation law

  • The symplectic density is conserved in the fluid and decays exponentially in the solid, while local energy is conserved in the fluid and dissipated only in the solid.In the solid, the symplectic density decays at rate 1/η; for Maxwell’s equations the energy loss is Ohmic dissipation.Source: §3.5 · p. 10

4 Structure-preserving integrators for the penalised class

The integrators split penalised evolution into exact dissipation and multi-symplectic transport, then compose the flows using Strang splitting. Under the stated conditions, the scheme has a discrete conformal conservation law, with geometry-local contraction in the solid and conservative behavior in the fluid.

4.1 Strang splitting into exact dissipation and multi-symplectic transport

  • Strang splitting combines an exactly integrated pointwise dissipative flow with a multi-symplectic method for the conservative PDE.The damped components are multiplied by e^(−χτ/η); for the wave equation velocity is damped with position and w frozen, while for Maxwell’s equations E is damped with H frozen.Source: §4.1 · pp. 10–11
  • The composed method is second-order accurate, and its step-size constraint comes only from the conservative substep, not the stiff penalty parameter.The stiff factor χ/η is confined to the exactly integrated dissipative substep.Source: §4.1 · p. 11

4.2 Discrete conformal multi-symplectic conservation law

  • When χ is constant on a grid point’s stencil, the scheme satisfies an exact discrete conformal conservation law, with density contracting by e^(−Δt/η) in the solid interior and conserved in the fluid, up to discrete flux.The result assumes the compatibility condition and uses fluxes evaluated at the intermediate stage.Source: §4.2 · p. 12
  • The wave equation’s flux can be expressed in original variables, whereas Maxwell’s flux mixes damped and undamped fields and uses the intermediate-stage formulation.Near solid boundaries where χ jumps across a stencil, the law uses local grid values; smoothed indicators are covered in the sense of backward error analysis.Source: §4.2 · p. 12

5 Conformal symplectic neural operators

The conformal symplectic neural operator combines exact geometry-dependent dissipation with learnable multi-symplectic evolution. Its conformal law holds for every admissible geometry, while a rigorous operator-setting approximation theory remains beyond this paper's scope.

5.1 From SympNets to structure-preserving operator learning

  • Learnable kick-and-drift shear layers are exact Hamiltonian flows, so their compositions preserve the multi-symplectic conservation law.The fields are represented in K-conjugate pairs, including (u, v) for waves and (H, E) for Maxwell’s equations.Source: §5.1 · p. 12

5.2 Architecture

  • The CSNO combines an exact, geometry-dependent penalisation semigroup with a learnable multi-symplectic evolution operator.Geometry may also condition the learnable potentials, but must not enter the symplectic structure.Source: §5.2 · p. 13
  • For every parameter and admissible geometry, the CSNO obeys an exact pointwise conformal law with a flux term, and its spatially integrated symplectic density contracts as prescribed by penalisation.The flux is from the interior multi-symplectic operator at the intermediate stage.Source: §5.2 · p. 13
  • Supplying the geometry indicator as input lets one trained CSNO apply to arbitrary domain shapes within grid resolution.When geometry conditions the learnable potentials, the symplectic structure must remain independent of that conditioning.Source: §5.2 · p. 13
  • A rigorous approximation theory for the operator setting is beyond the paper’s scope, although the authors expect the finite-dimensional density result to generalise.The authors identify this theory as ongoing work.Source: §5.2 · p. 13

6 Numerical experiments

The experiments test penalised wave and Maxwell scattering, structure-preserving integration, and geometry-conditioned operator learning. Across these cases, the proposed methods reproduce energy budgets and the learned conformal symplectic operators track reference energy on held-out geometries.

6.1 Validation of the penalisation: 1D reflection

  • 2.02, 2.07, and 2.32 error ratios under successive ∆t halvings confirm second-order splitting, while decreasing η improves the rigid-wall approximation.Penetration into the solid decreases as η decreases.Source: §6.1 · p. 14

6.2 Structure-preserving integration: 2D scattering by a circular obstacle

  • 8.7 × 10−3 energy-budget residual: the split scheme tracks the exact 2D scattering budget, while explicit Euler grows about 100 orders by T = 6 and splitting stays stable uniformly in η.The residual is reported at the level of the trapezoidal time-quadrature.Source: §6.2 · pp. 14–15

6.3 Operator learning: wave propagation past an airfoil

  • On held-out airfoil geometries, CSNO tracks ground-truth energy across the prediction window, whereas FNO energy increases and its predictions develop unnatural oscillations by t = 4.Both models capture gross scattered-field features, but their spectral and energy behavior diverges over time.Source: §6.3 · p. 15, Figure 4 · p. 17

6.4 Maxwell’s equations: conducting obstacles in 2D and 3D

  • The TM Maxwell scattering pattern coincides with the acoustic-wave case, and the fluid-region relative L2 difference between the two computed solutions decreases essentially linearly with ∆t.The reported differences are 0.154, 0.077, and 0.051 on 128², 256², and 384² grids, respectively.Source: §6.4 · p. 16, Figure 5 · p. 18
  • 1.0 × 10−2 energy-budget residual: the conformal split-Yee scheme tracks the 3D Maxwell Poynting budget, while explicit Euler exhibits catastrophic energy growth.The split-Yee scheme is also reported to remain stable uniformly in η.Source: §6.4 · p. 17
  • 0.073 ± 0.017 vs 0.988 ± 0.015 final-frame electric-field relative L2 error, and 0.5% vs 56% final-energy deviation, favor CSNO over FNO on 3D Maxwell rollouts.These test-set results are reported for rollouts on held-out geometries.Source: §6.4 · p. 18

7 Conclusion and future work

The conclusion identifies a multi-conformal symplectic structure for compatible Brinkman-penalised systems and presents structure-preserving integration and operator learning for arbitrary geometries. Experiments show energy-budget tracking, while the authors identify nonlinear theory and broader computational settings as open directions.

Conclusion

  • Compatible Brinkman-penalised systems have multi-conformal symplectic structure with damping rate χ(x)/η when penalised components pair with unpenalised ones through the symplectic matrix.The paper identifies wave equations with classical Brinkman friction and Maxwell equations with artificial Ohmic conductivity as satisfying the compatibility condition.Source: §7 · pp. 18–19
  • The conformal split-Yee scheme tracks the exact Poynting budget in the 3D conducting-sphere problem, while explicit Euler exhibits catastrophic energy growth.The comparison uses the same lattice and time step.Source: Figure 7 · p. 19
  • Strang-split integrators and conformal symplectic neural operators reproduce the conformal conservation law and local energy budget on arbitrary geometries without specialised meshing or retraining.The neural operators receive geometry through an indicator function and satisfy the conformal conservation law by construction.Source: §7 · p. 19

Future work

  • Nonlinear Hamiltonian PDEs in the penalised class and convergence rates as η → 0 that respect the conformal structure remain open theoretical directions.The authors also identify variational and geometric structures underlying multi-conformal symplectic PDEs for further study.Source: §7 · p. 19

Conclusion

  • For conductivity-penalised 3D Maxwell equations with randomly sampled conducting obstacles, on a held-out geometry, CSNO tracks ground-truth electromagnetic energy within 0.5% at final time, while FNO energy decays spuriously below half the true value.Source: Figure 8 · p. 20

Future work

  • Future computational and learning directions include complex-shape electromagnetic scattering, coupled electromagnetic-acoustic architectures, approximation theory, moving geometries, and dissipative fluid models.The proposed extensions are presented as natural next targets or directions for further study.Source: §7 · p. 20

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