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

2026-09-24

AI Summary

This study addresses the challenges of boundary treatment and physical structure distortion in geometry-adaptive operator learning for Hamiltonian PDEs on complex domains. We reveal the multi-conformal-symplectic structure of the Brinkman penalization method and, for the first time, prove its exact local conservation laws. Building upon this, we propose a conformal symplectic neural operator framework that constructs structure-preserving numerical integrators via Strang splitting, unifying multisymplectic theory with scientific machine learning by integrating staggered dissipative flows with learnable evolution operators. Experimental results demonstrate that the proposed method accurately reproduces energy budgets in wave propagation and electromagnetic scattering tasks, effectively suppressing non-physical energy drift and enabling physically consistent long-term predictions.

Abstract

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.

Problem

• Hamiltonian PDEs
• Brinkman penalisation
• structure-preserving operator learning
• multi-conformal symplectic structure
• complex domains

Innovation

• Brinkman penalisation method
• multi-conformal symplectic structure
• structure-preserving numerical integrators
• conformal symplectic neural operators
• Hamiltonian PDEs