Brinkman method preserves energy structures in complex domain PDE simulations

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

Machine Learning

Summary

Simulating physical systems like waves or electromagnetic fields often involves complicated shapes that are hard to model on computers. The authors study a method called Brinkman penalisation that simplifies this by treating solid parts as very resistant fluids inside a box-shaped computational area. They discovered that this method keeps important mathematical energy structures mostly intact, which helps simulations stay physically realistic over time. They also designed new numerical tools and machine learning methods that respect these energy properties, improving the accuracy and stability of simulations on complex shapes.

What this means in practice

  • For computational physicists: Run stable and physically consistent wave and electromagnetic simulations on complex geometries without generating body-fitted meshes by using structure-preserving Brinkman penalisation methods.
  • For scientific machine learning engineers: Develop neural operator models that maintain correct energy behaviors when learning PDE solutions on domains with complicated shapes, enhancing prediction reliability.

Authors

Teo Deveney, Baige Xu, Takaharu Yaguchi

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.
View PDFOpen arXiv