Elucidating the Conformal Structure of the Brinkman Penalisation Method for Geometry-Adapted, Structure-Preserving Operator Learning of Hamiltonian PDEs
Read the original on arXiv Machine Learning →The paper investigates how the Brinkman penalisation method, which embeds complex domain boundary-value problems into a simple computational box, preserves a multi-conformal symplectic structure for multi-symplectic Hamiltonian PDEs under a specific compatibility condition. It demonstrates that this leads to an exact local conservation law where the multi-symplectic two-form is conserved in the fluid region and decays exponentially inside the solid. Building on these findings, the authors propose structure-preserving numerical integrators via Strang splitting and conformal symplectic neural operators that interleave exact dissipative flows with learnable multi-symplectic evolution operators, and validate their approach with numerical experiments on wave and electromagnetic scattering.
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