Conformal Symplectic Structure of Brinkman Penalisation for Structure-Preserving Operator Learning of Hamiltonian PDEsNew

This paper investigates the theoretical foundations of combining Brinkman penalization with operator learning for Hamiltonian PDEs. The authors demonstrate that multi-symplectic Hamiltonian PDEs regularized by Brinkman-type penalization preserve a multi-conformal symplectic structure when a specific compatibility condition linking the symplectic matrix and penalization projection is satisfied. This mathematical result yields an exact local conservation law, enabling structure-preserving operator learning on complex domains without body-fitted mesh generation. The approach embeds boundary-value problems on intricate geometries into simple computational boxes by modeling solid regions as highly dissipative media.