阐明用于哈密顿偏微分方程的几何自适应、保结构算子学习的 Brinkman 惩罚方法的共形结构

中文摘要 AI 辅助

Brinkman 惩罚方法通过将固体区域建模为强耗散介质，将复杂域上的边值问题嵌入到一个简单的计算盒子中，从而避免了贴体网格生成。我们表明，在连接辛矩阵与惩罚投影的相容性条件下，经 Brinkman 型惩罚正则化的多辛哈密顿偏微分方程保持多共形辛结构。这产生了一个精确的局部守恒定律，在该定律下，多辛二形式在流体区域中守恒，并在固体内部呈指数衰减。具有 Brinkman 摩擦的线性波动方程和具有人工欧姆电导率的麦克斯韦方程满足该条件，并具有显式的修正哈密顿密度。在此基础上，我们提出了 (i) 通过 Strang 分裂实现的保结构数值积分器，其满足离散共形守恒定律，以及 (ii) 共形辛神经算子，其将精确耗散流与可学习的多辛演化算子交错，从而实现几何相关的算子学习。关于波和电磁散射的数值实验表明，我们的方法再现了正确的局部能量预算，并避免了非物理的能量漂移，为复杂域上物理一致的科学机器学习提供了一个有原则的框架。

英文摘要

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.

发表机构

• University of Bath（巴斯大学）
• RIKEN（理化学研究所）
• Kyushu University（九州大学）

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