AI 中文总结
该研究针对偏微分方程学习中算子与几何等因素纠缠的问题，提出几何感知LegONet，通过在环境谱域预训练物理机制，利用采样边界约束定义流形，改变域时仅改变坐标接口，实现任意域偏微分方程学习，能保持边界残差并得出预测定律。
AI 中文摘要
学习的偏微分方程求解器通常将控制算子与用于训练的几何、边界条件和离散化纠缠在一起。这限制了在新域上应用相同物理时的重用性，也使物理定律发现依赖于几何。我们引入了几何感知LegONet（gLegONet），它是类似乐高积木的算子学习的边界流形扩展。物理机制在环境谱域上作为模块化变分块预先训练一次。对于目标几何，采样的边界约束定义了一个仿射允许流形。其质量正交切坐标用于演化动力学并直接评估候选定律发现特征。改变域只改变代数坐标接口，而不改变学习的算子块。这将任意域偏微分方程学习从特定于几何的重新训练或软惩罚实施转变为可重用机制的边界保证组装。在对未见域的正向模拟和稀疏识别测试中，该方法将边界残差保持在代数容差附近，并从短期观测中得出预测性控制定律。
英文摘要
Learned PDE solvers often entangle governing operators with the geometry, boundary conditions, and discretization used for training. This limits reuse when the same physics is posed on new domains, and it also makes physical-law discovery geometry-dependent. We introduce Geometry-aware LegONet (gLegONet), a boundary-manifold extension of Lego-like operator learning. Physical mechanisms are pretrained once as modular variational blocks on an ambient spectral domain. For a target geometry, sampled boundary constraints define an affine admissible manifold. Its mass-orthonormal tangent coordinates are used to evolve the dynamics and evaluate candidate law-discovery features directly. Changing the domain therefore changes only an algebraic coordinate interface, not the learned operator blocks. This converts arbitrary-domain PDE learning from geometry-specific retraining or soft penalty enforcement into boundary-guaranteed assembly of reusable mechanisms. In forward simulations and sparse identification tests on unseen domains, the method maintains boundary residuals near the algebraic tolerance and yields predictive governing laws from short-time observations.
