EqGINO: 面向3D PDE的等变几何信息傅里叶神经算子
EqGINO: Equivariant Geometry-Informed Fourier Neural Operators for 3D PDEs
AI总结：
提出EqGINO框架，通过在谱域强制执行各向同性，实现离散对称性的精确等变，并泛化到任意连续旋转，有效建模3D PDE的坐标不变物理规律。
摘要：
用于3D偏微分方程（PDE）的深度学习代理通常难以在几何变换下泛化，因为它们严重依赖于特定的坐标系。虽然等变网络提供了一种解决方案，但它们通常依赖于空间域中的局部操作，使得对PDE动力学至关重要的全局感受野计算成本高昂。相反，傅里叶神经算子（FNO）高效地捕获全局交互，但由于谱群卷积的过高成本，在其中建立3D等变性仍然不切实际。为弥合这一差距，我们引入了EqGINO，一个在谱域中强制执行各向同性的几何鲁棒框架。通过设计，EqGINO保证对离散化计算域固有的离散对称性具有精确等变性。除了这种离散保证外，我们的结构先验使得即使在有限数量的SE(3)变换训练样本下，也能有效泛化到任意连续方向。因此，我们的方法在复杂的非规则3D几何上鲁棒地建模坐标不变的物理定律。我们的代码可在此https URL获取。
英文摘要：
Deep learning surrogates for 3D Partial Differential Equations (PDEs) often fail to generalize across geometric transformations because they depend heavily on specific coordinate systems. While equivariant networks offer a solution, they typically rely on local operations in the spatial domain, making the global receptive field, which is essential for PDE dynamics, computationally expensive. Conversely, Fourier Neural Operators (FNOs) efficiently capture global interactions, yet establishing 3D equivariance within them remains impractical due to the prohibitive cost of spectral group convolutions. To bridge this gap, we introduce EqGINO, a geometrically robust framework that enforces isotropy in the spectral domain. By design, EqGINO guarantees exact equivariance to the discrete symmetries inherent to the discretized computational domain. Beyond this discrete guarantee, our structural prior enables effective generalization to arbitrary continuous orientations even with a limited number of SE(3)-transformed training samples. Consequently, our method robustly models coordinate-invariant physical laws on complex irregular 3D geometries. Our code is available at https://github.com/sung-won-kim/EqGINO
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
