让3D物理方程模型在任意方向下都稳定，突破传统深度学习的几何局限。
EqGINO: Equivariant Geometry-Informed Fourier Neural Operators for 3D PDEs
- 在频域构建各向同性结构，实现3D等变性
- 仅用少量旋转样本即实现连续方向泛化
- 适用于复杂不规则3D几何上的物理模拟
针对3D偏微分方程（PDE）的深度学习代理模型在几何变换下泛化能力差的问题，现有方法或依赖局部操作导致全局感受野计算昂贵，或在傅里叶神经算子（FNO）中难以实现3D等变性。本文提出EqGINO，通过在频域设计各向同性结构，确保离散对称性下的精确等变性。该结构先验使模型在仅使用少量SE(3)变换训练样本时，仍能有效泛化至任意连续方向。实验表明，该方法可在复杂不规则3D几何上稳健建模坐标不变的物理规律。
原文摘要 · Abstract (English)
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
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