让卷积网络自动适应复杂物理对称性，提升方程预测精度。
Conditional Clifford-Steerable CNNs for PDE Modeling
- 用输入依赖的核结构增强模型对称性表达能力
- 在流体与相对论电动力学任务中超越标准方法
- 适合需要高对称性建模的物理模拟研究者
我们提出条件型克利福德可平移卷积神经网络（C-CSCNN），统一融合任意伪欧几里得群的等变性，显著提升标准CSCNN的表达能力。发现标准形式的核基不完整，限制了模型容量。为此，通过引入输入特征场的等变表示来扩充核，并推导出输入依赖核的等变约束，利用隐式参数化高效求解。在多个偏微分方程预报任务（包括流体动力学和相对论电动力学）上进行实证验证，结果表明该方法始终优于标准CSCNN，且性能达到当前最优基准水平。
原文摘要 · Abstract (English)
We introduce Conditional Clifford-Steerable CNNs (C-CSCNNs), a unified framework that incorporates equivariance to arbitrary pseudo-Euclidean groups and significantly improves the expressivity of standard CSCNNs. We show that the kernel basis of the standard formulation is incomplete, limiting model capacity. To address this, we augment the kernels with equivariant representations of the input feature field. We derive the equivariance constraint for these input-dependent kernels and show how it can be solved efficiently via implicit parameterization. We empirically validate on multiple PDE forecasting tasks, including fluid dynamics and relativistic electrodynamics, where our method consistently outperforms standard CSCNNs and performs on par with state-of-the-art baselines.
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