Conditional Clifford-Steerable CNNs with Complete Kernel Basis for PDE Modeling
This work addresses a specific bottleneck in equivariant neural networks for PDE modeling, offering an incremental improvement over existing methods.
The authors tackled the limited expressivity of Clifford-Steerable CNNs due to an incomplete kernel basis by proposing Conditional Clifford-Steerable Kernels that augment kernels with equivariant representations from input features, resulting in improved performance on PDE forecasting tasks like fluid dynamics and relativistic electrodynamics.
Clifford-Steerable CNNs (CSCNNs) provide a unified framework that allows incorporating equivariance to arbitrary pseudo-Euclidean groups, including isometries of Euclidean space and Minkowski spacetime. In this work, we demonstrate that the kernel basis of CSCNNs is not complete, thus limiting the model expressivity. To address this issue, we propose Conditional Clifford-Steerable Kernels, which augment the kernels with equivariant representations computed from 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 demonstrate an improved expressivity of the resulting framework on multiple PDE forecasting tasks, including fluid dynamics and relativistic electrodynamics, where our method consistently outperforms baseline methods.