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Clifford-Steerable Convolutional Neural Networks
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Proposes Clifford-Steerable CNNs that process multivector fields on pseudo-Euclidean spaces with E(p,q)-equivariant convolutions, implementing O(p,q)-steerable kernels implicitly via Clifford group equivariant networks.
Problem
Equivariant CNNs for fields on pseudo-Euclidean spaces (e.g. Minkowski spacetime) need O(p,q)-steerable kernels, which are not solved analytically for non-compact mixed-signature groups.
Approach
Implement O(p,q)-steerable convolution kernels implicitly as a CGENN kernel network followed by an O(p,q)-equivariant kernel head built from weighted geometric products, so steerability holds by construction without deriving irreps or Clebsch-Gordan coefficients.
Key results
- CS-ResNets outperform all baselines on all three tasks, especially in R^3 and R^{1,2}.
- On Navier-Stokes, CS-ResNet trained on 64 simulations beats basic ResNet and FNOs trained on 80x more data.
- On relativistic Maxwell R^{1,2}, the basic ResNet does not approach CS-ResNet performance even with 16x more data.
- Empirical E(2) equivariance error of CS-CNNs is on par with analytical e2cnn steerable CNNs (up to numerical artefacts).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use CS-ResNet instead of basic ResNet or FNO | Navier-Stokes forecasting · Navier-Stokes R² | MSE: 64 training simulations; 80× more data | Helped |
| Use CS-ResNet instead of basic ResNet | Maxwell forecasting · Maxwell R¹˒² | MSE: 16× more data | Helped |
| Use CS-ResNet instead of the baselines | Physical dynamics forecasting · Navier-Stokes R²; Maxwell R³; Maxwell R¹˒² | MSE | Helped |
| Use an equivariant steerable model instead of non-equivariant models | Models evaluated on Navier-Stokes R² · Navier-Stokes R² | Relative O(2)-equivariance error | Helped |
| Use the weighted Cayley-table parameters in the kernel head (ablation) | CS-CNN · Not specified | Not specified | Helped |
| Include a scalar orbital-shell component in the kernel-network input (ablation) | CS-CNN · Not specified | Kernel scaling to grid range | Helped |
| Add a second consecutive convolution to recover missing kernel degrees of freedom (ablation) | CS-CNN with O(2)-steerable kernels · O(2) kernel expressiveness | Kernel degrees of freedom | Helped |
Surprising findings
- Single CS-CNN kernel spaces are incomplete: they miss some angular-frequency-2 kernels and coupled radial parts, though composing two convolutions recovers them.
Limitations
- Kernel spaces are not complete; some degrees of freedom are missing in a single convolution.
- Non-compact O(p,q) orbits are truncated by finite grid kernels, breaking equivariance for large transformations.
- Only multivector field types are covered; more general O(p,q) representations are not handled.
Experimental setup
Datasets: incompressible Navier-Stokes 2D simulations Maxwell 3D simulations relativistic Maxwell R^{1,2} simulations
Benchmarks: Navier-Stokes 2D forecasting Maxwell 3D forecasting relativistic Maxwell R^{1,2} forecasting
Metrics: mean squared error relative equivariance error
Baselines: Conventional ResNet Clifford ResNet Steerable ResNet Fourier Neural Operator G-FNO
Builds on: Clifford group equivariant neural networks steerable CNNs implicit steerable kernels
Closest papers by meaning
- Conditional Clifford-Steerable CNNs for PDE Modeling (ICML 2026 · similarity 0.94)
- General E(2)-Equivariant Steerable CNNs (NeurIPS 2019 · similarity 0.89)
- Implicit Convolutional Kernels for Steerable CNNs (NeurIPS 2023 · similarity 0.89)
- A Program to Build E(N)-Equivariant Steerable CNNs (ICLR 2022 · similarity 0.89)
- A Probabilistic Approach to Learning the Degree of Equivariance in Steerable CNNs (ICML 2024 · similarity 0.85)
- Building Transformation Layers for Riemannian Neural Networks (NeurIPS 2026 · similarity 0.85)
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