Clifford-Steerable CNNs

• Clifford-Steerable CNNs are deep learning architectures that achieve exact equivariance under pseudo-Euclidean isometries using multivector-valued feature fields.
• They employ implicit, group-steerable convolutional kernels via Clifford group-equivariant networks to model complex physical phenomena in tasks like fluid dynamics and electrodynamics.
• The method integrates FFT-based optimizations and conditional kernel techniques to restore full harmonic bases, offering enhanced accuracy and stability despite higher computational costs.

Clifford-Steerable Convolutional Neural Networks (CS-CNNs) are a class of deep learning architectures designed to achieve exact equivariance under pseudo-Euclidean isometry groups, unifying the treatment of translational, orthogonal, and Lorentzian symmetries within a single convolutional framework. These networks process feature fields whose values are multivectors in Clifford algebras, allowing natural and explicit representations for a wide range of physical quantities. CS-CNNs implement implicit, group-steerable convolutional kernels via Clifford group-equivariant neural networks, and have demonstrated state-of-the-art performance on partial differential equation (PDE) forecasting tasks including fluid dynamics and relativistic electrodynamics (Zhdanov et al., 2024, Szarvas et al., 15 Oct 2025).

1. Pseudo-Euclidean Symmetry and Clifford Algebra

A central aspect of CS-CNNs is their principled equivariance to the isometry group $E(p,q)=\mathbb{R}^{p+q}\ltimes O(p,q)$, where $O(p,q)$ acts as the pseudo-orthogonal (Lorentz) group on $\mathbb{R}^{p,q}$, a vector space with metric signature $(p,q)$; for example, $(3,0)$ for classical Euclidean space and $(1,3)$ for Minkowski spacetime.

The Clifford algebra $\mathrm{Cl}(p,q)$, generated by orthogonalizing basis vectors $e_i$ under the bilinear form $\eta^{p,q}$, provides the representation space for the multivector-valued feature fields directly encoding scalar, vector, and higher-grade quantities. The $O(p,q)$ action is lifted to the entire algebra via the natural grading-preserving algebra automorphism $O(p,q)$0, enabling precise equivariant transformations for all geometric components present in the data (Zhdanov et al., 2024, Szarvas et al., 15 Oct 2025).

2. Equivariant Convolutions and Steerable Kernel Parameterization

A convolutional layer $O(p,q)$1 is $O(p,q)$2-equivariant if and only if its kernel $O(p,q)$3 satisfies the steerability relation:

$O(p,q)$4

CS-CNNs realize such steerable kernels via an implicit parameterization. Specifically, a Clifford group-equivariant neural network (CGENN) $O(p,q)$5 produces c-channel multivector outputs. These are passed through a fixed, linear "kernel head" $O(p,q)$6 that expands multivector entries into $O(p,q)$7-linear endomorphisms of $O(p,q)$8 by weighted, partial geometric product expansions, with learnable mixing weights $O(p,q)$9:

$\mathbb{R}^{p,q}$0

Together, equivariance of $\mathbb{R}^{p,q}$1 and $\mathbb{R}^{p,q}$2 ensures that their composition $\mathbb{R}^{p,q}$3 satisfies the group steerability constraint (Zhdanov et al., 2024).

3. Network Architecture, Implementation, and Optimization

The principal components of CS-CNNs include:

• Input/Output: Channel-wise multivector fields $\mathbb{R}^{p,q}$4 decomposed into grades.
• Steerable convolution: $\mathbb{R}^{p,q}$5.
• Nonlinearity: Grade-wise geometric-product gating, e.g., $\mathbb{R}^{p,q}$6 with a scalar nonlinearity $\mathbb{R}^{p,q}$7 on the grade-0 part.
• Residual blocks: Combination of steerable convolution, group normalization, activation, and skip connections.

Optimization techniques include pre-computation of geometric-product Cayley tables for efficient kernel expansion, the use of FFT-based (Fast Fourier Transform) convolution and backpropagation, and the implementation of CGENNs with grade-structured linear and geometric-product layers (Zhdanov et al., 2024).

4. Kernel Basis Completeness and Conditional Kernels

The standard CS-CNN parameterization does not realize a complete basis of equivariant steerable kernels. For example, in the O(2)-equivariant vector-to-vector setting, harmonic components at frequency two (cos$\mathbb{R}^{p,q}$8, sin$\mathbb{R}^{p,q}$9) are missing from the representable space when kernels depend solely on the spatial offset $(p,q)$0, limiting expressivity.

Conditional Clifford-Steerable CNNs (C-CSCNNs) solve this by incorporating global, translation-invariant summaries $(p,q)$1 of the input field as additional kernel arguments:

$(p,q)$2

Here, $(p,q)$3 is typically the mean pooled multivector over spatial regions, ensuring both translation and $(p,q)$4-equivariance. The conditional kernel $(p,q)$5 is parameterized via an O($(p,q)$6)-equivariant network $(p,q)$7 and the kernel head $(p,q)$8 as before, restoring the missing harmonics and the full kernel basis in a single layer (Szarvas et al., 15 Oct 2025).

5. Empirical Performance in PDE Modeling

Empirical evaluations demonstrate that CS-CNNs and their conditional variants achieve robust, data-efficient results on diverse PDE forecasting tasks including Navier–Stokes (R$(p,q)$9), shallow-water equations, non-relativistic and relativistic Maxwell equations in both Euclidean and Minkowski space:

Task	Metric	Best C-CSCNN Result	Comparison
Navier–Stokes	1-step MSE	C-CSCNN halves MSE of plain CS-CNN with 512 trajectories	Outperforms ResNet, Transolver, Swin-Transformer, FNO
Shallow Water SWE	5-step relative L$(3,0)$0	C-CSCNN-L: 2.94% (55M params); C-CSCNN-S: 3.51% (10M)	Better or competitive with much larger baselines
Maxwell (3D)	1-step field MSE	C-CSCNN halves error vs. CS-CNN and outperforms FNO/G-FNO	-
Maxwell (R$(3,0)$1)	1-stepfield MSE	Similar data-efficiency gains across data regimes	-

In all tasks, both CSCNNs and C-CSCNNs maintain relative equivariance error $(3,0)$2 and error maps demonstrate lower transient and stable long-horizon errors compared to baselines. Notably, C-CSCNNs achieve improved data efficiency, enabling smaller models to match or surpass much larger transformer- and FNO-based architectures (Szarvas et al., 15 Oct 2025, Zhdanov et al., 2024).

6. Limitations, Computational Cost, and Extensions

CS-CNNs currently support only full multivector (Cl($(3,0)$3)) representations and do not directly realize all $(3,0)$4 irreducible representations (e.g., spinors). Some kernel degrees of freedom require stacking multiple layers due to the implicit parameterization, and practical implementation on noncompact $(3,0)$5 groups necessitates grid truncation, leading to approximate equivariance under large boosts.

Feature dimensionality scales as $(3,0)$6 per channel (with $(3,0)$7), and geometric-product layers are computationally more expensive than conventional ReLU nonlinearities, though tractable up to $(3,0)$8. Prospective extensions include applying alternative Clifford-based nonlinearities, conditioning at local or hierarchical levels, adaptations to pseudo-Riemannian manifolds, and the construction of physics-informed architectures, e.g., using steerable PDE operators. Conditioning on global or learned multivector summaries further improves the kernel basis, enabling stable and expressive modeling of a broader class of physical phenomena (Szarvas et al., 15 Oct 2025, Zhdanov et al., 2024).

7. Impact and Future Research Directions

Clifford-Steerable CNNs provide a unified, E($(3,0)$9)-equivariant deep learning framework grounded in Clifford algebra and group theory, enabling exact symmetry-preserving modeling of fields with diverse geometric content across varied pseudo-Euclidean spaces. Conditional kernel mechanisms resolve basis incompleteness while maintaining computational tractability, expanding applicability to challenging PDEs and physical forecasting problems with notable gains in accuracy, efficiency, and long-term stability.

Possible future research includes: local/hierarchical or adaptive conditioning, expansion to curved manifold scenarios via parallel-transport convolution, and application domains such as molecular dynamics, geophysical flows, and high-energy physics on space-times of different signatures. This suggests continued investigation of Clifford-algebraic neural architectures and generalized symmetry-informed networks for scientific machine learning (Szarvas et al., 15 Oct 2025, Zhdanov et al., 2024).