CS-CNNs: Clifford-Steerable Convolutional Nets

• CS-CNNs are neural architectures that leverage Clifford algebras and pseudo-Euclidean symmetry to enforce E(p,q)-equivariance.
• They employ an implicit kernel parameterization via equivariant MLPs, enhancing data efficiency and artifact-free modeling in complex physical systems.
• Conditional extensions (C-CSCNNs) address kernel-basis incompleteness, improving expressivity and performance in applications like fluid dynamics and electrodynamics.

Clifford-Steerable Convolutional Neural Networks (CS-CNNs) are a family of neural architectures for learning on multivector fields defined over pseudo-Euclidean spaces $\mathbb{R}^{p,q}$, engineered to be equivariant with respect to the isometries of these spaces, specifically the pseudo-Euclidean group $E(p,q) = \mathbb{R}^{p+q} \rtimes O(p,q)$. These networks feature an implicit parametrization of steerable kernels via Clifford algebra-valued equivariant neural networks, enabling $E(p, q)$-equivariance for arbitrary $p, q$, including physically relevant cases such as $E(3)$ on $\mathbb{R}^3$ (Euclidean space) and the Poincaré group $E(1,3)$ on Minkowski spacetime. CS-CNNs impose exact physical symmetry constraints, yielding significant improvements in data efficiency and invariance for PDE surrogate modeling in physics, such as fluid dynamics and electrodynamics (Zhdanov et al., 2024, Szarvas et al., 15 Oct 2025).

1. Mathematical Foundation: Clifford Algebras and Pseudo-Euclidean Symmetry

Clifford algebras $\mathrm{Cl}(p,q)$ underpin the signal representations in CS-CNNs. For a real vector space $V = \mathbb{R}^{p+q}$ with bilinear form $\eta$ of signature $E(p,q) = \mathbb{R}^{p+q} \rtimes O(p,q)$0, $E(p,q) = \mathbb{R}^{p+q} \rtimes O(p,q)$1 is constructed as the associative algebra generated by an orthonormal basis $E(p,q) = \mathbb{R}^{p+q} \rtimes O(p,q)$2, with relations

$E(p,q) = \mathbb{R}^{p+q} \rtimes O(p,q)$3

where $E(p,q) = \mathbb{R}^{p+q} \rtimes O(p,q)$4 encodes the signature. A general multivector $E(p,q) = \mathbb{R}^{p+q} \rtimes O(p,q)$5 decomposes as

$E(p,q) = \mathbb{R}^{p+q} \rtimes O(p,q)$6

where each grade (scalar, vector, bivector, etc.) transforms under the spin representation: $E(p,q) = \mathbb{R}^{p+q} \rtimes O(p,q)$7 acts via $E(p,q) = \mathbb{R}^{p+q} \rtimes O(p,q)$8 for $E(p,q) = \mathbb{R}^{p+q} \rtimes O(p,q)$9, extended gradewise.

Feature fields in CS-CNNs are maps $E(p, q)$0 equipped with the natural representation $E(p, q)$1 of $E(p, q)$2. The $E(p, q)$3 group acts by pulling back space and transforming Clifford algebra coefficients:

$E(p, q)$4

where $E(p, q)$5 is the standard Clifford action. This formalism naturally models all geometric field components that arise in physics, such as electromagnetic vector and bivector fields (Zhdanov et al., 2024, Szarvas et al., 15 Oct 2025).

2. Steerable Convolution: Equivariance Constraints and Kernel Parameterization

A convolutional layer in this framework has the form

$E(p, q)$6

where the kernel $E(p, q)$7. Full $E(p, q)$8-equivariance requires (cf. Weiler & Cohen)

$E(p, q)$9

This steerability constraint ensures outputs commute with $p, q$0 symmetries. For compact groups like $p, q$1 explicit kernel bases exist, but for non-compact $p, q$2 (e.g., Lorentz group), constructing an explicit steerable basis is intractable.

CS-CNNs circumvent this by parameterizing $p, q$3 implicitly:

$p, q$4

where $p, q$5 is an $p, q$6-equivariant MLP (Clifford group equivariant neural network, CGENN), and $p, q$7 is a fixed $p, q$8-equivariant "kernel head" that lifts multivector blocks into $p, q$9. Equivariance follows by construction, as both $E(3)$0 and $E(3)$1 are designed to commute with $E(3)$2 actions (Zhdanov et al., 2024).

3. Kernel Basis Completeness and Conditional Extensions

An important limitation, established in (Szarvas et al., 15 Oct 2025), is that the original implicit parameterization does not generate a complete basis for $E(3)$3-equivariant kernels in general. For example, in the $E(3)$4 case, the original kernel space lacks the Fourier angular modes corresponding to $E(3)$5 and $E(3)$6; only angular frequency 0 appears in the scalar channel. This restricts expressivity, as certain irreducible representations are omitted.

To resolve this, Conditional Clifford-Steerable Convolutions ("C-CSCNNs", Editor's term) are introduced, allowing the kernel to depend not only on the relative coordinate $E(3)$7, but also on global or local statistics $E(3)$8 (typically mean-pooled inputs) of the feature field:

$E(3)$9

with $\mathbb{R}^3$0 being $\mathbb{R}^3$1-equivariant in both arguments. The generalized steerability constraint is

$\mathbb{R}^3$2

This framework provably recovers the missing harmonics; for $\mathbb{R}^3$3, $\mathbb{R}^3$4-dependent combinations restore all angular frequencies, spanning the equivariant kernel space when $\mathbb{R}^3$5 varies over $\mathbb{R}^3$6 (Szarvas et al., 15 Oct 2025). A plausible implication is that, in high-dimensional settings or complex geometric tasks, the conditional formulation is essential for full expressivity.

4. Architectural Organization and Computational Considerations

A typical CS-CNN stack comprises several key components:

• Embedding Layer: Inputs (e.g., scalars, field components) are linearly mapped into $\mathbb{R}^3$7 channels using Clifford embedding plus a small MLP.
• Residual Blocks: Each block contains a CS-Convolution (computing $\mathbb{R}^3$8 using the kernel network and head with a local support grid), channel-wise GroupNorm, and a nonlinear activation (e.g., grade-wise gating $\mathbb{R}^3$9 with GELU CDF).
• Final Readout: Reduces Clifford channels to scalar or geometric output types.

Parameter and FLOP scaling are governed by the Clifford algebra dimension $E(1,3)$0, channel counts $E(1,3)$1, and kernel network depth. For a kernel network of $E(1,3)$2 layers:

• Kernel network parameters: $E(1,3)$3
• Kernel head parameters: $E(1,3)$4
• Total FLOPs per layer: $E(1,3)$5, with $E(1,3)$6 support points

Conditioning incurs negligible additional cost: only one global pool per block. Implementations use JAX/Flax/XLA and ensure equivariance up to discretization artifacts through grid masking (Zhdanov et al., 2024, Szarvas et al., 15 Oct 2025).

5. Empirical Evaluation in Physical PDE Modeling

CS-CNNs and their conditional extensions have been validated on benchmarks central to physics-informed learning:

• Navier–Stokes (2D incompressible): Predicting velocity and pressure.
• Shallow Water Equation (2D): Forecasting combined height/velocity fields.
• Maxwell Equations: Both in $E(1,3)$7-D Euclidean and $E(1,3)$8-D relativistic settings (spacetime-multivector fields).

Training: Input consists of several past field states; prediction is for either one-step-ahead or $E(1,3)$9-step rollout. Loss is mean squared error; metrics include MSE and relative $\mathrm{Cl}(p,q)$0 error. Baselines include standard ResNet, Clifford ResNet, $\mathrm{Cl}(p,q)$1-steerable CNNs, Fourier Neural Operator (FNO), group-equivariant FNO, and transformer-type models (Zhdanov et al., 2024, Szarvas et al., 15 Oct 2025).

Empirical results:

• CS-CNNs achieve 2–10$\mathrm{Cl}(p,q)$2 lower MSE than non-equivariant baselines, especially in low-data regimes. For Navier-Stokes, CS-CNNs match performance with 64 trajectories, versus 5120 for a ResNet.
• C-CSCNNs demonstrate further gains in accuracy, with relative $\mathrm{Cl}(p,q)$3 error outperforming all baselines, including U-Net, FNO, Swin Transformer, and large-scale vision transformers. For the 5-step shallow water rollout, C-CSCNN with 10M parameters achieves 3.51% relative $\mathrm{Cl}(p,q)$4 error, below FNO's 3.97%.
• Equivariance error is numerically negligible ($\mathrm{Cl}(p,q)$5), confirming symmetry enforcement.
• Only CS-CNNs and C-CSCNNs maintain stability and invariance under Lorentz boosts for Maxwell equations in Minkowski space, while other models break physical symmetry.

Model	#Params	Rel $\mathrm{Cl}(p,q)$6 (SWE-5)
DilResNet	4.2M	13.20%
U-Net$\mathrm{Cl}(p,q)$7	148M	5.68%
FNO	268M	3.97%
CViT-L	92M	1.56%
C-CSCNN (10M)	10M	3.51%
C-CSCNN (55M)	55M	2.94%

6. Mechanisms Underlying Gains in Efficiency and Physical Consistency

The performance advantages of CS-CNNs derive from several interlocking properties:

• Exact $\mathrm{Cl}(p,q)$8-equivariance: The function space is restricted to symmetry-compatible solutions, decreasing sample complexity and increasing generalization.
• Unified geometric representation: Multivector features allow encoding scalars, vectors, and higher-grade objects without cumbersome manual design.
• Implicit kernel parametrization: Avoids explicit (and intractable) group theoretic kernel construction for non-compact groups, enabling practical network building for Lorentzian and other physically important scenarios.
• Continuous and differentiable kernel networks: Prevents aliasing and allows smooth steerable filters.
• Residual stacking: Successive application of CS-Convs recovers missing higher angular frequency modes, marginally mitigating expressivity loss of a single layer in the original, non-conditional scheme.

Although the original CS-CNN architecture omits parts of the full steerable kernel basis, as shown in (Szarvas et al., 15 Oct 2025), C-CSCNNs fully remedy this kernel incompleteness by including feature-dependent conditional terms in the kernel. This design achieves both algebraic completeness in the equivariant space and empirical data efficiency.

7. Limitations, Generalizations, and Outlook

The initial CS-CNN framework, while effective, exhibits kernel-basis incompleteness when only the position variable is used. The conditional formulation, by input-adaptive kernels, resolves this at minimal computational cost and ensures exact $\mathrm{Cl}(p,q)$9-equivariance to machine precision. Implementations are robust to discretization, but boundary effects (e.g. from grid masking) may break perfect equivariance.

CS-CNNs and C-CSCNNs present a unified, scalable methodology for symmetry-enforcing deep architectures in physics-informed learning, applicable to arbitrary pseudo-Euclidean geometries. The conditional extension is crucial for full expressivity in convolutions, as established by both formal analysis (harmonic decomposition, group representation) and empirical results on challenging PDE datasets (Zhdanov et al., 2024, Szarvas et al., 15 Oct 2025).