Clifford Kolmogorov-Arnold Networks

We introduce Clifford Kolmogorov-Arnold Network (ClKAN), a flexible and efficient architecture for function approximation in arbitrary Clifford algebra spaces. We propose the use of Randomized Quasi Monte Carlo grid generation as a solution to the exponential scaling associated with higher dimensional algebras. Our ClKAN also introduces new batch normalization strategies to deal with variable domain input. ClKAN finds application in scientific discovery and engineering, and is validated in synthetic and physics inspired tasks.

💡 Research Summary

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The paper introduces Clifford Kolmogorov‑Arnold Networks (ClKAN), a generalization of the Kolmogorov‑Arnold Network (KAN) to arbitrary Clifford algebras Cl(p,q,r). While the original KAN works on real‑valued functions and the Complex‑valued KAN (CVKAN) extends it to the complex domain, many scientific and engineering problems require higher‑dimensional hyper‑complex representations (e.g., electromagnetic fields, weather models, multi‑joint robot kinematics). ClKAN addresses this gap by defining the network’s building blocks directly in the Clifford algebra, using the geometric product for all linear operations.

Two families of radial basis functions (RBFs) are proposed. The “naïve” RBF ϕ₁(x)=exp(−‖x‖²) maps a Clifford element to a real scalar, potentially discarding multivector information. The “Clifford” RBF ϕ₂(x)=x·exp(−‖x‖²) retains the full multivector structure by scaling the input element with a scalar kernel value, thus preserving grades (scalar, vector, bivector, etc.) throughout the network. Both RBFs are combined with a Clifford‑adapted SiLU activation, where the SiLU function is applied component‑wise and learnable weight and bias are multivector‑valued.

A major obstacle is the exponential growth of grid points when constructing the KAN expansion in high‑dimensional Clifford spaces. To mitigate this, the authors replace the full uniform grid with a Randomized Quasi‑Monte‑Carlo (RQMC) Sobol grid. By sampling grid points from a scrambled Sobol sequence, the hypercube