CVKAN: Complex-Valued Kolmogorov-Arnold Networks

TL;DR

CVKAN introduces a complex-valued Kolmogorov-Arnold Network by learning edge-wise complex radial-basis functions, complemented by a complex residual activation and multiple complex batch normalization schemes to maintain fixed-grid inputs. The approach preserves the intrinsic interpretability of Kolmogorov-Arnold networks while leveraging complex-valued representations, leading to improved stability and competitive accuracy on complex-valued symbolic tasks, physical formulae, and knot classification. Across synthetic and real-world-like datasets, CVKAN demonstrates parameter efficiency and shallower architectures, along with a visualization toolkit that aids explainability. The work points toward applications in complex-valued scientific problems and future extensions to hypercomplex algebras such as quaternions.

Abstract

In this work we propose CVKAN, a complex-valued Kolmogorov-Arnold Network (KAN), to join the intrinsic interpretability of KANs and the advantages of Complex-Valued Neural Networks (CVNNs). We show how to transfer a KAN and the necessary associated mechanisms into the complex domain. To confirm that CVKAN meets expectations we conduct experiments on symbolic complex-valued function fitting and physically meaningful formulae as well as on a more realistic dataset from knot theory. Our proposed CVKAN is more stable and performs on par or better than real-valued KANs while requiring less parameters and a shallower network architecture, making it more explainable.

Figures (5)

• Figure 1: Visualization of a weighted sum of three rbf with a grid in the interval $[-2, 2]$ and grid points at $g_0=-2, g_1=0, g_2=2$.
• Figure 2: Visualization of the complex RBFs with a grid in the interval $[(-2 -2i), (2+2i)]$ and $G=3$ grid points per dimension resulting in $3\cdot3=9$ grid points.
• Figure 3: Visualization of a ckan for symbolic function fitting. Height of the graphs show magnitude of the learned function, colormap encodes the phase. Both functions on the lower edges represent quadratic functions with a constant offset. They sum to $z_1^2 + z_2^2$ since their offsets cancel out, which can be seen on the roots, which are identical besides a 90° rotation. The upper function represents a typical shape of a complex square function near the middle with some error artifacts on the edges of the grid.
• Figure 4: Visualization of the first layer of a ckan on the knot dataset with the opacity of edges and nodes proportional to their relevance score. For better readability only the function corresponding to the most relevant blue input node for feature merid_translat_c is shown. In our visualization tool the individual functions and edges can be interactively selected by double-clicking on the corresponding nodes or by specifying the edge index $(l,i,j)$ to view only the function of edge $E_{l,j,i}$ in detail.
• Figure 5: Confusion matrix for ckan on the knot dataset normalized to probabilities within each row.