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This paper introduces geometry-constrained Kolmogorov-Arnold Networks (KANs) that leverage learnable edge functions derived from Banach duality, allowing the geometry of the model to be learned from data rather than being predetermined. The results show that these networks outperform fixed-basis models across various symbolic-regression tasks, particularly excelling in the presence of measurement noise. Notably, the learned scalar exponent per edge provides interpretability and reveals consistent geometric relationships across different targets and dimensions, enhancing the model's adaptability and performance.
Geometry-constrained KANs not only outperform traditional fixed-basis models in symbolic regression but also adaptively learn their geometric properties, revealing deeper insights into the data structure.
Kolmogorov-Arnold Networks (KANs) replace fixed activations in deep architectures with learnable univariate edge functions, making the choice of edge parametrisation central. Existing variants rely on fixed bases such as splines, polynomials, or Fourier features, which impose a function-space geometry before data are observed. We introduce geometry-constrained KANs, a family of edge activations derived from Banach duality maps in which the geometry itself is learned through a scalar exponent $p>1$ per edge. This exponent controls the qualitative response: sub-Euclidean values produce sharp, threshold-like behaviour reminiscent of the $\ell_1$ (LASSO) geometry, $p = 2$ recovers the linear regime, and larger values produce flatter responses near the origin. Across 50 symbolic-regression targets ($40$ from the AI Feynman benchmark plus $10$ synthetic stress tests), geometry-constrained KANs match or beat every fixed-basis baseline on median NRMSE (Banach-KAN $0.030$, tying Chebyshev and improving on splines); on average rank Banach-KAN is best on the $18$-equation core ($2.00$) and statistically tied with the strongest spline on the full benchmark ($2.32$ vs. $2.34$). The clearest gains appear under measurement noise: as $\sigma$ grows from $0$ to $1$, $\ell^p$-KAN degrades only $3.7\times$ -- below even a cross-validated spline ($\approx 11\times$) -- while an unregularised spline degrades $21.6\times$; Banach-KAN degrades $8.8\times$, comparable to a tuned spline but far more stable than the unregularised one. Banach-KAN also takes the most per-equation wins in the small-sample regime, with fixed-basis models catching up only as the training set grows. Learned exponents provide an interpretable, relative signal: at a fixed initialisation they reveal a consistent, target-dependent geometric ordering across equation families and input dimensions.
