RuleKAN: Addressing the Symbolic Expressivity Gap in KANs for Symbolic Regression

Symbolic regression tries to turn data into analytic equations that carry useful meaning. Kolmogorov–Arnold Networks (KANs) are attractive for this purpose because their learned one-dimensional edge functions can be replaced by analytic formulas. The same flexibility creates a problem: each KAN edge can absorb structure that a scientist may want the equation to expose. For example, a single learned curve can approximate $e^x\sin x$, and a two-branch fuzzy-rule can be reproduced without displaying its gate and branches. We study which symbolic capacities are needed to recover those structures and more, introducing two related method families: RuleKAN trains a KAN to propose interacting variables and then searches explicit sums of products of analytic factors; RuleSISP searches the same language without the learned-support restriction. Bounded extensions of these methods add whole-expression powers, ratios, and one additional composition level. Across $30$ analytic tasks spanning products, fuzzy rules, powers and ratios, nested functions, canonical KAN targets, and physics equations, the RuleKAN family performs best overall (NRMSE $1.07\times 10^{-3}$; mean rank $4.97$) against baseline methods like PySR, SR-KAN, Operon, etc. It has significantly lower predictive error than Operon, SR-KAN, and the other six KAN-based baselines (FDR-adjusted $q<0.001$), and wins on about 66% of tasks against PySR. Notably, on fuzzy rules, RuleSISP recovers the generating structure significantly better than PySR despite its competitive predictive error (mean expanded-rule F1 $0.446$ vs. $0.000$; $q=0.0084$). The results show that symbolic recovery is not one search problem, separating distinct bottlenecks in factorization, interaction coverage, whole-expression operations, symbolic depth, and recovery of the intended structure.