Softly Symbolifying Kolmogorov-Arnold Networks

Abstract

Kolmogorov--Arnold Networks (KANs) offer a promising path toward interpretable machine learning: their learnable activations can be studied individually while collectively fitting complex data accurately. In practice, however, trained activations often lack *symbolic fidelity*, learning pathological decompositions with no meaningful correspondence to interpretable forms. We propose Softly Symbolified Kolmogorov--Arnold Networks (S2KAN), which integrate symbolic primitives directly into training. Each activation draws from a dictionary of symbolic and dense terms, with learnable gates that sparsify the representation. Crucially, this sparsification is differentiable, enabling end-to-end optimization, and uses an optimization objective inspired by minimum description length. S2KAN can discover interpretable activation functions while retaining dense splines for flexible approximation. We introduce a quantitative measure of symbolic fidelity, assessing how closely learned activations can be approximated by short expressions from a specified symbolic language. Across real-world prediction tasks, dynamical-system forecasting, and synthetic symbolic benchmarks, S2KAN generally achieves higher symbolic fidelity and often far smaller models, while maintaining competitive predictive accuracy on real-world data and low prediction errors on most symbolic benchmark configurations. We also examine the models' learned responses on scientific data, including the contributions of their symbolic components. Together, these results show that incorporating symbolic structure during training can yield compact, accurate KAN models whose learned responses can be examined through their symbolic components.