<p>The Kolmogorov–Arnold representation theorem demonstrates that multivariate continuous functions can be articulated as superpositions of univariate functions and their summation, indicating neural networks with inherently decomposable structures. Recent Kolmogorov–Arnold Networks (KANs) exhibit encouraging empirical precision and interpretability; nonetheless, the domain lacks a cohesive framework delineating the conditions under which these models are data efficient, the influence of architectural decisions on approximation, and the quantification of interpretability. This study presents Kolmogorov–Arnold Neural Networks (KANNs) as a representation-theoretic basis for interpretable deep learning. We characterize KANNs as compositional models constructed from learnable univariate operators organized in Kolmogorov–Arnold–type superpositions, deriving approximation bounds for Hölder-smooth targets that correlate error rates with width, depth, and basis smoothness. We present an intrinsic interpretability index founded on variance decomposition into principal effects and interaction remainders, resulting in a standard set of one-dimensional “laws” along with residual coupling. We then develop a functional spline-parameterized KANN featuring identifiability regularization and a limited aim that promotes disentangled, human-auditable components. Across the studied regression, PDE surrogate, and scientific law discovery tasks, KANNs frequently achieve better parameter efficiency than width-matched baselines while yielding more structured functional decompositions. These results support KANNs as a promising and mathematically grounded option for scientific and engineering machine learning when the target mapping exhibits sufficient latent decomposability.</p> Graphical abstract <p></p>