Kolmogorov-Arnold Networks (KANs): Towards Interpretable and Efficient Function Approximation Beyond MLPs

Kolmogorov-Arnold Networks (KANs) represent a paradigm shift in neural network architectures, drawing inspiration from the Kolmogorov-Arnold representation theorem. Unlike traditional Multi-Layer Perceptrons (MLPs) that employ fixed activation functions at nodes, KANs utilize learnable activation functions parameterized as splines on network edges. This fundamental architectural change eliminates the need for linear weight matrices, replacing every weight parameter with univariate functions. Through rigorous mathematical analysis and algorithmic development, we demonstrate that KANs achieve superior performance in terms of accuracy, parameter efficiency, and interpretability. Our theoretical contributions include convergence proofs, approximation bounds, and scaling laws that establish KANs as viable alternatives to MLPs. Empirical evaluations across function fitting, partial differential equation solving, and scientific discovery tasks validate the practical advantages of KANs, achieving 100× parameter efficiency improvements while maintaining comparable or superior accuracy. This work provides comprehensive algorithmic frameworks, mathematical proofs, and implementation strategies for KAN architectures, establishing their potential to revolutionize deep learning paradigms.