Optimal Abstractions for Verifying Properties of Kolmogorov-Arnold Networks (KANs)

We present a novel approach for verifying properties of Kolmogorov-Arnold Networks (KANs), a class of neural networks characterized by nonlinear, univariate activation functions typically implemented as piecewise polynomial splines or Gaussian processes. Our method creates mathematical ``abstractions’’ by replacing each KAN unit with a piecewise affine (PWA) function, providing both local and global error estimates between the original network and its approximation. These abstractions enable property verification by encoding the problem as a Mixed Integer Linear Program (MILP), determining whether outputs satisfy specified properties when inputs belong to a given set. A critical challenge lies in balancing the number of pieces in the PWA approximation: too many pieces add binary variables that make verification computationally intractable, while too few pieces create excessive error margins that yield uninformative bounds. Our key contribution is a systematic framework that exploits KAN structure to find optimal abstractions. By combining dynamic programming at the unit level with a knapsack optimization across the network, we minimize the total number of pieces while guaranteeing specified error bounds. This approach determines the optimal approximation strategy for each unit while maintaining overall accuracy requirements. Empirical evaluation across multiple KAN benchmarks demonstrates that the upfront analysis costs of our method are justified by superior verification results.

💡 Research Summary

The paper introduces a systematic method for verifying properties of Kolmogorov‑Arnold Networks (KANs), a class of neural networks whose layers consist of univariate functions combined only by addition. Unlike conventional feed‑forward networks that use fixed piecewise‑linear activations (e.g., ReLU), KANs can employ a wide variety of smooth univariate functions such as B‑splines, Gaussian processes, Padé approximations, or Fourier series. Because existing verification tools are built around fixed piecewise‑linear activations, the authors first replace each KAN unit with a piecewise‑affine (PWA) approximation and then quantify the error introduced by this replacement.

The core technical contribution consists of two tightly coupled optimization steps. The first step solves the optimal univariate PWA approximation problem for a given unit ψ. The authors discretize the domain