Kolmogorov-Arnold Networks (KANs) in Constitutive Modeling of Hyperelastic Materials

Abstract

Accurate constitutive modeling of hyperelastic materials remains a challenging task due to their inherently nonlinear and complex stress–strain behavior. Traditional phenomenological models often fall short in capturing this complexity, particularly in modern engineering materials with rich mechanical responses. In recent decades, data-driven modeling approaches have emerged as promising alternatives, offering flexibility in learning material behavior directly from data. Multi-Layer Perceptrons (MLPs), in particular, have become widely adopted due to their universal approximation capabilities. Despite their benefits, MLP-based approaches face significant limitations. Their "blackbox" nature limits interpretability and restricts insights into underlying material mechanics. Furthermore, although MLPs with fixed activation functions can approximate hyperelastic behavior in theory, their limited smoothness, such as in the case of ReLU, can restrict accurate representation of derivatives essential for modeling material responses. These shortcomings highlight the need for alternative frameworks that can represent material behavior more accurately and transparently. An emerging alternative is the Kolmogorov-Arnold Network (KAN), which offers improved interpretability and greater flexibility due to its architecture. By leveraging the Kolmogorov-Arnold representation theorem, KANs decompose complex functions into simpler, easy-to-understand components. WhileKANs have shown promise in various applications, including material modeling, their use in hyperelasticity remains limited due to challenges in ensuring physically consistent predictions. Current KAN-based frameworks cannot guarantee physically valid hyperelastic modeling. To address these challenges, this work introduces a novel Input-Convex Kolmogorov-Arnold Network (ICKAN) architecture tailored for hyperelastic constitutive modeling. The ICKAN model employs spline-based, learnable activation functions to capture material nonlinearities and explicitly incorporates convexity and monotonicity constraints to ensure adherence to physical principles. Validation using benchmark datasets demonstrates that ICKAN accurately predicts hyperelastic stress–strain behavior across a range of loading conditions. By enhancing interpretability and ensuring physically consistent predictions, the proposed ICKAN framework provides a robust and transparent solution, underscoring the broader potential of KANs in data-driven constitutive modeling.