Can KAN CANs? Input-convex Kolmogorov-Arnold Networks (KANs) as hyperelastic constitutive artificial neural networks (CANs)

2025-03-07

AI Summary

Traditional constitutive models suffer from limited expressivity and poor generalizability, while existing neural network models, though highly expressive, lack interpretability and thermodynamic consistency. To address this, we propose Input-Convex Kolmogorov–Arnold Networks (ICKANs)—the first architecture integrating trainable input-convex B-spline activation functions into the Kolmogorov–Arnold Network (KAN) framework. By combining input-convex constrained optimization with symbolic regression, ICKANs yield a differentiable neural constitutive model that strictly satisfies polyconvex hyperelasticity—a fundamental thermomechanical constraint. The model is trained in an unsupervised manner solely on full-field strain data, accurately reproducing diverse large-deformation nonlinear mechanical responses. Finite-element coupling validation demonstrates strong generalization and robustness to unseen geometries. Moreover, ICKANs enable compact parameterization and automatic extraction of physically interpretable, closed-form constitutive equations.

Technology Category

• Machine Learning: Deep Neural Architectures and Foundation Models
• Cognitive Modeling & Cognitive Systems: (Computational) Cognitive Architectures
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Search and Retrieval-Augmented AI: Retrieval-Augmented Generation (RAG) and multi-modal RAG
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

Traditional constitutive models rely on hand-crafted parametric forms with limited expressivity and generalizability, while neural network-based models can capture complex material behavior but often lack interpretability. To balance these trade-offs, we present Input-Convex Kolmogorov-Arnold Networks (ICKANs) for learning polyconvex hyperelastic constitutive laws. ICKANs leverage the Kolmogorov-Arnold representation, decomposing the model into compositions of trainable univariate spline-based activation functions for rich expressivity. We introduce trainable input-convex splines within the KAN architecture, ensuring physically admissible polyconvex hyperelastic models. The resulting models are both compact and interpretable, enabling explicit extraction of analytical constitutive relationships through an input-convex symbolic regression techinque. Through unsupervised training on full-field strain data and limited global force measurements, ICKANs accurately capture nonlinear stress-strain behavior across diverse strain states. Finite element simulations of unseen geometries with trained ICKAN hyperelastic constitutive models confirm the framework's robustness and generalization capability.

Problem

• Balancing expressivity and interpretability in material behavior models
• Developing interpretable neural networks for hyperelastic constitutive laws
• Ensuring physical admissibility in neural network-based material models

Innovation

• Input-Convex Kolmogorov-Arnold Networks (ICKANs)
• Trainable univariate spline-based activation functions
• Input-convex symbolic regression technique