用可解释的神经网络学习材料非线性力学行为，兼顾精度与物理合理性。
Can KAN CANs? Input-convex Kolmogorov-Arnold Networks (KANs) as hyperelastic constitutive artificial neural networks (CANs)
- 基于Kolmogorov-Arnold表示，用可训练的单变量样条函数构建模型。
- 通过单调输入凸约束，确保模型满足材料力学的多凸性要求。
- 适合需要高可解释性的力学建模场景，如材料设计与仿真验证。
传统本构模型依赖人工构造的参数形式，表达能力与泛化性受限；而神经网络模型虽能捕捉复杂材料行为，却常缺乏可解释性。为平衡此矛盾，本文提出单调输入凸柯尔莫哥洛夫-阿诺德网络（ICKAN），用于学习多凸超弹性本构关系。ICKAN利用柯尔莫哥洛夫-阿诺德表示，将模型分解为可训练的单变量样条激活函数的复合形式，实现丰富表达力。在KAN架构中引入可训练的单调输入凸样条，确保各向同性可压缩超弹性模型具有物理可接受的多凸性。所得模型紧凑且可解释，可通过单调输入凸符号回归技术显式提取解析本构关系。在全场应变数据和有限全局力测量下进行无监督训练，ICKAN准确捕捉了多种应变状态下的非线性应力-应变行为。对未见几何结构的有限元模拟验证了该框架的鲁棒性与泛化能力。
原文摘要 · Abstract (English)
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 monotonic 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 monotonic input-convex splines within the KAN architecture, ensuring physically admissible polyconvex models for isotropic compressible hyperelasticity. The resulting models are both compact and interpretable, enabling explicit extraction of analytical constitutive relationships through a monotonic input-convex symbolic regression technique. 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.
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