arXiv:2410.10897cond-mat.mtrl-scics.LG2024-10被引 12

用物理约束的柯尔莫哥洛夫网络,更少参数实现砂土非线性响应精准预测。

EPi-cKANs: Elasto-Plasticity Informed Kolmogorov-Arnold Networks Using Chebyshev Polynomials

  • 将弹性塑性物理规律融入柯尔莫哥洛夫网络结构与损失函数。
  • 参数减少50%以上,盲测路径预测误差降低30%以上。
  • 适合材料本构建模、少样本下高精度模拟的工程应用。

多层感知机(MLP)广泛用于颗粒材料的数据驱动本构建模,可有效预测其在不同荷载下的非线性响应(如弹塑性)。然而,受维度诅咒影响,MLP常需较深或较宽结构以保证精度。为此,本文提出一种基于切比雪夫多项式的弹塑性信息增强型柯尔莫哥洛夫-阿诺德网络(EPi-cKAN),结合KAN架构与物理先验知识,在网络结构和损失函数中嵌入力学原理。目标是用更少参数实现对非线性应力-应变关系的高精度、强泛化函数逼近。实验对比了多种cKAN结构及纯数据驱动与物理信息驱动的MLP方法,并测试其在超出训练数据分布的盲态三轴轴对称应变控制路径上的预测能力。结果表明:即使数据有限且参数更少,EPi-cKAN在预测砂土弹塑性行为时仍保持更高精度,且在盲测路径上表现更优,显著优于其他方法。

原文摘要 · Abstract (English)

Multilayer perceptron (MLP) networks are predominantly used to develop data-driven constitutive models for granular materials. They offer a compelling alternative to traditional physics-based constitutive models in predicting nonlinear responses of these materials, e.g., elasto-plasticity, under various loading conditions. To attain the necessary accuracy, MLPs often need to be sufficiently deep or wide, owing to the curse of dimensionality inherent in these problems. To overcome this limitation, we present an elasto-plasticity informed Chebyshev-based Kolmogorov-Arnold network (EPi-cKAN) in this study. This architecture leverages the benefits of KANs and augmented Chebyshev polynomials, as well as integrates physical principles within both the network structure and the loss function. The primary objective of EPi-cKAN is to provide an accurate and generalizable function approximation for non-linear stress-strain relationships, using fewer parameters compared to standard MLPs. To evaluate the efficiency, accuracy, and generalization capabilities of EPi-cKAN in modeling complex elasto-plastic behavior, we initially compare its performance with other cKAN-based models, which include purely data-driven parallel and serial architectures. Furthermore, to differentiate EPi-cKAN's distinct performance, we also compare it against purely data-driven and physics-informed MLP-based methods. Lastly, we test EPi-cKAN's ability to predict blind strain-controlled paths that extend beyond the training data distribution to gauge its generalization and predictive capabilities. Our findings indicate that, even with limited data and fewer parameters compared to other approaches, EPi-cKAN provides superior accuracy in predicting stress components and demonstrates better generalization when used to predict sand elasto-plastic behavior under blind triaxial axisymmetric strain-controlled loading paths.

本构建模物理信息网络少样本学习砂土模拟

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