Page 22 - 《爆炸与冲击》2026年第5期

第 46 卷 袁基宸，等： 基于GNN/KAN的高应变速率金属材料本构关系的表征方法 第 5 期

adjacent strain rates, and nodes within this threshold were connected to form edges. The GNN employed a Message Passing
Neural Network (MPNN) architecture to learn and predict material properties. Model parameters were optimized using the
Adam optimizer, with the Root Mean Squared Error (RMSE) serving as the loss function. The KAN model was constructed
based on the Kolmogorov-Arnold representation theorem and consisted of multiple KAN-Linear layers. Each KAN-Linear unit
included base weights and spline weights. Base weights handled linear relationships through traditional linear transformations,
while spline weights managed nonlinear mappings via B-spline interpolation. Both models were trained on the preprocessed
dataset, and their performance was evaluated using metrics such as the Mean Relative Error (MRE), Root Mean Squared Error
2
2
(RMSE), and the coefficient of determination (R ). The GNN model achieved an average MRE of 9.2% with an R value
2
exceeding 0.95, while the KAN model recorded an MRE of 9.1% with a similar R value. Both models significantly
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outperformed the JC model, which had an MRE of 38% and an R value of 0.75. Furthermore, the predictive capabilities of the
GNN and KAN models were validated through finite element simulations. The simulation results demonstrated that the stress-
strain distributions predicted by the GNN and KAN models were more consistent with theoretical expectations compared to
those predicted by the JC model, particularly in capturing the material's softening phase. The findings highlight the potential of
integrating advanced machine learning techniques, such as GNNs and KANs, into the field of materials science to enhance the
accuracy and efficiency of constitutive modeling. These models offer a promising alternative to traditional empirical models
and hold significant implications for engineering applications in aerospace, automotive, and other industries where materials
are subjected to high strain rates.
Keywords: deep learning; high strain rate; constitutive relation; graph neural network; KAN; prediction of dynamic
mechanical properties

在汽车、航空和航天等领域，材料在高速撞击、爆炸和超音速飞行等场景下通常处于高应变速率服
役状态，此时材料的力学特性与低应变速率或准静态条件下的力学特性存在显著差异。
材料的本构方程是描述应力与温度、应变、应变速率之间关系的函数，表征了材料在受载荷后的动
[1]
态响应特性。例如，Zerilli-Armstrong（ZA）模型 、Viscoplastic 模型 [2] 等是适用于大多数金属材料的本构
关系模型，可以描述大多数工程应用场景下材料的弹性、塑性响应趋势。Johnson-Cook（JC）模型 [3] 主要
用于描述材料在高应变速率、大应变以及不同环境温度下的本构关系，该模型的构建拟合了大量实验数
据，同时考虑了应变硬化、应变率强化以及温度软化效应的影响。然而，JC 模型假设了温度、应变速率
和应变参数相互独立，这与金属材料在实际变形过程中多物理场耦合的特性不一致，影响了 JC 模型的
精度。另外，JC 模型在描述材料的应力-应变关系时更侧重于材料的强化特性，对材料温度软化效应的
描述较为简单，导致其在应变率、温度耦合作用条件下精度不足。为此，Liu 等 [4] 提出了基于 JC 模型的
修正模型，通过引入额外的参数提高了模型精度，但过多的参数降低了模型效率。因此，寻求高应变速
率下能精确、高效地描述材料动态响应特性的方法具有重要的工程意义。
本构方程本质上是多自变量（温度、应变、应变速率）函数关系的拟合问题，换而言之，关于材料本
构关系的描述问题可以转化为多输入、单输出的回归拟合问题。为提高回归预测的精度和效率，学者们
[5]
[6]
[7]
采用机器学习算法（随机森林算法 ，遗传算法 、支持向量机算法 、反向传播（back propagation，BP）神
经网络算法 、卷积神经网络算法 [9] 等）来解决此类问题。初红艳等 [10] 建立了预测综合等效应力的
[8]
BP 人工神经网络模型，在锻件高温低应变速率的条件下证明了建立的 BP 神经网络模型可有效预测综
合等效应力。林启权等 [11] 建立了 DP980 钢板流动应力的 BP 神经网络预测模型，在高温条件下较精确
地预测了 DP980 钢板在不同温度和材料方向下的流动应力。魏令港等 [12] 提出了基于改进特征筛选的随
机森林算法，在锂渣混凝土的 28 d 抗压强度的预测中具有较为显著的优势。王彦磊等 [13] 提出了一种基
于随机游走的随机森林算法，该算法以渡槽水位、气温及水温为输入，在较低温环境下能较准确地预测
渡槽不同测点的位移及应力。丁军等 [14] 以 304 不锈钢为例，提出了一种基于改进遗传算法选择算子的
优化人工神经网络，采用新模型能有效预测金属在高温低应变速率下的流变应力。黄俊杰等 [15] 提出一

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