Evaluation of BP Neural Network Algorithms for Predicting Elastic Buckling Loads on Cold-Formed Steel Components
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摘要: 弹性屈曲临界荷载是准确评价冷弯型钢构件承载力的重要指标. 利用人工神经网络(artificial neural networks, ANNs)模型对冷弯C型截面轴压构件的屈曲临界载荷进行了预测，将影响屈曲的几何参数和有限条法所得的计算结果作为数据集，对神经网络模型进行了训练、验证和测试. 基于最优化理论，采用6种不同的优化算法进行了模型的训练，并比较了不同算法的网络模型性能. 通过随机网格搜索确定最优超参数，使用3种统计参数来评估训练后的人工神经网络的性能，以得到最适合预测屈曲临界荷载的神经网络模型. 结果表明：Levenberg-Marquardt(L-M)算法在非线性最小二乘问题上相较于其他算法具有更高的准确性，多次训练后，L-M算法使模型预测误差非常小，而其他算法在准确度上不及L-M算法.Abstract: The elastic buckling critical load is a crucial indicator for accurately assessing the load-bearing capacity of cold-formed steel components. The artificial neural networks were used to predict the buckling loads on cold-formed flanged steel columns, with geometric parameters and finite strip method results as the dataset. Six optimization algorithms based on the optimization theory were applied to train the networks, with their performances compared. Optimal hyperparameters were determined through random grid search. Three statistical parameters were used to evaluate the networks' post-training performances. The Levenberg-Marquardt (L-M) algorithm demonstrates higher accuracy in nonlinear least squares problems, significantly reducing prediction errors after multiple trainings, and outperforming other algorithms.
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图 2 冷弯型钢C型构件截面尺寸示意图
Figure 2. Schematic diagram of sectional dimensions of cold-formed lipped C-type channel steel columns
图 5 神经网络(输入层、隐藏层、输出层)
Figure 5. The neural network (input layer, hidden layer, output layer)
图 6 MSE和计算时长随神经元个数的变化
Figure 6. MSE and computation time changes with the number of neurons
图 7 神经网络模型预测结果与CUFSM计算结果对比
Figure 7. Comparison between neural network model predictions and CUFSM calculation results
图 8 GPU加速的SCG算法与无GPU加速的L-M算法预测值的MAPE对比
Figure 8. Comparison of MAPE values between the GPU-accelerated SCG algorithm and the non-GPU-accelerated L-M algorithm
表 1 冷弯型C型截面尺寸范围(单位: mm)
Table 1. Ranges of sectional dimensions of cold-formed lipped channel steels (unit: mm)
component min max Hc 41.15 406.40 Bc 31.75 88.90 Dc 4.78 25.40 t 0.48 3.15 
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表 2 各训练算法运行时长
Table 2. Runtime lengths of various training algorithms
training algorithm average time/s ratio min time/s max time/s GDX 1.29 1.00 1.16 1.41 GDM 3.93 3.05 0.32 4.18 GD 4.13 3.21 4.10 4.19 SCG 9.36 7.27 7.81 37.75 BFG 24.80 19.26 16.93 167.68 L-M 132.10 102.62 110.54 135.84
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表 3 各训练算法验证集误差
Table 3. Validation set errors for various training algorithms
training algorithm average MSE ratio min MSE max MSE L-M 2.01E4 1.00 1.67E4 2.56E4 BFG 4.48E4 2.16 3.89E4 5.74E4 SCG 5.69E4 2.74 4.95E4 6.74E4 GDX 8.73E3 42.10 6.20E3 1.30E2 GD 8.67E2 417.85 6.92E2 1.16E1 GDM 8.99E2 433.52 5.09E2 2.02E1
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表 4 BFG和SCG训练算法时长对比
Table 4. Comparison of training durations between BFG and SCG algorithms
training algorithm average time/s ratio min time/s max time/s BFG 40.23 1.00 22.46 64.90 SCG 45.27 1.13 19.40 208.06 L-M 363.35 9.03 107.76 1 359.262
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表 5 BFG和SCG验证集MSE对比
Table 5. Comparison of validation set MSE between BFG and SCG algorithms
training algorithm average MSE ratio min MSE max MSE BFG 3.40E4 1.00 2.40E4 8.6E4 SCG 3.40E4 1.00 2.70E4 4.5E4 L-M 1.95E4 0.57 1.68E4 2.33E4
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表 6 BFG和SCG epoch对比
Table 6. Comparison of epochs between BFG and SCG algorithms
training algorithm average epoch ratio min epoch max epoch BFG 1 191.90 1.00 657 1 957 SCG 2 358.55 1.98 1 247 4 185 L-M 1 344.75 1.13 394 5 000
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