NEURAL NETWORK ALGORITHM FOR NONLINEAR STRUCTURAL SEISMIC RESPONSE
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摘要: 提出一种基于长短期记忆(long short-term memory, LSTM)神经网络模型计算非线性结构地震响应的新方法，采用单向多层堆叠式LSTM架构，并借助滑动时间窗实现递推计算。改进了模型预测效果的评价指标，可考虑响应在不同幅值区间的敏感性差异，避免了传统评价指标的相位敏感问题。利用实测地震动和多层框架结构进行了新方法的验证，给出了网络超参数的取值原则，并讨论了不同工况下模型的泛化能力。结果表明，LSTM模型的计算精度较好、对地震动类型具有鲁棒性。由于神经网络模型便于分布式、云部署的特点，该方法可在城市区域地震响应快速模拟等传统数值方法受限的应用场景发挥作用。Abstract: A new method based on the long short-term memory (LSTM) neural network model is proposed for calculating seismic responses of nonlinear structures. It adopts a unidirectional multilayer stacked LSTM architecture and recursively calculates structural responses using a sliding time window. Updated accuracy evaluation indexes are suggested to take into account the sensitivity difference in different response amplitude ranges and avoid the phase sensitivity issue of traditional ones. The new method is validated by multi-layer frame structures subjected to measured ground motions, the principles for selecting network hyperparameters are given, and the generalization ability of the method for different conditions is discussed. The results indicate that the LSTM model achieves good computational accuracy and is robust to various type of ground motions. Due to the cloud deployment feature of the neural network model, the new method is expected to contribute to application scenarios where traditional numerical methods are limited, such as rapid simulation of seismic response in urban areas.
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图 6 例1中4条地震动响应时程的预测结果
Figure 6. Four prediction results of the seismic response time history in Example 1
图 7 例2中4条地震动响应时程的预测结果
Figure 7. Four prediction results of the seismic response time history in Example 2
图 9 M1-50模型的测试集预测结果分布
Figure 9. Distribution of prediction results of the test set of the M1-50 model
图 10 代表性地震动的傅里叶幅值谱与加速度时程
Figure 10. Fourier amplitude spectrum and acceleration time history of representative ground motions
图 11 滤波前后16号地震动预测结果对比
Figure 11. Comparison of prediction results of No. 16 ground motion before and after filtering
表 1 LSTM网络结构
Table 1. LSTM network structure
层 类型 激活函数 输出形状 Input InputLayer − (None, n, f+1) LSTM1 LSTM tanh (None, n, 200) LSTM2 LSTM tanh (None, n, 200) LSTM3 LSTM tanh (None, 200) FC Dense Linear (None, s×f) 注：n为输入时间序列长度；s为输出时间序列长度；f为预测自由度数量。 
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表 2 10层框架结构参数与恢复力模型
Table 2. Parameters and restoring force model of a 10-story frame structure
楼层 1 2 3~6 7~9 10 层高h/m 4.60 4.00 3.20 3.20 3.20 质量/(×105 kg) 1.52 1.45 1.36 1.32 1.15 弹性模量/(×104 MPa) 3.25 3.25 3.15 3.00 3.00 屈服强化系数α1 0.40 屈服后强化系数α2 0.10 开裂位移xc h/550 屈服位移xy h/70 极限位移xp h/45
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表 3 地震动数据集
Table 3. Earthquake ground motion dataset
数据集 地点 持时/s 数量 训练集 East Japan 300 27 测试集 Imperial Valley 37.06~50.08 24 Coalinga 58.16 2 Whittier Narrows 40.00 2 Superstition Hills 29.85 2
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表 4 6层框架结构参数与恢复力模型
Table 4. Parameters and restoring force model of a 6-story frame structure
楼层 1 2 3~6 层高h/m 4.60 4.00 3.20 质量/(×105 kg) 1.52 1.45 1.36 弹性模量/(×104 MPa) 3.25 3.25 3.15 屈服强化系数α1 0.60 屈服后强化系数α2 0.20 开裂位移xc h/800 屈服位移xy h/80 极限位移xp h/40
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表 5 窗口大小取值分析的模型评价结果
Table 5. Model evaluation results of window size analysis
LSTM模型 M1-25 M1-50 M1-75 M1-100 M1-125 M2-50 M2-75 M2-100 模型参数 窗口大小 25 50 75 100 125 50 75 100 训练集 train1 train1 train1 train1 train1 train2 train2 train2 测试集 test1 test1 test1 test1 test1 test2 test2 test2 评价指标(1层) 峰值百分误差/(%) 10.8700 9.3000 10.0300 12.9100 10.2500 13.9700 15.5400 15.7900 P相关系数 0.7021 0.8697 0.8155 0.8399 0.7487 0.7628 0.7252 0.7663 WRMSE/m 0.0020 0.0015 0.0017 0.0017 0.0019 0.0040 0.0035 0.0034 加权决定系数 0.7425 0.8639 0.8137 0.8226 0.7625 0.2995 0.6970 0.7292 评价指标(5层) 峰值百分误差/(%) 16.3400 10.4400 10.9600 10.7600 19.5100 16.7800 15.2000 12.4600 P相关系数 0.6208 0.8560 0.8298 0.8363 0.7136 0.7618 0.7336 0.7739 WRMSE/m 0.0121 0.0081 0.0084 0.0084 0.0109 0.0205 0.0169 0.0169 加权决定系数 0.4798 0.8219 0.7927 0.7963 0.5212 0.1327 0.6969 0.7266 评价指标(9层) 峰值百分误差/(%) 12.4000 8.3600 10.1100 10.2100 16.2500 15.1600 15.4700 12.7500 P相关系数 0.6412 0.8649 0.8170 0.8405 0.7301 0.7678 0.7351 0.7750 WRMSE/m 0.0153 0.0103 0.0122 0.0110 0.0137 0.0257 0.0221 0.0223 加权决定系数 0.5695 0.8471 0.7712 0.8147 0.6360 0.3349 0.7015 0.7324 评价指标(均值) 峰值百分误差/(%) 13.2000 9.3700 10.3700 11.2900 15.3400 15.3000 15.4000 13.6700 P相关系数 0.6547 0.8635 0.8208 0.8389 0.7308 0.7642 0.7313 0.7717 WRMSE/m 0.0098 0.0067 0.0074 0.0070 0.0089 0.0167 0.0142 0.0142 加权决定系数 0.5973 0.8443 0.7925 0.8112 0.6399 0.2557 0.6985 0.7294
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表 6 不同地震动与结构的响应主周期
Table 6. Response main period under different ground motions and structures
PGA/(m/s2) 0.5 2.5 4 数据集 train test train test train test 结构1 0.762 0.764 0.878 0.839 0.950 0.923 结构2 0.494 0.483 0.517 0.512 0.564 0.552 注：响应主周期为响应时程主频率的倒数，表示响应的波长特征，单位为s。增加PGA=0.5 ${\rm{m/}}{{\rm{s}}^2}$的对照组以说明结构在线性状态下的响应波长。
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表 7 地震动泛化能力分析的模型评价结果
Table 7. Model evaluation results of earthquake generalization ability analysis
LSTM模型 M1-50 M1-100 M2-100 M3-100 模型参数 窗口大小 50 100 100 100 训练集 train1 train1 train2 train3 测试集 test1 test2 test1 test2 test1 test2 test1 test2 评价指标(1层) 峰值百分误差/(%) 9.3000 16.5900 12.9100 13.4800 16.6500 15.7900 13.7300 12.1300 P相关系数 0.8697 0.7090 0.8399 0.8035 0.7892 0.7663 0.8384 0.8029 WRMSE/m 0.0015 0.0037 0.0017 0.0031 0.0020 0.0034 0.0018 0.0029 加权决定系数 0.8639 0.6617 0.8226 0.7742 0.7526 0.7292 0.8152 0.7537 评价指标(5层) 峰值百分误差/(%) 10.4400 12.2300 10.7600 13.8200 15.3500 12.4600 14.7800 10.9800 P相关系数 0.8560 0.6672 0.8363 0.8061 0.7851 0.7739 0.8338 0.7918 WRMSE/m 0.0081 0.0198 0.0084 0.0154 0.0101 0.0169 0.0088 0.0153 加权决定系数 0.8219 0.5844 0.7963 0.7663 0.7216 0.7266 0.7976 0.7158 评价指标(9层) 峰值百分误差/(%) 8.3600 11.7900 10.2100 14.7600 15.1100 12.7500 13.7500 10.8300 P相关系数 0.8649 0.6827 0.8405 0.8063 0.7912 0.7750 0.8390 0.8022 WRMSE/m 0.0103 0.0252 0.0110 0.0203 0.0132 0.0223 0.0117 0.0197 加权决定系数 0.8471 0.6362 0.8147 0.7716 0.7395 0.7324 0.8059 0.7302 评价指标(均值) 峰值百分误差/(%) 9.3700 13.5400 11.2900 14.0200 15.7000 13.6700 14.0900 11.3100 P相关系数 0.8635 0.6863 0.8389 0.8053 0.7885 0.7717 0.8371 0.7990 WRMSE/m 0.0067 0.0162 0.0070 0.0129 0.0084 0.0142 0.0074 0.0126 加权决定系数 0.8443 0.6274 0.8112 0.7707 0.7379 0.7294 0.8062 0.7332
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表 8 结构模型泛化能力分析的模型评价结果
Table 8. Model evaluation results of structural model generalization ability analysis
LSTM模型 M1-25 M1-25 M1-50 M1-50 M2-25 M2-50 M2-75 M2-100 M3-100 模型参数 窗口大小 25 25 50 50 25 50 75 100 100 训练集 train1 train1 train1 train1 train2 train2 train2 train2 train3 测试集 test1 test2 test1 test2 test2 test2 test2 test2 test2 评价指标(1层) 峰值百分误差/(%) 7.7200 11.9500 7.4900 8.4400 12.1700 12.6200 9.2700 8.3800 9.2800 P相关系数 0.9042 0.8534 0.9309 0.8884 0.8010 0.8043 0.9044 0.8783 0.8714 WRMSE/m 0.0010 0.0021 0.0008 0.0017 0.0022 0.0025 0.0017 0.0017 0.0020 加权决定系数 0.8992 0.8599 0.9269 0.9013 0.8247 0.7769 0.8974 0.8706 0.8560 评价指标(3层) 峰值百分误差/(%) 6.5700 9.0200 7.1600 6.6600 11.2300 12.2900 9.9900 7.5600 8.3900 P相关系数 0.9016 0.8550 0.9297 0.8878 0.7853 0.8130 0.8980 0.8901 0.8785 WRMSE/m 0.0029 0.0059 0.0023 0.0049 0.0064 0.0071 0.0048 0.0048 0.0055 加权决定系数 0.8994 0.8657 0.9223 0.9041 0.7976 0.7881 0.9012 0.8962 0.8690 评价指标(6层) 峰值百分误差/(%) 6.2200 7.7200 6.5600 5.6400 9.8100 11.5400 9.6200 7.0600 8.2500 P相关系数 0.9024 0.8544 0.9302 0.8875 0.8135 0.8067 0.8953 0.8554 0.8732 WRMSE/m 0.0041 0.0084 0.0033 0.0070 0.0085 0.0104 0.0069 0.0072 0.0080 加权决定系数 0.8976 0.8643 0.9244 0.9038 0.8324 0.7812 0.8994 0.8873 0.8663 评价指标(均值) 峰值百分误差/(%) 6.8400 9.5600 7.0700 6.9100 11.0700 12.1500 9.6300 7.6700 8.6400 P相关系数 0.9027 0.8543 0.9303 0.8879 0.7999 0.8080 0.8992 0.8746 0.8744 WRMSE/m 0.0027 0.0055 0.0021 0.0045 0.0057 0.0067 0.0045 0.0046 0.0052 加权决定系数 0.8988 0.8633 0.9245 0.9031 0.8183 0.7820 0.8993 0.8847 0.8637
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