MLP-ANN model for predicting uniaxial compressive strength of rocks based on the rebound method
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摘要:
岩石单轴抗压强度是岩土工程中的重要参数，合理确定其数值对工程设计至关重要。本文提出了一种基于多层感知机的人工神经网络（MLP-ANN）模型，用于预测岩石单轴抗压强度。该模型以岩性、节理面、施密特锤回弹高度和纵波波速为输入参数，采用最大最小归一化进行参数标准化，并通过
k 折交叉验证提高模型的泛化能力。为优化模型性能，文章探讨了神经元数量、数据分割比例和激活函数对预测结果的影响。经对比验证，研究确定了最优模型配置：神经元数量为8，训练集与测试集比例为8∶2，激活函数选用Tanh函数。模型预测值与实际值对比分析结果表明，最优模型的平均绝对误差为3.500 MPa，均方根误差为5.836 MPa。结果表明，该模型预测误差较小，预测准确率较高，具有较好的实用性。Abstract:Objective The uniaxial compressive strength (UCS) of rock is an important parameter in geotechnical engineering, as well as accurately determining its value is crucial for engineering design.
Methods This study proposed a machine learning model based on a multi-layer perceptron-Artificial Neural Network (MLP-ANN) to predict the UCS of rock. The model takes lithology, joint surfaces, Schmidt hammer rebound height, and P-wave velocity as input parameters, and applies min-max normalization to standardize these parameters. Additionally, k-fold cross-validation is used to improve the model’s generalization ability. To further optimize model performance, the paper explores the impact of the number of neurons, data splitting ratio, and activation function on prediction results.
Results Through comparative validation, the study determines the optimal model configuration: 8 neurons, a training-to-testing ratio of 8∶2, and the Tanh activation function. The comparison between predicted and actual values shows that the optimal model achieves an average absolute error of 3.500 MPa and a root mean square error of 5.836 MPa.
Conclusion These results indicate that the model has a small prediction error and high accuracy, illustrating good practicality.
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图 6 MLP-ANN基本框架
xi为第i个输入变量；ni, j为第i层、第j个神经元；yi为第i个目标值
Figure 6. Basic framework of MLP-ANN
图 8 通过室内实验获取岩样相关参数（SRH、vp、 UCS）相对频率直方图和累计频率分布曲线
μg为平均值；q0为标准差
Figure 8. Histograms and cumulative distribution curves of parameters related to rock samples (SRH, vp, UCS) obtained by laboratory tests
图 10 不同激活函数损失变化图（隐藏层神经元数量=8）
Figure 10. Plot of loss variation with different activation functions (number of neurons in hidden layer = 8)
图 11 数据分割百分比对MLP-ANN预测性能影响
Figure 11. Effect of data segmentation percentage on MLP-ANN prediction performance
表 1 室内实验结果
Table 1. Results of experimental tests
岩性 数量 参数 最小值 平均值 中值 最大值 COV值/% 砂岩 56 ρ/(g·cm−3) 2.623 2.658 2.659 2.686 0.68 SRH 45.67 56.28 56.67 61.33 5.16 vp/(km·s−1) 3.319 3.747 3.718 4.287 6.66 μd 0.304 0.325 0.314 0.350 5.01 Ed/MPa 0.023 0.026 0.026 0.027 3.83 Gd/MPa 8.670 9.614 9.726 10.160 4.60 UCS/MPa 88.98 93.26 93.43 106.28 7.05 花岗岩 56 ρ/(g·cm−3) 2.780 2.788 2.787 2.794 0.09 SRH 60.67 67.43 67.33 76.33 3.92 vp/(km·s−1) 4.727 4.901 4.899 5.096 1.84 μd 0.347 0.361 0.359 0.381 2.89 Ed/MPa 0.037 0.040 0.039 0.042 3.80 Gd/MPa 13.576 14.470 14.192 15.674 4.70 UCS/MPa 134.33 148.88 149.06 168.59 4.57 片麻岩横节理 41 ρ/(g·cm−3) 2.639 2.652 2.653 2.655 0.12 SRH 62.33 68.24 68.67 72.00 3.27 vp/(km·s−1) 2.762 3.899 3.911 4.936 13.91 μd 0.336 0.356 0.353 0.363 1.40 Ed/MPa 0.031 0.032 0.032 0.033 1.24 Gd/MPa 11.431 11.823 11.875 12.215 2.20 UCS/MPa 97.13 121.75 122.41 137.42 6.94 片麻岩纵节理 55 ρ/(g·cm−3) 2.625 2.652 2.649 2.684 0.53 SRH 60.67 66.19 66.00 71.33 3.68 vp/(km·s−1) 2.652 3.637 3.688 4.395 12.91 μd 0.308 0.329 0.330 0.333 1.15 Ed/MPa 0.030 0.033 0.033 0.035 5.86 Gd/MPa 11.221 12.148 12.193 13.121 5.13 UCS/MPa 119.66 137.65 137.95 157.74 5.46 注：ρ为密度；SRH为施密特锤回弹高度；vP为纵波波速；μd为动泊松比；Ed为动弹性模量；Gd为动剪切模量；UCS为岩石单轴抗压强度，下同 
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表 2 激活函数种类
Table 2. Types of activation functions
序号 函数名称 表达式 图像 1 Sigmoid $ {f}{}{(}{x}{)=}\dfrac{{1}}{{1}{+}{{\mathrm{exp}}}{(-}{x}{)}} $ 
2 Tanh $ {f}{}{(}{x}{)}{=}\dfrac{{1-{\mathrm{exp}}}{(}{-2x}{)}}{{1+{\mathrm{exp}}}{(}{-2x}{)}} $ 
3 ReLU $ f(x)={\mathrm{max}}(0,x) $ 
4 Softmax $ f(x){=}\dfrac{{\mathrm{exp}}(x_{k})}{\displaystyle\sum_{i=1}^{n}{\mathrm{exp}}(x_{i})} $ 
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表 3 神经元数量与不同激活函数在训练集和测试集上的RMSE值比较
Table 3. Comparison of RMSE values between the number of neurons and different activation functions on the training and test sets
RMSE/MPa 隐藏层神经元数量 函数名称 Sigmoid ReLU Tanh Softmax 训练集 测试集 训练集 测试集 训练集 测试集 训练集 测试集 2 0.0580 0.0591 0.0617 0.0626 0.0564 0.0604 0.0562 0.0572 4 0.0576 0.0600 0.0599 0.0610 0.0553 0.0600 0.0562 0.0570 8 0.0570 0.0610 0.0577 0.0603 0.0550 0.0582 0.0560 0.0577 16 0.0570 0.0620 0.0570 0.0621 0.0554 0.0582 0.0560 0.0577 32 0.0576 0.0653 0.0576 0.0633 0.0554 0.0601 0.0561 0.0579 64 0.0596 0.0707 0.0555 0.0622 0.0596 0.0607 0.0562 0.0577
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表 4 测试集数据分配
Table 4. Data allocation for the testing set
岩性 数据量 数据量占比/% 测试集数据百分比/% 10 20 30 40 砂岩 56 26.9 6 12 17 22 花岗岩 56 26.9 6 12 17 22 片麻岩横节理 41 19.7 4 8 12 17 片麻岩纵节理 55 26.5 5 10 16 22 合计 208 100 21 42 62 83
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