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3D stochastic reconstruction of porous media based on attention mechanisms and residual networks

Abstract

Because of the complex internal porous structures, the reconstruction of porous media has encountered many difficulties in practical research and development. At present, numerical simulation methods are widely used in the field of reconstructing porous media, which can reconstruct results similar to the true pore structure, but generally are CPU-intensive and time-consuming. Recently, with the rapid development of deep learning, the powerful ability of feature extraction and structure prediction owned by deep learning can be transferred to reconstruct porous media. The convolutional neural network (CNN) is one of the classical methods in deep learning. A traditional CNN is unable to specially focus on the effective features in learning and possibly has degradation problem with the increase of layers. To address the degradation problem and make CNNs extract important features, residual networks and attention mechanisms are combined in CNNs to respectively alleviate network degradation and focus on the important features in the reconstruction of porous media. Besides, the bilinear interpolation is used to enlarge the size of compressed data. Compared with some traditional numerical reconstruction methods, CNN, GAN and some variants of GAN, our method has shown its practicability and effectiveness.

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Funding

This work is supported by the National Natural Science Foundation of China (Nos. 41672114, 41702148).

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Correspondence to Pengfei Xia.

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The authors declare that we have no conflict of interest.

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Appendices

Appendix 1: The main functions and parameters of our method

As shown in Fig. 5, our method is based on CNNs, combining a CBAM with ResNets as well as bilinear interpolation to realize porous media reconstruction. The specific parameters and functions of our method are shown in Table 

Table 9 Parameters and structures in our method

9. Some parameters and terms are described below.

  1. 1.

    Conv2d: a 2D convolution operation that is used to extract features of 2D data.

  2. 2.

    Hidden layer of MLP: the hidden layer in the fully-connected network MLP.

  3. 3.

    Output layer of MLP: the output layer in the fully-connected network MLP.

  4. 4.

    Units: the number of neural units in a layer of the network.

  5. 5.

    batch_normalization: a regularization function that normalizes the input data, alleviates the problem of scattered distribution of data features, and makes the network model more stable.

  6. 6.

    kernel_size: the size of a convolution kernel in a layer.

  7. 7.

    Stride: the step size of each convolution kernel.

  8. 8.

    leaky_relu: an activation function that alleviates the problem of gradient vanishing.

  9. 9.

    Filters: the number of filters in one layer of networks.

  10. 10.

    Scale: a magnification (scale > 1) or contraction (0 < scale < 1) factor for a bilinear interpolation, e.g. ‘scale = 2’ means that the original feature map is enlarged to twice using bilinear interpolation.

  11. 11.

    leaky_relu: an activation function.

  12. 12.

    Padding = '‘same’: “padding” means the filling of image edges, and “same” is a filling attribute, which means the output size and input size of the layer are same after they are divided by the stride size and rounded up.

Note that in Conv_1 and Conv_2 there are altogether six layers, and in Conv_3 and Conv_4 there are also exactly the same six layers. The reason is that in Conv_1 and Conv_2 the convolution operation gradually reduces the size of feature maps for six times; on the contrary, bilinear interpolation used as the reverse operation of reducing the size of feature maps enlarges the feature maps gradually for six times, too. Although convolution operations are used in Conv_3 and Conv_4, the size of feature maps is not influenced since their kernel size is 1 × 1 and the stride is 1.

Appendix 2: Discussion on the influence of different network layers

As shown in Table 9 and discussed in Sect. 2.3, Conv_1 and Conv_2 altogether have six layers; Conv_3 and Conv_4 also have the same number of layers because the same number of bilinear interpolation operations are used to offset the effect of convolution operations used for the reduction of feature maps in Conv_1 and Conv_2. Normally the layer numbers of Conv_1 (layer number = 1) and Conv_4 (layer number = 2) in our method are fixed. Let the total layer number in the Conv_1 and Conv_2 be i, so the total layer number in Conv_3 and Conv_4 is i, too. We will discuss the influence of i on the reconstruction time and quality.

Table

Table 10 The reconstruction time for only one result using different i layers

10 shows the reconstruction time for only one result when i = 3, 4, 5, 6, respectively. Note that when i = 3, since the layer numbers of Conv_1 (layer number = 1) and Conv_4 (layer number = 2) are fixed, it can be inferred that the layer number of Conv_2 is 2 and that of Conv_3 is 1. Similarly, when i = 4, the layer number of Conv_2 is 3 and that of Conv_3 is 2; when i = 5, the layer number of Conv_2 is 4 and that of Conv_3 is 3; when i = 6, Conv_2’s layer number is 5 and Conv_3’s is 4.

It can be seen from Table 10 that with the increase of i, the reconstruction time also increases. Figures 

Fig. 15
figure15

MPC curves the TI and reconstructions using different i layers in three directions

15a–c show MPC curves of using different i layers in three directions. As shown in Fig. 15, it can be seen that the reconstructed result is closest to TI when i = 6. In real reconstruction, the reconstruction time and quality should be balanced. Generally, increasing the number of layers can improve the reconstruction quality, but the reconstruction time also increases dramatically. In our experiment, i = 6 seems to be an appropriate balance between reconstruction quality and time.

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Zhang, T., Xia, P. & Lu, F. 3D stochastic reconstruction of porous media based on attention mechanisms and residual networks. Stoch Environ Res Risk Assess (2021). https://doi.org/10.1007/s00477-021-02092-z

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Keywords

  • Porous media
  • Reconstruction
  • Convolutional neural network
  • Convolution block attention module
  • Residual network