# Estimating Contact Force Chains Using Artificial Neural Network
## 3 Results  
### Predicting Contact Force Chains  
The dimensions of each feature in the training set are 40,000 \(\times\) 180, where 40,000 is the number of records and 180 is the number of bins. Note that all bins are independent of each other. The ANN model is trained and tested in each bin. The \(R\) values of all bins for the training set are shown in Figure 6. We can see that the trained models perform well in all bins, with the maximum and minimum \(R\) values being 0.975 and 0.709, respectively. The median value and standard deviation of \(R\) for the training set are 0.859 and 0.05, respectively. This result indicates that there is a nonlinear functional relationship between the input (i.e., five particle-scale features) and output (i.e., contact force). This result also means that the anisotropy of CFC in sheared granular systems represented by a rose diagram can be estimated separately using the proposed ANN model, which was impossible before such a technique was developed. Compared with other bins, the bins in the ranges of [30\({}^{\circ}\), 60\({}^{\circ}\)] and [120\({}^{\circ}\), 150\({}^{\circ}\)] have smaller values of \(R\) mainly because of the development of the shear band. Such a result is reasonably expected, because as [PERSON] [16] noted, higher frequencies of contact gain and contact loss occur within the shear bands, which leads the trained model to be less effective in the directions aligned with the shear band. Figure 6: The distributions of \(R\) for the training and testing sets and the distribution of DTT. Two bins (i.e., [10\({}^{\circ}\), 11\({}^{\circ}\)] and [5\({}^{\circ}\), 6\({}^{\circ}\)]) with good model performance in the training set and a bin ([42\({}^{\circ}\), 43\({}^{\circ}\)]) located within the shear band with a smaller value of \(R\) were selected for comparing their training output with the ground truth. The comparison results are shown in Figure 7. For the bins of [10\({}^{\circ}\), 11\({}^{\circ}\)] and [5\({}^{\circ}\), 6\({}^{\circ}\)], we can see that the estimated contact forces agree well with the ground truth derived from the DEM simulations. Along the horizontal direction, the contact force decreases before shear band forms and remains stable afterward. For the bin of [42\({}^{\circ}\), 43\({}^{\circ}\)], we can see that the predicted contact forces match the fluctuated ground truth, although there is some difference between them. The above results demonstrate again the high capability and robustness of the proposed ANN model. Figure 8 displays the network outputs vs. targets for the three selected bins in the training set. Ideally, the data should fall along the 45\({}^{\circ}\) line, where the network outputs are equal to the targets. For the current problem, the matches are reasonably good in the bins of [10\({}^{\circ}\), 11\({}^{\circ}\)] and [5\({}^{\circ}\), 6\({}^{\circ}\)], with the \(R\) values being 0.975 and 0.963, respectively. In the bin of [42\({}^{\circ}\), 43\({}^{\circ}\)], with the \(R\) value of 0.728, the accuracy level of the model prediction of the contact forces is also acceptable. The \(R\) values of all bins for the testing set are also shown in Figure 6. It can be seen that the tested model in the bin of [148\({}^{\circ}\), 149\({}^{\circ}\)] having the \(R\) value of 0.967 shows the best performance among all models, while the model in the bin of [94\({}^{\circ}\), 95\({}^{\circ}\)] having the \(R\) value of 0.556 shows the worst performance. The median value and standard deviation of \(R\) are 0.827 and 0.09 for the testing set, respectively. Overall, the distributions of the \(R\) values of both the tested and trained models show a general trend of decreasing performance levels from the horizontal direction to the vertical direction, with the lowest performance seen in the bin ranges of [30\({}^{\circ}\), 60\({}^{\circ}\)] and [120\({}^{\circ}\), 150\({}^{\circ}\)] which are approximately aligned with the shear band orientation (Figure 2d).