# Estimating Contact Force Chains Using Artificial Neural Network
## 5 Conclusions  
In this study, a multilayer feedforward ANN model based on a detailed set of 2D DEM simulation data was developed to estimate, with a high degree of accuracy, the anisotropy of CFC in a quasi-statically sheared granular system. The main contributions and conclusions from this study are stated as follows:  
In a quasi-statically sheared granular system, the five particle-scale features, i.e., particle coordination number, size, \(x/y\)-velocity, and spin, all contain predictive information of the CFC. An excellent model performance manifested in a close match between the rose diagrams of CFC from the ANN predictions and DEM simulations is obtained. The training result shows that the proposed ANN model is less effective in the directions aligned with the shear band.  
The results from the parametric study demonstrate that the combined features of particle size and coordination number have a dominating influence on the CFC estimation. The importance of the input features can be sorted (in the descending order) according to their model performance in the estimation of CFC as: particle size \(>\) coordination number particle y-velocity > particle x-velocity > particle spin. The performance of the ANN model is also influenced by the data size, but the amount of data needed to train the ANN model is still a controversial issue which has not yet been solved.  
The major contribution of this study lies in the following three aspects: (1) making a pioneering attempt to propose an ANN model for predicting the evolution of CFC within a sheared granular system; (2) providing valuable information about the key particle-scale features that must be carefully selected for the ML analysis; (3) suggesting ML as a promising tool for understanding and quantifying the complex mechanical behaviors of granular materials. Although only biaxial tests were conducted in this study, it represents the first feasible and effective step of a comprehensive research project to carry out machine learning investigations in the micromechanics of granular materials. Future enhancement of the model may include optimizing the structure of the proposed ANN model to better reflect the nonlinear relationship between the input and output variables, identifying more effective features as the input variable and using a larger dataset to train the ANN model. A similar approach can also be applied to the study of other physical quantities that cannot be measured directly from the experiment, such as the frictional state, energy input and dissipation within a sheared granular material. Another promising direction is to develop ML-assisted constitutive models to better quantify and predict the stress-strain behavior of granular materials.  
[PERSON] proposed the idea and edited the manuscript; [PERSON] performed the DEM simulations and wrote the manuscript; Both authors have read and agreed to the published version of the manuscript.  
This study was supported by General Research Fund Grant Nos. CityU 11201020, CityU 11213517 and CityU 11207321 from the Research Grants Council of the Hong Kong SAR and Research Grant No. 51779213 from the National Science Foundation of China.  
Not applicable.  
**Data Availability Statement:** Some or all data, models, or code generated or used during the study are available from the corresponding author by request.  
The authors would like to express gratitude to [PERSON] and [PERSON] for their valuable advice in constructing the ANN model.  
The authors declare no conflict of interest.