# Estimating Contact Force Chains Using Artificial Neural Network
## 4 Discussion  
Although the interparticle contact force is mathematically defined by the contact model adopted in the DEM modeling [36], the evolution of CFC cannot be analytically computed because the complex kinematics of the entire particle assembly cannot be easily predicted. That facilitates the implementation of ML algorithms in the estimation of CFCs. It is anticipated that the proposed ANN model captures the evolutions of contact force very well using the adopted five particle-scale features. This is because input and output have an obvious causal relationship based on the physics-based modeling [36]. Another important reason for selecting the five particle-scale features for the model development  
Figure 16: A comparison between predicted contact force chains and ground truth for ML11 at strain (a) 3.75%; (b) 7.5%; (c) 11.25%; (d) 15%. Figure 15: The cumulative probability of \(R\) with different data sizes. is that all these features can be readily acquired from micro-CT data and therefore will facilitate the future extension of the model to the micro-CT results [16]. Among the five particle-scale features, the particle size shows the highest level of importance for CFC prediction when investigating the effects of the features. This result was well expected as the particle size is an inherent soil characteristic that provides insights into the mechanical behavior of natural sands [43; 44]. [PERSON] and [PERSON] [45], [PERSON] [46] and [PERSON] et al. [47] demonstrated that the coordination number is strongly affected by the particle size, and the distributed coordination numbers correspond to different types of contacts between the small and large components. The CFCs estimation results in Figure 12 also present a similar result, in which the performance of coordination number shows a similar effect to those observed in particle size. It should be mentioned that this study serves as the starting point of a comprehensive research project which aims to develop and apply ML methods to granular material research. Our ultimate goal is to develop a powerful and robust ML model that is able to predict the macroscopic behaviors of a sheared sand, such as initial stiffness, maximum stress ratio or even stress-strain relationship, purely based on the input from micro-CT. Establishing a ML model to predict CFC using a DEM database represents the first step of a fundamental and rigorous approach to achieve such a goal. Another point that is worthy of an explanation is that only the spatial distribution of CFC (i.e., anisotropy of CFC) is examined while the heterogeneity of CFC is not considered in this paper. The reason for this is that theoretical studies [48; 49] have shown that a strong correlation usually exists between the anisotropy of CFC and the shear stress ratio of a loaded granular assembly, which means that we will be able to predict the stress ratio of the granular assembly once the anisotropies of CFC and material fabric become available. However, the heterogeneity of CFC, which is due to the inherent non-uniformity of any granular material, does not need to be specially investigated because it offers little value to our quantitative prediction of the macroscopic properties of granular materials in future. It must be emphasized that due to the major hindrance of measuring interparticle contact forces experimentally, the current way of using DEM data for constructing the ML model is the only feasible approach, while the ANN model proposed to solve CFC prediction problem can be replaced by some other ML algorithms such as long short-term memory network [50]. For instance, [PERSON] and [PERSON] [34] adopted a DEM-based ML approach to predict the constitutive relationships of 2D granular materials using CNN and bidirectional long short-term memory neural network. In their study, the contact fabric was simplified and approximated using a second Fourier component and characterized by two parameters.