## 2 Modeling
### Neural Network Surrogate Models for Time-Varying Mesh Stiffness  
The mesh of the face gear and pinion is a nonconjugate contact, where the contact position is a point that expands into an elliptical contact area with respect to the magnitude of the load when bearing the load. Moreover, as the load increases, the number of teeth simultaneously engaged in the face gear mesh pair increases, and the shape of the mesh stiffness curve changes. Therefore, the mesh stiffness pair is closely related to the load. In the CFGSTTS, the elastic support stiffness of the gears affects the load distribution among the gears. The elastic support stiffness changes, and the normal contact force of the gear pair changes accordingly, leading to a change in the mesh stiffness. Consequently, a load-dependent time-varying meshing stiffness model was developed to further consider the time-varying meshing stiffness. In our previous study, we used the strain energy approach to determine the mesh stiffness [35]. The upper face gear contains the web. The face gear's parameters are listed in Table 1. The specific loading and simulation procedure is described in the literature [35]. Notably, in this study, the contact stiffness of the tooth contact pair was set to 10 times the default value, which was 1.0; the convergence tolerances of the displacement and force were set to 1/100 of the default values, which were 0.05 and 0.005, respectively. The computational cost increased, but a smoother mesh stiffness curve could be acquired. Figure 1: Schematic diagram of concentric face gear split-torque transmission system: **(a)** 3D schematic diagram; **(b)** 2D schematic diagram. Figure 2: Power flow direction of CFGSTTS: **(a)** input gear; **(b)** idler gear; **(c)** tail gear. Through quasistatic finite element analysis, the meshing stiffness data, including the load, pinion rotation angle, and stiffness values, were obtained for 1 cycle under 42 sets of loads in the range of 100 to 8000 Nm. Surrogate models can be used to quickly provide a solution instead of a real system by seeking the response relationship between the input and output variables. For complex nonlinear problems, the multilayer feedforward neural network is a more commonly used surrogate model. A multilayer feedforward neural network is a network model with a hierarchical structure composed of simple neural units, which has a strong simulation ability for nonlinear systems. In this study, 42 sets of data were used as training samples to construct a neural network with the topology shown in Figure 3, including input, hidden, and output layers. The input layer contained  
The generalization ability of feedforward neural networks refers to the ability of the network to describe the correct input-output relationship for nontrained samples in the same sample set, which is used to characterize the prediction ability of the learning model for unknown data. To verify the generalization ability of the load-related time-varying mesh stiffness proxy model, the prediction values of the neural network were calculated for input loads of 250 and 1600 Nm. The results in Figures 4b and 5b show that the output of the agent model was almost the same as that of the finite element simulation, indicating that the agent model had a strong generalization capability. Figure 6 illustrates the meshing stiffness at a fixed load and the meshing stiffness as a function of the load. With the support stiffness shown in Table 2, the static meshing forces of Input Gear 1 with the lower and upper face gears were 8967 and 12,588 N, respectively. Furthermore, based on this meshing force, the meshing stiffness of Input Gear 1 with the lower face gear Kh16g and the upper face gear Kh17g was calculated for a given load. Kh16 and Kh17 are the meshing stiffnesses calculated based on the meshing stiffness proxy model. Figure 6 shows that Kh16 is different from Kh16g and that Kh17 and Kh17g differ in value.