## 2 Characteristics of Surface Potential Distribution for DC Grounding Electrode
### Introduction of OSDM Model
#### 3.1.2 Input and Output Variable Selection  
According to the sample data set of the previous section, the regression prediction model is constructed to realize the accurate prediction of the surface potential distribution of the DC grounding electrode under different breakage conditions. The current distribution matrix \(I\) = \([I_{1},I_{2},I_{3},\ldots,I_{k},\ldots,I_{Q}]^{T}\) of all the feeder rods of the DC grounding electrode is taken as the input part of the prediction model. The corresponding surface potential matrix \(U\) = \([U_{1},U_{2},U_{3},\ldots\,U_{P}]^{T}\) is the output part of the prediction model. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Number**} & \multicolumn{4}{c}{**Input**} & \multicolumn{4}{c}{**Output**} \\  & \(\boldsymbol{I_{1}}\) & \(\boldsymbol{I_{2}}\) & \(\boldsymbol{I_{3}}\) & \(\boldsymbol{I_{4}}\) & \(\ldots\) & \(\boldsymbol{U_{1}}\) & \(\boldsymbol{U_{2}}\) & \(\boldsymbol{U_{3}}\) & \(\boldsymbol{U_{4  
During the operation of the DC grounding electrodes, the surface potential distribution exhibits complex spatial response characteristics when breakage failures occur at different locations of the feeder rod. This mapping relationship between failure locations and surface potentials demonstrates high-dimensional nonlinear characteristics [28]. On the one hand, the spatial coordinates of the fracture point and the potential value of the P monitoring points constitute a multidimensional solution space; meanwhile, the electromagnetic field distribution in the grounding electrode domain is influenced by multiphysics factors, including material properties, soil stratification, and current coupling, making it challenging to establish explicit functional relationships. To solve the problem of low computational efficiency and the long time of traditional co-simulation methods, this study adopts a data-driven modeling strategy to construct a radial basis function neural network (RBFNN) surrogate model, as illustrated in Figure 14, which replaces conventional finite element models. The selection of RBFNN [29; 30; 31] is mainly based on the following considerations: Firstly, the network can effectively fit the complex relationship between the fault and response of grounding electrode fracture through the nonlinear mapping characteristics of the Gaussian kernel function, and its universal approximation theorem theoretically guarantees the approximation ability to any continuous function. Secondly, the local response characteristics of RBFNN significantly improve the training speed and effectively avoid the problem of gradient disappearance. Finally, the network structure has mathematical homology with support vector machine, which is suitable for dealing with regression problems with small samples and high-dimensional output. The specific configuration of the network architecture is as follows: the input layer parameter is the \(Q\)-dimensional current distribution matrix \(I=[I_{1},I_{2},I_{3},\ldots,I_{k},\ldots,I_{Q}]^{T}\), which characterizes the working current of each feeder rod. The number of neurons in the hidden layer is equal to the number of samples in the training set. The output layer parameters are the \(P\)-dimensional potential matrix \(U=[U_{1},\,U_{2},\,U_{3},\,\ldots,\,U_{P}]^{T}\), corresponding to the surface circumferential measurement potential value of the DC grounding electrode. MATLAB R2020b programming software is used to realize the training and predictions of the model. The training set and test set of model prediction samples under different types are set as shown in Table 4.  
The absolute error between the actual value and the predicted value of the test set under some DC grounding electrode breakage types, and the root mean square error (RMSE) of the test set, are illustrated in Figures 15 and 16.