## 4 Method
### EIT Inverse Model
#### 4.1.1 Improved Levenberg-Marquardt Method  
LM algorithm has recently been utilized in the adaptation of arbitrarily connected neural (ACN) networks [41]. These ACN networks are capable of tackling more intricate problems while employing fewer neurons [42]. The LM algorithm is particularly effective for minimizing the loss function of the sum of squares error type. As a result, it offers fast training for neural network models that involve this type of error [43]. Implementations of the LM algorithm necessitate the computation of the Jacobian matrix, which is directly related to the total number of training patterns. Due to its rapid local convergence rates, the LM algorithm and its variations find extensive applications in solving non-smooth equation systems. It proves invaluable in addressing various concerns, including but not limited to non-linear complementarity, variational inequality, Karush-Kuhn-Tucker (KKT) non-linear programming, as well as mechanics and engineering problems [44]. By combining the advantages of stable gradient descent (GD) and the fast convergence near the extreme point of the GN, the LM algorithm overcomes their respective drawbacks. Specifically designed for solving the non-linear inverse problem, the LM method follows a two-step approach, involving the linearization of the non-linear inverse problem and the regularization of the iterative scheme [45]. Our proposed ILM algorithm is improved by the method in [46; 47; 48]. At the beginning, this method was employed to solve the non-linear least squares multiplication problem [49]. For regular issues of non-linear  
\[F(x)=0 \tag{10}\]  
where \(F(x)\): \(\mathbf{R}^{n}\longrightarrow\mathbf{R}^{m}\), and the diagram of proposed LM algorithm is shown in Figure 3.  
The results obtained from running the Broyden-Fletcher-Goldfarb-Shanno (BFGS), LM, and improved LM algorithms simultaneously are shown in Figure 4. Figure 4a displays the iteration time required to achieve the same termination condition for the objective function value. Figure 4b reveals that the ILM algorithm is similar to the classic LM algorithm in the early iterations but significantly diverges after a certain number of iterations. The ILM algorithm converges each parameter to the target value faster, resulting in faster running speed and significantly fewer iteration steps to reach the same termination condition. In summary, the ILM algorithm provides a search direction closer to the target descent direction, with faster convergence speed and fewer iteration steps. Figure 3: The diagram of the ILM algorithm. The following objective function can be minimized in order to generate the inverse EIT problem:  
\[\min_{x\in X=R^{n}}\|F(x)-y\|^{2} \tag{11}\]  
Assuming that \(F\) is Frechet differentiable and denoting its derivative as \(F(x)\), the adjoint operator is \(F^{\prime}(x)^{*}\), and the algorithm follows a basic form. We start with an initial guess value \(x_{0}\in X\), in the \(k\)-th step, let  
\[x_{k+1}= \begin{cases}x_{k}+d_{k}&\quad k=0,1,\dots\\ x_{k}&\quad\text{otherwise}\end{cases}  \tag{12}\]  
where \(d_{k}\) is the result of performing the proposed Newton step. \[d_{k}=\big{(}F^{\prime}(x_{k})^{*}F^{\prime}(x_{k})+\mu_{k}I\big{)}^{-1}F^{ \prime}(x_{k})^{*}(y-F(x_{k})) \tag{13}\]  
where \(\mu_{k}>0\) is called the LM parameter. The selection of the parameter \(\mu\) is the most crucial stage in the LM algorithm. When the objective function (15) does not contain explicit regularisation, \(Reg(\sigma)=0\). The quadratic term of Equation (17) has a Taylor expansion that is given by  
\[F(\sigma_{0}+\phi\sigma)=F(\sigma)+F^{\prime}(\sigma)(\phi\sigma)+\frac{1}{2} F^{\prime\prime}(\sigma)(\phi\sigma)^{2} \tag{14}\]  
To facilitate the substitution of GCNs for CNNs in the model-based learning method aimed at addressing non-linear inverse problems, the data represented on the finite element grid are converted into graph data. Graph data consist of nodes connected by edges. This study focuses on undirected, unweighted homogeneous graphs.