## 2 Design Algorithm
### The Architecture of the Proposed ANN Model  
The proposed model includes a multi-layer feed forward network neural network (ANN), which can predict the desired output parameters by considering the circuit parameters. The proposed model works as a surrogate model to predict the parameters of the proposed filter. Figure 2 shows the proposed MLP structure for the defined artificial network. According to this figure, the input parameter is connected to output nodes by two hidden layers, including seven neurons in each hidden layer. Additionally, the values of fc (GHZ), \(\zeta\) (dB/GHz), BW (MHZ), and RL (dB), which are desirable resonator parameters, are considered the output of the proposed ANN model.  
A flowchart of the proposed structure of the ANN is illustrated in Figure 3. As shown, within the first step, the structure of the network, the hidden layer numbers, and the neuron numbers are considered, due to the input and output parameters and the problem conditions. In the next steps, the weights and biases of the ANN are defined randomly; then, as explained in the flow chart, these values are calculated using the proposed ANN model. Finally, after reaching the desired accuracy, the ANN model is created.  
In a multi-layer feed-forward neural network, the values of neurons in the hidden layers are updated through a process called forward propagation and backpropagation. The first step is initialization, in which the weights and biases of the neural network are  
Figure 3: The flowchart of the proposed structure of the ANN.  
Figure 2: Proposed structure of the ANN model, including two hidden layers with seven neurons in each hidden layer.  
randomly initialized. The next step is forward propagation, in which input values are provided to the input layer, the weighted sum of inputs for each neuron in the first hidden layer is computed, and an activation function is applied to the computed sum, producing an output value for each neuron in the first hidden layer. The same process is repeated for the second hidden layer, using the output values from the first hidden layer as inputs. Then, the weighted sum of inputs for each neuron in the output layer is computed, and an activation function is applied to the computed sum, thus generating the final output values of the neural network [46; 47]. In the proposed ANN model, the hyperbolic tangent activation function ('tansig') is used for the first and the second hidden layers, while the linear activation function ('purelin') is used for the output layer.  
The next step is backpropagation, in which the error between the predicted output values and the actual output values is calculated; then, the weights and biases in the output layer are updated using the calculated error and an optimization algorithm. The next step is backpropagating the error to the second hidden layer; then, the error contribution of each neuron in the second hidden layer is calculated, based on the weights connecting it to the output layer and the propagated error from the output layer. In the next level, the weights and biases are updated in the second hidden layer, using the calculated error contribution and the optimization algorithm. By repeating the same process for the first hidden layer, the error from the second hidden layer is backpropagated, and the corresponding weights and biases are updated [47; 48].  
Then, the forward propagation and backpropagation steps are repeated for each training sample in the training set. Additionally, the weights and biases are adjusted iteratively, based on the error calculated for each sample. During training, the weights and biases are updated gradually, allowing the neural network to learn the underlying patterns and to optimize its predictions. By iteratively updating the weights and biases through forward propagation and backpropagation, the neural network adjusts its internal parameters to minimize the error between the predicted and actual output values, ultimately improving its ability to make accurate predictions [49].