This article introduces the deep neural network method into the field of high-dimensional microwave modeling. Deep learning is nowadays highly successful in solving complex and challenging pattern recognition and classification problems. This article investigates the use of deep neural networks to solve microwave modeling problems that are much more challenging than that solved by the previous shallow neural networks. The most commonly used activation function in the existing deep neural network is the rectified linear unit (ReLU), which is a piecewise hard switch function. However, such a ReLU is not suitable for microwave modeling where the input-output relationships are smooth and continuous. In this article, we propose a new deep neural network to perform high-dimensional microwave modeling. A smooth ReLU is proposed for the new deep neural network. The proposed deep neural network employs both the sigmoid function and the smooth ReLU as activation functions. The new deep neural network can represent the smooth input-output relationship that is required for microwave modeling. An advanced three-stage deep learning algorithm is proposed to train the new deep neural network model. This algorithm can determine the number of hidden layers with sigmoid functions and those with smooth ReLUs in the training process. It can also overcome the vanishing gradient problem for training the deep neural network. The proposed deep neural network technique can solve microwave modeling problems in a higher dimension than the previous neural network method, i.e., shallow neural network method. Two high-dimensional parameter-extraction modeling examples of microwave filters are presented to demonstrate the proposed deep neural network technique.

This article presents an overview of artificial neural network (ANN) techniques for a microwave computer-aided design (CAD). ANN-based techniques are becoming useful for performing forward/inverse modeling for active/passive components to enhance a circuit design. With measured or simulated data of microwave devices, ANNs can be trained to learn relevant microwave relationships, which are, otherwise, computationally expensive or for which efficient analytical formulas are not available. Fundamental concepts of the ANN structure and training, such as feedforward neural networks (FFNNs), recurrent neural networks (RNNs)/dynamic neural networks (DNNs)/time-delay neural networks (TDNNs), deep neural networks, and neural network training and extrapolation, are described. Knowledge-based neural networks (KBNNs) are described for improving the accuracy and reliability of modeling and design optimization. Various advanced ANN techniques, such as neuro-transfer function (neuro-TF) modeling, neural network inverse modeling, and deep neural network modeling, are discussed. The existing and emerging applications of ANN in microwave CAD are identified, such as electromagnetic (EM)/multiphysics modeling, modeling of nonlinear circuits and transistors, filter design, very large-scale integration (VLSI) interconnects, oscillator, transmitter and receiver modeling, and CAD applications in such as gallium nitride (GaN) high electron-mobility transistor (HEMT), wireless power transfer (WPT), microelectromechanical system (MEMS), and substrate-integrated waveguide (SIW).

Space mapping is a recognized surrogate-based optimization method to accelerate the electromagnetic (EM) design. In this article, for the first time, space mapping is elevated from solving the problem of EM optimization to solving the problem of multiphysics optimization for high power microwave filters. Multiphysics analysis, which involves the EM domain with other physics domains, is increasingly important for high-performance microwave components to obtain an accurate system design. To speed up the multiphysics design, a space-mapping-based surrogate model including a coarse model and two mapping functions is proposed in this article. We propose to use EM single physics responses as the coarse model to provide good approximations to fine model multiphysics responses. To avoid repetitive EM simulations during the surrogate model training and optimization process, the coarse model is developed using an artificial neural network (ANN). Frequency mapping and explicit input mapping are further performed to develop the proposed surrogate model. Multiple EM and multiphysics training samples are evaluated in parallel to develop the surrogate model. A trust-region algorithm, tailored to the space-mapping-based multiphysics optimization technique, is proposed to improve the convergence. By exploiting the knowledge of the coarse model established by relatively inexpensive EM data, the proposed technique can provide a larger and more efficient optimization update in each optimization iteration, consequently obtaining optimal solutions faster than the existing multiphysics optimization without space mapping. Two examples of multiphysics optimization of high-power microwave filters are used to validate the proposed technique.

Artificial neural network technique has gained recognition as a powerful technique in microwave modeling and design. This paper proposes a novel deep neural network topology for parametric modeling of microwave components. In the proposed deep neural network, the outputs are S-parameters. The inputs of the proposed model include geometrical variables and the frequency. We divide the hidden layers in the proposed deep neural network topology into two parts. Hidden layers in Part I handle both the geometrical inputs and the frequency inputs while hidden layers in Part II only handle the geometrical inputs. In this way, more training parameters are utilized to specifically learn the relationship between the Sparameters and the geometrical variables, which are more complicated than that between the S-parameters and the frequency. The purpose is to reduce the total number of training parameters in the deep neural network model. New formulations are derived to calculate the derivatives of the error function with respect to training parameters in the deep neural network. Taking advantage of the calculated derivatives, we propose an advanced two-stage training algorithm for the deep neural network. The two-stage training algorithm can determine the number of hidden layers in both parts during the training process and guarantee that the proposed deep neural network model can achieve the required model accuracy. The proposed deep neural network can achieve similar model accuracy with less training parameters compared to the commonly used fully connected neural network. The proposed technique is demonstrated by two microwave parametric modeling examples.

This article proposes a novel training approach for parametric modeling of microwave passive components with respect to changes in geometrical parameters using matrix Padé via Lanczos (MPVL) and electromagnetic (EM) sensitivities. In the proposed approach, the EM responses of passive components versus frequency are represented by pole-zero-gain transfer functions. The relationships between the poles/zeros/gain in the transfer function and geometrical variables are learned by neural networks. To generate training data, we apply the MPVL algorithm to compute (or recompute) the poles/zeros each time we change the geometrical parameters. However, the indices of the poles/zeros after the recomputation may not have clear correspondences with those before the recomputation, posing additional challenges to predict the poles/zeros reliably for a new change of geometrical parameters. We propose a novel sensitivity-analysis-based pole-/zero-matching algorithm to obtain the correct correspondences between the poles/zeros at different geometrical parameter values. The proposed algorithm exploits the EM sensitivities, which provide useful information for the direction of movement of the poles/zeros, to predict the new positions of the poles/zeros for each change of geometrical parameters in the multidimensional parameter space. The predicted new positions are then used to guide the matching process of poles/zeros between different geometrical parameter values. Using the matched poles/zeros to train the neural networks allows us to have fast and reliable predictions for the poles/zeros subject to large geometrical variations, consequently increasing the accuracy and robustness of the overall model. Compared with the existing methods, the proposed approach can obtain better accuracy in challenging applications involving large geometrical variations. Three microwave examples are used to illustrate the proposed approach.

Electromagnetic (EM) topology optimization aims at finding the optimal shape and topology of the EM structure by adding or eliminating materials, such as metal within the design space. This optimization problem cannot be solved by the widely used surrogate-based optimization because different shapes and topologies cannot be related by changing the values of a given set of geometrical parameters. EM simulations are needed when the shape and topology change. This article proposes an efficient EM topology optimization technique for microwave component design. The finite element method (FEM) is used for EM simulation. We propose a new method to integrate the matrix Padé via Lanczos (MPVL) and the Householder formula. The proposed method combines the advantages of the MPVL and the Householder formula so that the effort of solving the large FEM matrix equation at many frequencies is reduced to the effort of solving only a small matrix problem at a single frequency point, thereby speeding up the topology optimization process. We further propose a new method to reduce the small matrix problem into an even smaller one by exploiting the inheritance pattern of a genetic algorithm (GA). New formulations are derived to exploit the common factors between new children EM structures and the parent EM structures in each generation of GA so that computational expense can be further reduced. The proposed technique incorporating advanced FEM-MPVL and the inheritance pattern of GA can greatly accelerate the topology optimization process. Three microwave examples are used to demonstrate the proposed technique.