When using Gaussian process (GP) machine learning as a surrogate model combined with the global optimization method for rapid optimization design of electromagnetic problems, a large number of covariance calculations are required, resulting in a calculation volume which is cube of the number of samples and low efficiency. In order to solve this problem, this study constructs a deep GP (DGP) model by using the structural form of convolutional neural network (CNN) and combining it with GP. In this network, GP is used to replace the fully connected layer of the CNN, the convolutional layer and the pooling layer of the CNN are used to reduce the dimension of the input parameters and GP is used to predict output, while particle swarm optimization (PSO) is used algorithm to optimize network structure parameters. The modeling method proposed in this paper can compress the dimensions of the problem to reduce the demand of training samples and effectively improve the modeling efficiency while ensuring the modeling accuracy. In our study, we used the proposed modeling method to optimize the design of a multiband microstrip antenna (MSA) for mobile terminals and obtained good optimization results. The optimized antenna can work in the frequency range of 0.69–0.96 GHz and 1.7–2.76 GHz, covering the wireless LTE 700, GSM 850, GSM 900, DCS 1800, PCS1900, UMTS 2100, LTE 2300, and LTE 2500 frequency bands. It is shown that the DGP network model proposed in this paper can replace the electromagnetic simulation software in the optimization process, so as to reduce the time required for optimization while ensuring the design accuracy.

At present, solving most of the problems concerning antennas relies on full-wave electromagnetic simulation software. However, using electromagnetic simulation software to analyze the antenna is not only complicated but also computationally expensive [

Convolutional neural network (CNN) is a type of feedforward neural network (FNN) that includes convolution calculations and has a deep structure, and it is also one of the representative algorithms of deep learning (DL). In DL, CNN can be understood as a deep neural network (DNN) that can reduce the dimension of data [

The basic structure of CNN consists of input layers, convolutional layers, pooling layers, fully connected layers, and output layers, among which the convolutional layers and the pooling layers usually have multiple layers according to the actual problem. The traditional CNN is composed of forward pass and back propagation, so BP is used to optimize the parameters of the NN and train the NN. We use PSO to optimize the parameters. Figure

Schematic diagram of convolutional layer and pooling layer.

Take Figure

The pooling layer is generally constructed on the next layer of the convolutional layer. It also consists of multiple feature surfaces, each of which corresponds to the unique feature surface of the previous layer. The feature of the pooling layer is that it does not change the number of feature surfaces. The number of neurons on the output feature surface of the pooling layer is calculated as follows:

The GP describes a functional distribution. It is a set of infinite random variables, and any subset of these variables conforms to the Gaussian distribution. Its properties can be determined by the average value function

Assume the finite data set

Joint Gaussian prior distribution composed of

On the premise that the testing point

The covariance function of the GP must meet the Mercer condition, that is, for any point set, a non-negative positive definite covariance matrix can be guaranteed. This study chooses the Ardmatern52 covariance function as the covariance function of the GP:

We adopted PSO algorithm for optimization. PSO algorithm is easy to implement, simple, with less parameters, and can effectively solve the global optimization problems [

The basic idea of the PSO algorithm is to accelerate each particle to approach the best position of itself and the swarm. In the solution space, the starting position and speed of the particles will be randomly set. During the iterative search process, the algorithm will record the best positions experienced by individual particles and swarms and the corresponding fitness function values. The speed and position update formula of the particle swarm algorithm is as follows:

The deep Gaussian process (DGP) network model is the combination of the CNN and the GP, which is shown in Figure

Deep Gaussian process network structure.

In the DGP network modeling method, the samples required for model training, that is, the training input and training output, can be obtained by the electromagnetic simulation software HFSS. In this paper, VBScript language is used to realize the data exchange between MATLAB software and HFSS software, which makes the acquisition of training data more concise and automatic. After obtaining the training data, it would be uniformly normalized. Assume that each group of input data is

The trained DGP network can finally be used for antenna optimization design. The process of optimization design is shown in Figure

Flowchart of optimal design.

In recent years, the 4G system of LTE has matured and developed worldwide, and the 5G communication technology has also been gradually and widely used [

The antenna in [_{11} less than 6 dB to cover the impedance bandwidth of 270 MHz (0.69 to 0.96 GHz) and 1.06 GHz (1.7 to 2.76 GHz), so that we can cover the wireless LTE 700, GSM 850, GSM 900, DCS 1800, PCS1900, UMTS 2100, LTE 2300, and LTE 2500 frequency bands.

Schematic diagram of the multiband microstrip antenna.

During the modeling process, the 20 size parameters of the antenna (as shown in Table

Optimization parameters of the multiband microstrip antenna.

Variable name | Variable range (unit: mm) | Variable name | Variable range (unit: mm) |
---|---|---|---|

8.5∼10 | PL4 | 3.5∼4.5 | |

15∼22 | PL5 | 12∼18 | |

6∼12 | PL6 | 5∼7 | |

1∼2 | PL7 | 7∼12 | |

3∼4 | PL8 | 5∼7 | |

1.5∼2.5 | Plg1 | 3.5∼4.5 | |

27∼29 | Lsg1 | 1∼3 | |

21∼24 | Lsg2 | 1∼3 | |

PL1 | 50∼54 | Lsg3 | 40∼45 |

PL2 | 4.5∼5.5 | Wsg | 1∼2 |

The proposed DGP network model used here has 3 convolutional layers and 2 pooling layers. The size of the convolution kernel of each convolutional layer is 1 × 2, the number of channels of convolutional layer 1 is 3, the number of channels of convolutional layer 2 is 1, the number of channels of convolutional layer 3 is 3, and the size of pooling layer of each layer is 1 × 2. Figure _{11} amplitude corresponding to the frequency points sampled in the frequency band. The specific frequency band range is 0.5 GHz–3 GHz, and the sampling interval is 0.04 GHz, with 63 frequency points in each group.

The DGP network model structure.

After training, we use PSO to optimize the design. The number of particles in the PSO algorithm is 20, the maximum number of iterations is 500, the acceleration constant is _{11} predicted by the proposed method and the simulation results of the electromagnetic simulation software HFSS, and Figure

Optimized parameters of the multiband microstrip antenna.

Variable name | Variable range (unit: mm) | Variable name | Variable range (unit: mm) |
---|---|---|---|

9.66 | PL4 | 4.02 | |

17.96 | PL5 | 17.66 | |

10.05 | PL6 | 6.27 | |

1.09 | PL7 | 11.79 | |

3.26 | PL8 | 6.75 | |

1.65 | Plg1 | 3.74 | |

27.56 | Lsg1 | 1.58 | |

22.32 | Lsg2 | 2.34 | |

PL1 | 52.11 | Lsg3 | 43.47 |

PL2 | 4.95 | Wsg | 1.068 |

S_{11} comparison results of the multiband microstrip antenna.

Field pattern of the antenna.

This study proposes a modeling method based on deep Gaussian process networks. In the framework of deep learning, this paper first utilizes the advantages of convolutional neural network and Gaussian process and creatively combines these advantages. Then, we take advantage of the convolutional neural network to reduce the input data dimension without losing data characteristics and finally use the Gaussian process adaptability to the nonlinear problem to predict antenna frequency, so as to guarantee the accuracy and reduce the calculation time, thereby improving the efficiency. Meanwhile, the proposed deep Gaussian process network model combined with the PSO algorithm is adopted to conduct optimization design. The optimized results are very close to the results obtained by the HFSS high-fidelity model simulation, indicating that the modeling method is sufficiently reliable. The optimized antenna size meets the requirements of the index, showing that the method has practical value in the antenna optimization design.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.

This work was supported by the National Natural Science Foundation of China (NSFC) under Grant no. 61771225, the Natural Science Foundation of Jiangsu Province of China under Grant no. BK20190956, and the Qinglan Project of Jiangsu Higher Education.