This study proposes a neural network-family competition genetic algorithm (NN-FCGA) for solving the electromagnetic (EM) optimization and other general-purpose optimization problems. The NN-FCGA is a hybrid evolutionary-based algorithm, combining the good approximation performance of neural network (NN) and the robust and effective optimum search ability of the family competition genetic algorithms (FCGA) to accelerate the optimization process. In this study, the NN-FCGA is used to extract a set of optimal design parameters for two representative design examples: the multiple section low-pass filter and the polygonal electromagnetic absorber. Our results demonstrate that the optimal electromagnetic properties given by the NN-FCGA are comparable to those of the FCGA, but reducing a large amount of computation time and a well-trained NN model that can serve as a nonlinear approximator was developed during the optimization process of the NN-FCGA.
Nowadays, with the improving performance of high-speed computing systems and progress in computational electromagnetics, the computationally intensive full-wave EM analysis (i.e. method of moments (MoM) [
Evolutionary computing is a branch of artificial intelligence (AI) and is based on direct and adaptive search techniques. The algorithms used in these evolutionary computing strategies are, in general, inspired by biological or sociological motivations. A significant advantage for using these strategies is that most evolutionary computing strategies are conceptually very straightforward and do not require any information pertaining to the gradient, which in turn allows the global optimality of objective functions that are rough, discontinuous, and multimodal. In particular, the genetic algorithm (GA) has been rapidly accepted in the electromagnetic research community and has been extensively studied in a variety of electromagnetic optimization problems, such as the optimization of broadband or multiband antennas [
The GA is originally proposed by Holland and his colleagues in 1975. It borrows the natural evolution mechanisms to find out the optimal solutions to the user-defined problems [
In this paper, we intend to propose a fast and effective optimization strategy-neural network-family competition genetic algorithm (NN-FCGA) to solve both the computing time and search result accuracy problems. The motivation behind the NN-FCGA is that it combines the good approximation performance of neural networks (NN) for the complicated analysis and the robust and effective optimum search ability of the FCGA to achieve the rapid and effective electromagnetic optimization.
This paper is organized in the following manner. The next section will introduce the GA, FCGA, NN, and NN-FCGA. In Section
As mentioned earlier, the GA was inspired by evolutionary genetics. In the traditional real-coded GA, each variable that requires optimization is regarded as a
Flowchart for (a) the GA and (b) the family competition in the reproduction process of the FCGA.
Recently, the concept of the family competition has been added to the reproduction process of evolutionary algorithms [
The settings used in the GA-based optimization algorithms (including the GA, FCGA, and NN-FCGA), unless otherwise noted, are detailed below and will be used throughout this study. In this study, the population size and total number of generations were set to 40 and 60, respectively. For the selection procedure, the
For the mutation procedure, an adaptive Gaussian mutation is used, where each gene is possibly replaced by an arbitrary number in the vicinity of the original individual with the mutation probability set to
Neural network (NN), which is also a branch of AI, is a computational model based on the concept of biological neural networks, as shown in Figure
Schematic diagram of a general back-propagation neural network.
The steps in the algorithm used in the NN-FCGA hybrid method for electromagnetic optimization are described as follows. In the initial population, many individuals are randomly generated and their fitness values are evaluated by performing the full-wave EM analysis. All individuals’ gene information and their corresponding fitness values are collected as the NN’s training samples to construct an NN approximation model. The values of all genes in an individual and their corresponding fitness values can be seen as the input and output of NN, respectively. In the next generation, the reproduction operation in the FCGA is performed to form a family. The fitness value of each family member is approximated by the NN instead of the relatively time-consuming full-wave EM analysis. The two fittest members are selected for use in creating a new generation. To improve the accuracy of NN, the fitness values of the elites (the champions in each of the families) are recalculated using the full-wave EM analysis. The updated fitness and gene information of the elites are then feedback to the NN in order to yield a more accurate approximation model.
Steps (2) to (4) are repeated until the stopping criterion of NN-FCGA is satisfied; in our case, the criterion is met when 60 generations have been generated.
In the
Our study utilizes a typical back-propagation neural network (BPNN) composed of three basic layers: input, hidden, and output. The diagram of the BPNN is shown in Figure
The various algorithms (GA, FCGA, NN, and FCGA) were written in the open-source program, Python; a computer intelligence technique optimization platform,
In this section, a low-pass filter and three polygonal electromagnetic absorbers as design examples are presented for the NN-FCGA. To provide a baseline for comparison, these examples are also presented for the FCGA and GA.
Filtering components have been widely used in microwave circuits. In the following, we will study the three GA-based optimization algorithms by applying them to a planar filter. A low-pass planar filter model [
Geometry and design parameters of the low-pass planar filter.
Here, the commercially available software CST Microwave Studio is used for the full-wave analysis and the calculated transmission and reflection coefficients are gathered for the fitness evaluation. This software employs the FDTD method coupled with the perfect boundary approximation and the finite integration technique [
Under the same predefined design variables and fitness function, the optimization was performed using the GA, FCGA, and NN-FCGA with the same empirical optimization parameters defined in the previous section. We note that the same initial population was used for all three cases. Figure
Optimized design parameters of the low-pass planar filter for the GA, FGGA, and NN-FCGA.
Ref [ | GA | FCGA | NN-FCGA | |
---|---|---|---|---|
0.4 | 0.570 | 0.035 | 0.030 | |
0.9 | 0.937 | 1.500 | 1.504 | |
0.5 | 0.086 | 0.032 | 0.032 | |
2.0 | 0.928 | 2.461 | 2.816 | |
0.03 | 0.025 | 0.076 | 0.053 | |
2.0 | 3.452 | 3.682 | 3.363 |
Best fitness plotted as a function of number of generations for the GA, FCGA, and NN-FCGA.
(a)Transmission coefficient versus frequency for the GA (red line), FCGA (green line), and NN-FCGA (blue line). (b) Optimized shapes for the GA, FCGA, and NN-FCGA.
It is clearly seen that the GA shows better results than the original design [
We now consider the issue of computation time. The execution time for the FDTD simulation for a particular filter shape is about 1.5 minute on a single processor. The training process and fitness evaluation for the NN take about 3 minutes and less than 0.1 second, respectively. Therefore, the computation time (on an HP Pentium IV-3.06 GHz PC with 3.37 GB RAM) for each generation of the GA and NN-FCGA are 60 minutes and 63 minutes, respectively; it is worth noting that both algorithms yield nearly the same computation time. In contrast, the computation time for each generation of the FCGA depends on the number of family members whose fitness value is required to be evaluated and it would linearly increase with increasing the number of children. The total computational time for GA, NN-FCGA, and FCGA are 60, 63, and 600 hours, respectively.
The role of NN in the NN-FCGA hybrid method is to act as a nonlinear function approximator, which can estimate the fitness value of a particular filter shape and replace the time-consuming FDTD simulation. The NN is first trained using the FDTD simulation results obtained from the initial population. Then, in each generation, the surviving individuals, which are chosen from the numerous families by the NN, are recalculated using the FDTD simulation. The results are then sent back to the NN in order to retrain the network model, thus yielding a more accurate one. After the network has been trained, the fitness value for an arbitrary individual can be readily obtained by the NN in a few microseconds. Since the NN takes only a small amount of time for training its network model and fitness evaluation, the NN-FCGA can save a large amount of computation time from fitness evaluation in the family competition process. This allows us to apply the family competition strategy to practical designs with complex and time-consuming models for the fitness evaluation. In short, as compared to the FCGA, the NN-FCGA can save a great deal of computation time, while maintaining the convergence speed and optimization performance. Therefore, the NN-FCGA appears to be the most efficient one among the three algorithms.
Moreover, this NN model can help us get more insight into the influence of each physical parameter on the fitness value. To illustrate this idea, we used the NN-FCGA optimized filter as an example to study the dependence of the fitness value on the width of the metallic patches
Dependences of the fitness value on the width of the metallic patches. The fitness values are obtained using the full-wave analysis (solid line) and the NN (dash line).
We also note that, in the relative high fitness region, especially for the
The anechoic chamber plays a very important role in the experimental characterization of antennas and scatters. Typically, the walls of the anechoic chamber were coated with a lossy material to simulate a free-space environment and backed with a conducting ground to avoid the outside interference. To avoid the reflections at the air-absorber interfaces, the front surface of the absorber is designed using a periodic and well-shaped structure. In order to achieve better absorbing performance, the optimization design of the absorber structures has been an active research topic for the past several years [
Geometry and design parameters of the electromagnetic absorber.
Figures
Best fitness plotted as a function of number of generations for (a) the TM polarization, (b) TE polarization, and (c)
(a)–(c) Reflection coefficient versus frequency for TM polarization, TE polarization,
Prediction performance of the NN for the TM polarization absorber design.
To summarize, two different types of electromagnetic designs, the low-pass planar filter and the electromagnetic absorber, are optimized using three GA-based optimization algorithms: the GA, the FCGA, and the proposed NN-FCGA. Among them, the GA has the worst optimum searching ability and the FCGA has the best optimum searching ability. However, the FCGA is extremely time-consuming, thus limiting the number of practical applications. There are several advantages that make the NN-FCGA a good alternative. First, the NN-FCGA can further improve the optimization results of the traditional GA, while the additional numerical optimization time is almost negligible. Although the NN-FCGA’s optimization results may not exceed those obtained using the FCGA, the NN-FCGA’s numerical optimization time is several times shorter than that for the FCGA. Moreover, a well-trained NN model can be obtained during the NN-FCGA’s optimization process. This model can help us get more insights into the influence of design parameters on the overall device performance. The design criteria can also be easily applied to any electromagnetic or other physical system whose objective characteristics can be known to the design parameters and the algorithm is thus completely general.
The authors would like to thank Professor C. T. Sun of Department of Computer Science, National Chiao Tung University, Taiwan, Dr. J. H. Tsai of Department of Simulation and Modeling, National Nano Device Laboratories, Taiwan, and Professor T. J. Yen of Department of Material Science and Engineering, National Tsin Hua University, Taiwan, for their fruitful discussion on evolutionary computing and full-wave analysis using CST Microwave Studio.