Surrogate Modeling for High-Frequency Design

The following sections are included:

• Preface
• About the Editors
• List of Contributors
• Acknowledgments
• Contents

The primary topic of the book is surrogate modeling and surrogate-based design of high-frequency structures. The purpose of the first two chapters is to provide the reader with an overview of the two most important classes of modeling methods, data-driven (or approximation), as well as physics-based ones. These are covered in Chapters 1 and 2, respectively. The remaining parts of the book give an exposition of the specific aspects of particular modeling methodologies and their applications to solving various simulation-driven design tasks such as parametric optimization or uncertainty quantification.

Data-driven models are by far the most popular types of surrogates. This is due to several reasons, including versatility, low evaluation cost, a large variety of matured methods, and — important from the point of view of practical utility — widespread availability through third-party toolboxes implemented in programming environments such as Matlab. This chapter covers the fundamentals of approximation-based modeling. We discuss the surrogate modeling flow, design of experiments, selected modeling methods (e.g., kriging, radial basis functions, support vector regression, or polynomial chaos expansion), as well as discuss model validation approaches. The presented material is intended to provide the readers who are new to the subject with the basics necessary to understand the remaining parts of the book. On the other hand, it is by no means exhaustive, and the readers interested in a more detailed exposition can refer to a rich literature of the subject (e.g., Queipo et al., 2005; Forrester and Keane, 2009; Biegler et al., 2014; Chugh et al., 2019; Jin, 2005; Santana-Quintero et al., 2010; Gorissen et al., 2009).

Chapter 1 was focused on data-driven (or approximation-based) modeling methods. The second major class of surrogates are physics-based models outlined in this chapter. Although they are not as popular, their importance is growing because of the challenges related to construction and handling of approximation surrogates for many real-world problems. The high cost of evaluating computational models, nonlinearity of system responses, dimensionality issues as well as combinations of these factors, may lead to a situation, where setting up a data-driven model is not possible or at least not practical. On the other hand, incorporation of the problem-specific knowledge, typically in the form of a lower-fidelity computational model, often alleviates the aforementioned difficulties. The enhancement of the low-fidelity models using a limited amount of high-fidelity data is the essence of physics-based surrogate modeling. This chapter provides a brief characterization of this class of surrogates, explains the concept and various types of low-fidelity models, as well as outlines several specific modeling approaches, also in the context of surrogate-assisted optimization.

This chapter provides an overview of parametric modeling of microwave components using combined neural network and transfer function (neuro-transfer function or neuro-TF). Transfer functions are used to represent the electromagnetic (EM) responses of passive components versus frequency. With the help of the transfer function, the nonlinearity of the neural network structure can be significantly decreased. We first introduce the neuro-TF modeling approach in rational format. Following that, we review the recent neuro-TF modeling approach in pole/zero format, where a pole/zero-matching algorithm is needed for addressing the issue of mismatch of poles and zeros before training the overall neuro-TF model. The neuro-TF modeling technique in pole/residue format is also reviewed. The orders of the pole–residue transfer functions may vary over different regions of geometrical parameters. A pole–residue tracking technique can be used to solve this order-changing problem. As a further advancement, we discuss the sensitivity analysis-based neuro-TF modeling technique. The purpose is to increase the model accuracy by utilizing EM sensitivity information and to speed up the model development process by reducing the number of training data required for developing the model. After the modeling process, the trained model can be used to provide accurate and fast prediction of the EM responses with respect to the geometrical variables and can be subsequently used in the high-level circuit and system design.

Antenna design automation via optimization continues to attract a lot of interest. This can be mainly attributed to the expected efficiency improvement that numerical optimization methods for antenna design offer in terms of the overall design time compared to traditional antenna design methodologies. Numerical optimization methods for antenna design still present several challenges in terms of their efficiency and optimization ability to handle contemporary antenna designs which are typically complex in terms of topology and performance requirements. Today, parallel computing and machine learning have been demonstrated as proficient ways of addressing the challenges of efficiency and optimization capability. In this chapter, the application of a state-of-the-art antenna optimization method which combines the advantages of parallel computing and machine learning via surrogate-based optimization (SBO) is presented. The method is the parallel surrogate model-assisted hybrid differential evolution for antenna optimization (PSADEA). The efficacy of PSADEA is demonstrated in this chapter through the design and optimization of a compact slotted monopole antenna for ultra-wide band (UWB) body-centric imaging applications and a novel compact quadruple-band indoor base station antenna for 2G/3G/4G/5G systems. The close agreement between the simulated and measured results for the fabricated prototypes of the PSADEA-synthesized designs for the two antennas validate the design solution quality of PSADEA. Further to this, the performance of PSADEA is compared with popular and commercially available Computer Simulation Technology Microwave Studio (CST-MWS) optimizers using the UWB slotted monopole antenna design problem. The outcomes of the comparisons reveal that PSADEA obtains very satisfactory design solutions repeatedly within an affordable computational cost in each run, whereas CST-MWS optimizers fail to obtain an adequate design solution in all runs.

As microprocessor design scales to nanometric technology, traditional post-silicon validation techniques are inappropriate to get a full system functional coverage. Physical complexity and extreme technology process variations introduce design challenges to guarantee performance over different process, voltage, and temperature (PVT) conditions. In addition, there is an increasingly higher number of mixed-signal circuits within microprocessors; many of them employed in high-speed input/output (HSIO) links. Improvements in signaling methods, circuits, and process technology have allowed HSIO data rates to scale beyond 10 Gb/s, where undesired effects can create multiple signal integrity problems. With all of these elements, post-silicon validation of HSIO links is tough and time-consuming. One of the major challenges in post-silicon electrical validation of HSIO links lies in the physical layer (PHY) tuning process, where equalization techniques are used to cancel these undesired signal integrity effects. Typical current industrial practices for PHY tuning require massive lab measurements, since they are based on exhaustive enumeration methods. In this chapter, dedicated surrogate-based modeling and optimization methods, including space mapping, are described to efficiently tune transmitter and receiver equalizers of HSIO links. The proposed methodologies are validated by laboratory measurements on realistic industrial post-silicon validation platforms.

Design of contemporary antenna systems is a challenging endeavor. The difficulties are partially rooted in stringent specifications imposed on both electrical and field characteristics, demands concerning various functionalities, but also constraints imposed upon the physical size of the radiators. Furthermore, conducting the design process at the level of full-wave electromagnetic (EM) simulations, otherwise dictated by reliability, entails considerable computational expenses. This is particularly troublesome for the procedures involving repetitive EM analyses, e.g., parametric optimization. Utilization of fast surrogate models as a way of mitigating this issue has been fostered in the recent literature. Notwithstanding, construction of reliable surrogates is hindered by highly nonlinear antenna responses and even more by the utility requirements: design-ready models are to be valid over wide ranges of operating conditions and geometry parameters. Recently proposed performance-driven modeling, especially the nested kriging framework, addresses these difficulties by confining the surrogate model domain to a region that encapsulates the designs being optimum with respect to the relevant figures of interest. The result is a dramatic reduction of the number of training samples needed to render a usable model.

This chapter discusses an enhancement of the nested kriging methodology that involves a variable-thickness domain. This is an important advancement over the basic framework with the major benefit being a further and significant (up to 70%) reduction of the training data acquisition cost without compromising the model predictive power. It is achieved while ensuring that the model domain covers the regions containing optimum designs for various sets of performance specifications. Our considerations are illustrated using two examples of microstrip antennas. Numerical results are supported by experimental validation of the selected designs obtained by means of the discussed models.

In recent years, surrogate-assisted optimization algorithm is commonly used in antenna and microwave component design optimization. Sampling method is the first step in data-driven surrogate model establishment. The purpose of the chapter is to explain and clarify various sampling methods and to inspire new implementations and applications. Four widely used conventional sampling methods are reviewed including the full factorial design, Monte Carlo sampling, Latin hypercube sampling, and space-infill sampling method. Two adaptive sampling methods are reviewed and explained. A hybrid sampling method considers both superiority of objective function values and uniformity of sample distribution, therefore, improving the performance of surrogate modeling and optimization at the same time. An adaptive sampling region updating strategy establishes model and performs optimization iteratively in local region(s). The size and the direction of movement of each local region are adaptively changed. As a result, the sampling efficiency is improved. These methods are valuable for EM-simulation-based data-driven surrogate modeling and optimization.

System design centering process searches for nominal values of system designable parameters which maximize the probability of satisfying the design specifications (yield function). Statistical design centering implements a statistical analysis method such as Latin hypercube sampling (LHS) for yield function estimation, and explicitly optimizes it. In this chapter, we introduce a new statistical design centering technique for microwave system design. The technique combines a modified surrogate-based derivative-free trust region (TR) optimization algorithm, and the generalized space mapping (GSM) technique. The modified TR algorithm is a derivative-free optimization algorithm that employs quadratic surrogate models to replace the computationally expensive yield function in the optimization process. The modified TR algorithm allows the shape of the TR to be dynamically adapted to a hyper-elliptic shape accompanied with the quadratic model. This improves the accuracy and convergence properties of the algorithm. Generally, TR algorithms exhibit global convergence features irrespective of the starting point setting. The new design centering approach utilizes the GSM technique to approximate the feasible region in the design parameter space with a sequence of iteratively updated space mapping (SM) surrogates. At each SM iteration, the modified TR algorithm optimizes the yield function for the current SM region approximation to get a better center. Two microwave circuit examples are used to show the effectiveness of the new design centering technique to obtain an optimal design in few SM iterations. In the design process, we employ Sonnet em for the bandstop microstrip filter design and CST Studio Suite for the ultra-wideband (UWB) multiple-input–multiple–output (MIMO) antenna.

Uncertainty quantification is an important aspect of engineering design, also pertaining to the development and performance evaluation of high-frequency structures systems. Manufacturing tolerances as well as other types of uncertainties, related to material parameters (e.g., substrate permittivity) or operating conditions (e.g., bending) may affect the characteristics of antennas or microwave devices. For example, in the case of narrow- or multi-band antennas, this usually leads to frequency shifts of the operating bands. Quantifying these effects is imperative to adequately assess the design quality, either in terms of the statistical moments of the performance parameters or the yield. Reducing the system sensitivity to parameter deviations is even more essential when increasing the probability of the system satisfying the prescribed requirements is of concern. The prerequisite of such procedures is statistical analysis, normally carried out at the level of full-wave electromagnetic (EM) analysis. Although necessary to ensure reliability, it entails considerable computational expenses, often prohibitive. Following the recently fostered concept of constrained modeling, this chapter discusses a simple technique for rapid surrogate-assisted yield optimization of high-frequency structures. The keystone of the approach is an appropriate definition of the optimization domain. This is realized by considering a few pre-optimized designs that represent the directions featuring maximum variability of the circuit responses (particularly the parts thereof that affect the yield value in the most significant way) with respect to its geometry parameters. Due to a small volume of such a domain, an accurate replacement model can be established therein using a small number of training samples, and employed to improve the yield. The implementation details are tailored to a particular type of device. Verification results obtained for several antenna structures and a miniaturized rat-race coupler indicate that the optimization process can be accomplished at low cost of a few dozen of EM simulations. The result reliability is validated through comparisons with EM-based Monte Carlo simulations.

RF circuit optimization is generally formed as a multi-objective optimization function. Weight setting is important can be guided by a principal component analysis (PCA) step that can also be used to screen parameters. This is illustrated on a circuit example. Two surrogate model-based optimization approaches are presented, one based on Bayesian Optimization, the second on a candidate search method. Bayesian optimization is best employed on all continuous domain problems while candidate search can be used on mixed integer-continuous problems. We implement co-Kriging in the candidate search algorithm to permit mixed fidelity electromagnetic evaluation. These algorithms are illustrated with several circuit examples.

In this chapter, we describe an advanced modeling method oriented towards the development of an automated user-friendly tuner calibration procedure based on adaptive sampling and modeling techniques. The potential shown by the proposed technique can be a stepping stone towards further developments that could be also introduced into commercial products. Pertinent numerical results are presented to validate the proposed method.

This chapter focuses on applying surrogate models in a low noise amplifier (LNA), which is one of the main building blocks of modern communication circuits. In microelectronics, surrogate modeling has been shown to yield promising results to describe complex design surfaces. Using this feature, surrogate models can be established to extract specific design insights that enable the development of integrated circuits with higher performance and reliability. Examples of surrogate modeling in LNAs covered here explore the relationships between the voltage gain, noise figure, as well as input and output reflection coefficients. In particular, the variability of the voltage gain has been characterized through a surrogate model. Input parameters are composed of the gate, source, and drain inductors, along with their quality factors. Results confirm that highly accurate surrogate models for performance metrics such as voltage gain if an LNA can be built and employed in circuit design optimization.

This chapter describes a recently developed multi-fidelity surrogate modeling method called polynomial chaos-based co-kriging (PC-co-kriging). PC-co-kriging is a multivariate extension of the PC-kriging method and its construction is similar to co-kriging. In particular, a PC-kriging model of a set of data points is created by using a polynomial chaos expansion (PCE) to model the global trend and, as in a regular kriging model, a Gaussian process to model the local deviations from the global trend. Two such PC-kriging models are created. One for the low-fidelity model data and another for the difference of the high- and low-fidelity model responses. The two models are combined using the same approach as in co-kriging. The PC-co-kriging modeling method is demonstrated on three different problems: response modeling of an analytical function, sensitivity analysis of ultrasonic testing measurement simulation model, and the optimal design under uncertainty of transonic airfoil shapes.

The following section is included:

• Index