Machine-learning-based global optimization of microwave passives with variable-fidelity EM models and response features

Maximizing microwave passive component performance demands precise parameter tuning, particularly as modern circuits grow increasingly intricate. Yet, achieving this often requires a comprehensive approach due to their complex geometries and miniaturized structures. However, the computational burden of optimizing these components via full-wave electromagnetic (EM) simulations is substantial. EM analysis remains crucial for circuit reliability, but the expense of conducting rudimentary EM-driven global optimization by means of popular bio-inspired algorithms is impractical. Similarly, nonlinear system characteristics pose challenges for surrogate-assisted methods. This paper introduces an innovative technique leveraging variable-fidelity EM simulations and response feature technology within a kriging-based machine-learning framework for cost-effective global parameter tuning of microwave passives. The efficiency of this approach stems from performing most operations at the low-fidelity simulation level and regularizing the objective function landscape through the response feature method. The primary prediction tool is a co-kriging surrogate, while a particle swarm optimizer, guided by predicted objective function improvements, handles the search process. Rigorous validation demonstrates the proposed framework's competitive efficacy in design quality and computational cost, typically requiring only sixty high-fidelity EM analyses, juxtaposed with various state-of-the-art benchmark methods. These benchmarks encompass nature-inspired algorithms, gradient search, and machine learning techniques directly interacting with the circuit's frequency characteristics.

kriging and co-kriging" section.The optimization process begins with the pre-screening stage, detailed alongside the construction of the initial (kriging) surrogate model in "Parameter space pre-screening.Initial surrogate model construction" section."Generating infill by means of nature-inspired optimization.Co-kriging surrogate" section expounds on the machine-learning framework involving co-kriging models and the infill criterion based on predicted objective function improvement.Lastly, "Complete optimization framework" section encapsulates the entire procedure using the pseudocode and a flow diagram.

Microwave design optimization. Problem statement
In this study, optimizing microwave circuits involves adjusting their independent variables, typically geometry parameters like component widths, lengths, and their spacings, consolidated within the parameter vector x (as illustrated in Fig. 1).The objective vector F t comprises the design goals, encompassing target operating parameters such as center frequency, power split ratio, and bandwidth.Evaluating the design x in relation to the target vector F t is accomplished through a scalar objective function U(x,F t ). Figure 2 provides several examples of common design scenarios along with the corresponding definitions of U. The primary type of microwave circuit responses are scattering parameters (cf.Fig. 1) 83 , along with the quantities that can be derived therefrom (e.g., the phase characteristic).
It is important to highlight that most real-world design challenges involve multiple objectives; that is, there is a need to enhance or regulate more than just one parameter or quantity.As most of available optimization algorithms are single-objective ones, multi-criterial problems are normally reformulated, e.g., by casting all but the primary objective into constraints 84 .Another popular option is scalarization using, e.g., a linear combination of goals 85 .Genuine multi-objective design is outside the scope of this paper.Now, we can express the microwave optimization task as a minimization task represented by the following form: where x * represents the sought-after optimal design, and X denotes the parameter space, typically an interval determined by the lower and upper variable bounds for x k , k = 1, …, n.

Response feature methodology
Full-wave computational models ensure reliable evaluation of microwave components, but they are CPU-intensive.This is a major hindrance to optimization procedures.The nonlinearity of circuit responses poses additional obstacles to globally exploring the parameter space.Figure 3 shows examples of frequency characteristics of a microstrip coupler at a number of randomly generated designs.Circuit optimization for a centre frequency of 1.5 GHz requires global search as local tuning initiated from most of the shown designs would not succeed.
The response feature techniques 86 were introduced to address the mentioned challenges by redefining the design problem using a discrete set of characteristic points (or features) from the system outputs 87 .This method leverages straightforward connections between the circuit's geometry/material parameters and the frequency and level coordinates of these points 80,[86][87][88][89] .Consequently, it regularizes the objective function, hastening optimization processes 87 , and simplifying behavioural modelling 89 .
The characteristics points are tailored to address specific design objectives 80 .Take, for example, a microwave coupler like the rat-race structure, and its responses depicted in Fig. 3. To regulate the center frequency, bandwidth of isolation characteristics, and power division ratio, the feature points can be defined as illustrated in Fig. 4, further detailed in the figure caption.Figure 4b demonstrates the straightforward connections between these feature points and the adjustable parameters.For further insights into this topic, additional discussions can be found in references such as 80 and 86 .In this work, we exploit these properties to facilitate a construction of surrogate models employed to enable and accelerate the global search process.At this point, it should be emphasized that utilization of response feature adds a layer of complexity in algorithm implementation, which is due to the necessity of defining and extracting the feature points.In practice, this requires implementation of a separate algorithm for scanning the frequency characteristics and identifying the feature points, here, realized in Matlab.On the other hand, especially when the parameter space is very large (broad ranges of geometry parameters) and highly-dimensional, the features might be difficult to identify.The latter normally occurs for heavily distorted responses at poor-quality designs.In practice, designs like these may be assigned high value of the objective function U F to mark them as low-quality.Nonetheless, the aforementioned issue constitutes a limitation of the proposed approach, yet can be mitigated to a certain extent by appropriate selection of the parameter space (e.g., based on prior experience with a given microwave circuit).
To integrate the response feature approach into the optimization process, we must establish suitable notation.Specifically, we will denote the feature points of the circuit under design as T being the vector of frequency and level coordinates, respectively.Table 1 provides a selection of example characteristics points for a microwave coupling circuit, focusing on the device's operating frequency and power split ratio.between operating conditions (extracted from the response features, here the center frequency f 0 and power split ratio K P ) and selected geometry parameter of the circuit.The plots are created using a set of randomly-generated designs.Only the points for which the corresponding characteristics allow for extracting the approximated operating parameters, as indicated above, are shown.Clear patterns are visible even though the trial points were not optimized whatsoever.
) the circuit's operating parameters, computed from f P (x).Following this, the design task posed in terms of characteristics points appears as: The analytical form of the function U F is similar to that shown in Fig. 2 but employs F o (x) rather than scattering parameters S kl .It can also be written in a compact manner as Here, the function U 0 accounts for the main objective.For clarification, consider a scenario, under which we seek to achieve the following: (i) align the coupler center frequency at the target value f t , (ii) establish the target power division ratio, and (iii) reduce |S 11 | (impedance matching) and |S 41 | (port isolation) at f t .
In this case, the main goal is to reduce f L.1 and f L.2 , cf.Table 1, i.e., we have U 0 (f The importance of the operating condition alignment is controlled by the scalar factor β. Typically, we set β = 100.It should be noted that conventional formulation (as shown in Fig. 2) is equivalent to (3) assuming that the optimum is attainable.Further, the second term in (3) (not present in standard formulation) acts as a regularization factor that facilitates identification of the optimum.

Variable-fidelity computational models
Reducing the EM analysis resolution-by decreasing the structural discretization density (alternative options discussed in 90 )-expedites the simulation process but compromises accuracy, as shown in Fig. 5.This accuracy loss can be rectified by appropriately adjusting the low-fidelity model, forming the basis of physics-based surrogate-assisted methodologies.One prominent example is space mapping 91,92 .The degree of acceleration achievable varies with the problem.For most microwave passive components, it ranges between 2.5 and six for the low-fidelity model, ensuring adequate representation of crucial circuit output details.
In this study, the low-and high-resolution models are denoted as R c (x) and R f (x), respectively.R c will be utilized to: (2) T Explanation of terms f f.1 and f f.2 -frequencies corresponding to the minima of |S 11 | and |S 41 | f L.1 and f L.2 -corresponding minima levels www.nature.com/scientificreports/ • Pre-screen the parameter space, i.e., to generate a set of random observables, from which those with extract- able feature points will be selected for further processing; • Construct the initial surrogate model using kriging (cf."Surrogate modelling using kriging and co-kriging" section).
Subsequent optimization steps will employ the low-fidelity samples into a co-kriging surrogate (cf."Surrogate modelling using kriging and co-kriging" section), which will also incorporate R f data acquired during the algorithm run.
At this point, it should be emphasized that the pre-screening process in large parameter spaces (broad ranges of geometry parameters, high dimensionality) may be a limiting factor in terms of the computational efficiency.This is because is large spaces, the percentage of random designs with extractable features might be low, meaning that acquisition of a required number of accepted design would incur considerable expenses.One way of mitigating this issue is appropriate selection of the search space, based on engineering experience and prior knowledge about the circuit at hand, so that reasonably narrow parameter ranges can be established.On the other hand, the aforementioned issue would be also detrimental to any global search algorithm, whether it is a direct method of surrogate-assisted one.

Surrogate modelling using kriging and co-kriging
Kriging and co-kriging interpolation 93,94 will be used as the modelling methods of choice within the machine learning framework proposed in this work.Both techniques are briefly recalled below.
We consider two datasets: , which consists of the parameter vectors x Bc (k) and the corresponding circuit outputs; Tables 2 and 3 outline both kriging and co-kriging models s KR (x) and s CO (x), respectively.Note that the cokriging surrogate incorporates the kriging model s KRc set up based on the low-fidelity data (X Bc , R c (X Bc )), and s KRf generated on the residuals (X Bf , r); here r = R f (X Bf ) -ρ⋅R c (X Bf ).The parameter ρ is included in the Maximum Likelihood Estimation (MLE) of the second model 94 .If the low-fidelity data at X Bf is unavailable, one can use an approximation R c (X Bf ) ≈ s KRc (X Bf ).The same correlation function is utilized by both models (cf.Tables 2 and 3).Table 2.An outline of the kriging surrogate modeling.More information can be found in the literature, e.g., 94 .

Model component Analytical form
Model formulation where M is a N Bc × t model matrix of X Bc , whereas F is a 1 × t vector of the evaluation point x (t is the number of terms used in the regression function 91 ) Bc ), ..., ψ(x, x and |Ψ| is the determinant of Ψ.In practice, a Gaussian correlation function (P = 2) is often employed, as well as F = [1 … 1] T and M = 1 Table 3.An outline of the co-kriging surrogate modeling.More information can be found in the literature, e.g., 94 .

Model component Analytical form
Model formulation The initial stage of the machine learning process described in this paper involves screening the parameter space, conducted at the level of low-fidelity EM models.Subsequently, an initial surrogate model is developed, utilizing the low-fidelity data collected during this phase.The pre-screening stage operates at the response feature level (refer to "Response feature methodology" section), enabling a sizeable curtailment of the number of samples compared to analysing the complete frequency characteristics of the designed circuit.The primary aim of the initial surrogate s (0) (x) is to capture the behaviour exhibited by the frequency and level coordinates of the feature points delineated for the optimized circuit.We have The model is rendered by means of kriging 95 (cf."Surrogate modelling using kriging and co-kriging" section, Table 2).The training data pairs are denoted as {x Bc (j) ,f P (x B (j) )}, j = 1, …, N init .The points are generated randomly in the parameter space X using independent joint uniform probability distributions; only the observables with extractable response features are included into the training set.The dataset size N init is adjusted to secure a sufficient surrogate model accuracy.For quantification, we utilize the relative RMS error, wherein the acceptance threshold E max serves as a controlling parameter within the process.Given the characteristics of the response feature, the demand for training points to establish a dependable model remains minimal, typically ranging from fifty to one hundred.Depending on the parameter space's dimensionality and ranges, the ratio of accepted observables falls between twenty to seventy percent.Consequently, the actual count of low-fidelity EM simulations required to compile the dataset {x Bc (j) ,f P (x B (j) )} varies from 1.5N init to 5N init .In practice, the process of rejecting (possibly significant) subset of the random samples acts as a pre-selection mechanism: it enables the identification of promising regions within the parameter space, concentrating the search process on these areas while excluding others.The pre-screening procedure has been summarized in Table 4.

Generating infill by means of nature-inspired optimization. Co-kriging surrogate
The core part of the optimization run consists of iterative generation of candidate solutions x (i) , i = 1, 2, …, to the problem (2), (3), and construction of the refined surrogate models s (j) , j = 1, 2, … .This stage is carried out using the high-resolution model R f .Each iteration produces a new design using the current surrogate s (i) , which is a co-kriging model (cf."Parameter space pre-screening.Initial surrogate model construction" section) constructed using the low-fidelity dataset {x Bc (j) , f P (x Bc (j) )}, j = 1, …, N init , and the high-fidelity dataset {x f (j) , f P (x f (j) )}, j = 1, …, i, consisting of the samples accumulated until iteration i.The formulation of the sub-problem (5) is the same as for the task (2), except that the response features are obtained from s (i) .Optimization is conducted in a global sense using the particle swarm optimization (PSO) algorithm 97 .PSO was chosen as a representative nature-inspired method, which also belongs to the most popular ones.Yet, any other population-based technique can be used instead because operating at the level of fast surrogate does not imposes any practical constraints on the computational budget expressed as the number ( 4) Parameter space pre-screening and initial surrogate model construction.A pseudocode.

Input parameters:
x Design space X (interval [l u], w here l and u are low er and upper bounds for designable parameters; other constraints are possible per designer's needs); x Required modelling error Emax; 2. Set j = 0; 3. Generate a sample xtmp X using uniform probability distribution; 4. Evaluate circuit characteristics at xtmp using low -fidelity EM analysis; 5. Extract the feature vector fP(xtmp); assign fP(xtmp) = 0 if features are not extractable (cf.Fig. 3 In view of machine learning, generating candidate designs using (5) corresponds to the infill criterion being the predicted objective function improvement 98 .Upon completing the pre-screening stage, the parameter space subset containing the optimum has been presumably identified, therefore, the search process can now be focused on its exploitation, rather than enhancing the overall metamodel's accuracy.The algorithm is stopped either due to convergence in argument, i.e., ||x (i+1) -x (i) ||< ε or if no improvement of the objective function has been detected over the last N no_improve iterations, whichever occurs first.The default values of the termination thresholds are ε = 10 -2 and N no_improve = 10.

Complete optimization framework
The proposed global optimization framework is outlined in Table 5 as a pseudocode and depicted in Fig. 6 as a flow diagram.The pre-screening stage unfolds within Steps 2 and 3, while the heart of the search processcreating infill points and refining the surrogate model-is executed across Steps 4 through 8. Step 9 verifies the termination criteria.
Let us discuss the control parameters compiled in Table 6.It is essential to note that there are only three parameters, with two determining the termination conditions.Their primary function is to govern the resolution of the optimization process.The third parameter manages the accuracy of the initial surrogate.The default value of E max corresponds to ten percent of the relative RMS error, which is a mild condition.A small number of control parameters is an advantage of the method as it eliminates the need for tailoring the setup to the particular being solved.In fact, identical arrangement will be used for all verification experiments described in "Verification experiments and benchmarking" section.
The full-wave electromagnetic simulations were performed on Intel Xeon 2.1 GHz dual-core CPU with 128 GB RAM, using CST Microwave Studio.The optimization framework has been implemented in MATLAB.The particle swarm optimizer and CST simulation software communicate through a Matlab-CST socket.The kriging and co-kriging surrogate models were set up using the SUMO toolbox of 99 .As mentioned earlier, the underlying optimization engine is particle swarm optimizer PSO 100 , which is one of the most representative nature-inspired population-based algorithms.PSO processes a swarm of N particles (parameter vectors) x i and velocity vectors v i , which stand for the position and the velocity of the ith particle, respectively.These are updated as follows: Table 5. Pseudocode of the ML framework for feature-based global parameter tuning of microwave components introduced in this study.

Input parameters:
x Target operating frequencies Ft (cf."Microwave design optimizatio n. Problem statement" section); x Definition of the response features fP and the objective function UF (cf."Response feature methodology" section); x Design space X (interval [l u], w here l and u are low er and upper bounds for designable parameters); x Required modelling error Emax; x Termination thresholds and Nno_improve; 2. Generate the set of initial samples {xBc (k) ,fP(xBc (k) )}k = 1,…,Ninit, as described in "Parameter space pre-screening.Initial surrogate model construction" section (cf.Fig. 4), using low -fidelity simulation data; 3. Construct (initial) kriging surrogate model s (0) (x); 4. Set i = 0; 5. Use the PSO algorithm to obtain infill point xf (i+1) by solving (5): 8. Construct the co-kriging surrogate model s (i) (x) using the updated dataset; 9. if ||x (i) -x (i-1) || < H OR no objective function improvement for Nno_improve iterations Go to 11; end 10.Go to 5; 11.Return x * = x (i) ; where r 1 and r 2 are vectors whose components are uniformly distributed random numbers between 0 and 1; • denotes component-wise multiplication.In our numerical experiments we use a standard setup: • Size of the swarm N = 10, • Maximum number of iterations k max = 100, • Control parameters, χ = 0.73, c 1 = c 2 = 2.05, cf. 100 .
As indicated in (7), the first step of altering the positions x i of the particles is the adjustment of the velocity vector, which is partially stochastic.There are three components therein, one being the current velocity, the second fostering particle relocation towards its local best position x i * , and the third one pushing the particle towards global best position g found so far by the swarm.The mentioned setup of control parameters is the most widely used one, typically recommended in the literature 99 .

Verification experiments and benchmarking
The validation of the optimization algorithm outlined in "Global machine-learning microwave optimization using response features and multi-resolution computational models" section involves numerical testing with two microstrip circuits.For comparison, these devices are also optimized with the use of several benchmark methods that include a gradient-based search with random starting point, a particle swarm optimizer (PSO), but also two machine-learning frameworks: (i) a procedure that directly handles frequency characteristics of the ( 6)  www.nature.com/scientificreports/circuit, and (ii) the algorithm of "Global machine-learning microwave optimization using response features and multi-resolution computational models" section exclusively using high-fidelity EM simulations.We chose these specific methods to showcase the multimodal nature of the design challenges and to validate the significance of the algorithmic tools embedded in the proposed procedure-particularly the use of response features and variable-fidelity simulations.The performance evaluation criteria encompass the optimization process's reliability (quantified as a success rate, i.e., the proportion of algorithm runs yielding acceptable outcomes), design quality, and computational efficiency.The material in this section is arranged as follows: "Verification circuits" section offers insights into the verification structures.The experimental setup and results are outlined in "Experimental setup.Numerical results" section, while "Discussion" section delves into the characteristics of the techniques considered and encapsulates the overall performance of the proposed approach.

Verification circuits
The machine learning procedure introduced in this study is demonstrated using two planar structures: • A miniaturized rat-race coupler (RRC) with meandered transmission lines (Circuit I) 101 ; • A dual-band power divider with equal division ratio (Circuit II) 102 .
The circuit topologies are depicted in Figs.7a and 8a, respectively.Essential data for both circuits, including substrate parameters, design variables, and design goals, are detailed in Figs.7b and 8b.The computational models are simulated in CST Microwave Studio, employing the time-domain solver.The low-fidelity models are representations with coarser discretization compared to the high-fidelity versions.For specific details regarding the number of mesh cells and simulation times for R c and R f , please refer to Table 7.    Random initial design, response gradients estimated using finite differentiation, termination criteria based on convergence in argument and reduction of the trust region size 103 III Machine learning algorithm (cf."Global machine-learning microwave optimization using response features and multi-resolution computational models" section) Algorithm similar to that of "Global machine-learning microwave optimization using response features and multi-resolution computational models" section Initial surrogate set up to ensure relative RMS error not higher than 10% with the maximum number of training samples equal to 400; Optimization based on processing the antenna frequency characteristics (unlike response features in the proposed procedure); Infill criterion: minimization of the projected objective function improvement 98

IV
Feature-based machine learning algorithm utilizing high-fidelity EM simulations only

Algorithm highlights
Surrogate model constructed at the level of response features; Optimization process only uses high-fidelity EM simulations; Infill criterion: minimization of the predicted objective function 98

Discussion
The data presented in Tables 10 and 11 are examined here to evaluate the efficiency of the suggested machinelearning approach and to juxtapose it with benchmark techniques.We're particularly focused on evaluating the following metrics: the search process's reliability assessed through the success rate, the design quality evaluated using the merit function value, and the cost efficiency of the global search process.

9.
Exemplary runs of the proposed machine-learning framework.Shown are: S-parameters of Circuit I (Case 1) at the designs produced by the proposed technique (top), and the evolution of the objective function value (bottom): (a) run 1, (b) run 2. The iteration counter starts after constructing the initial surrogate model.Target operating frequency, here, 1.8 GHz, marked using the vertical lines.www.nature.com/scientificreports/Another point for discussion is the effect of incorporating the response features and variable-fidelity simulation models.The observations are as follows: • Search process reliability Reliability is measured using the success rate (right-hand-side columns of Tables 10  and 11), which is the number of algorithm runs yielding the designs that feature operating parameters being close to the targets.For the proposed method, the success rate is 10/10, just as it is for both benchmark machine learning methods (Algorithms III and IV) and PSO (Algorithm I) set up with the budget of 1000 objective function calls.However, PSO working with the computational budget of 500 function evaluations  does not perform as well.Also, for Circuit II, which is a more challenging of the two test problems, the success rate of PSO is 9/10 even the budget of 1000 function calls.This indicates that direct nature-inspired optimization needs higher budgets (e.g., > 2000) to ensure the perfect score.At the same time, gradient-based optimization routinely fails (e.g., the success rate is only 2/10 for Circuit II, Case 1), which corroborates multimodality of the considered design tasks.• Design quality The solutions produced using the presented procedure exhibit comparable quality to those produced by other benchmark methods, as indicated by the average objective function value with the exception of gradient-based search, where the failed algorithm runs degrade the average.In terms of absolute numbers, all machine learning methods deliver results that are sufficiently good for practical applications, i.e., matching and isolation below − 20 dB, and power division close to the required values of 0 dB and 3 dB for Circuits I and II, respectively • Computational efficiency The efficiency of the proposed framework is by far the best across the entire bench- mark set and the considered test problems.We omit comparison with gradient-based algorithms as they were included in the benchmark solely to highlight the necessity of global optimization for the test problems considered.The average computational cost, quantified in terms of equivalent high-fidelity EM simulations, is approximately sixty.This signifies considerable savings compared to PSO (with the budget of 1000 function calls) are between 90 and 95 percent, depending on the test case.The savings over Algorithm II (machine learning processing full circuit responses) are about 85 percent on the average, and about 50 percent over Algorithm III (feature-based machine learning working at single-fidelity EM level).Regarding the latter, the average savings are 45 and over sixty percent for Circuit I and II, respectively, because the time evaluation ratio between the high-and low-fidelity model is more advantageous for the latter (5.2 versus 2.4).• The above numbers confirm that the employment of both response features and variable-fidelity are instru- mental in expediting the search process without being detrimental to its reliability.The speedup obtained due to variable-fidelity modeling is significant also because the majority of the circuit evaluations are associated with the parameter space pre-screening, which, in the proposed methods, is conducted using the low-fidelity system representation.A better perspective of the computational benefits due to the mentioned mechanisms can be provided by considering the acceleration factors.For example, the proposed methods is over 17 times faster than PSO, almost eight times faster than Algorithm III, and over twice as fast as Algorithm IV.
The overall performance of the presented methodology is quite promising.The variable-fidelity feature-based machine learning yields consistent results at remarkably low computational cost.Consequently, it seems to be suitable for replacing less efficient global optimization methods in the field of high-frequency EM-driven design.At this point, we should also discuss its potential limitation, which is related to the response feature aspect of the procedure.On one hand, defining and extracting characteristic points from the system outputs for a specific structure and design task adds complexity to the implementation process, albeit not in its core segment, which remains problem independent.
On the other hand, for highly-dimensional problems and very broad parameter ranges, the number of random observables generated in the pre-screening stage may be large as compared to those that exhibit extractable features.Such factors would compromise the computational efficiency of the methods, impacting their overall performance.Naturally, these same factors would also impede the efficacy of all benchmark methods.Yet, assuming that the parameter space is defined by an experienced designer, i.e., it is not excessively large, the risk of the occurrence of the aforementioned issue is rather low.

Conclusion
This paper introduced a machine-learning framework designed for the efficient global optimization of passive microwave components.Our methodology integrates several crucial mechanisms pivotal for achieving competitive reliability and minimizing search process expenses.These mechanisms encompass the response feature approach, parameter space pre-screening, and the utilization of variable-fidelity EM simulations.The response feature method helps in regularizing the objective function landscape, consequently reducing the necessary dataset size for constructing precise surrogate models.Pre-screening of the parameter space aids in the initial identification of the most promising regions, while variable-resolution models contribute to additional computational acceleration.Within this framework, both low-and high-fidelity simulation data are combined into a unified surrogate model using co-kriging.The optimization process itself focuses on rapid identification of the optimum design, which is facilitated by the infill criterion applied in our framework (predicted objective function improvement).The particle swarm optimization (PSO) algorithm serves as the underlying search engine.Numerical verification experiments involving two microstrip components illustrate the superior performance of the proposed technique, showcasing its ability to achieve superior design quality, reliability, and computational efficiency.The CPU savings versus nature-inspired optimization are up to 95 percent (average acceleration factor of 17), 85 percent over the machine learning procedure working directly with circuit frequency characteristics (acceleration factor of eight), and 50 percent over the feature-based machine learning algorithm that only uses high-fidelity EM models (acceleration factor of two).The mean running cost corresponds to sixty high-resolution EM simulations.This level of expenses is comparable to local optimization.Consequently, the presented framework seems to be an attractive alternative to both conventional and surrogate-assisted global optimization procedures (including machine-learning algorithms) utilized so far in high-frequency engineering.A primary objective for future work involves extending the method to encompass other types of microwave components, such as filters.This extension would involve automating the procedures for defining and extracting feature points.Furthermore, utilization of alternative machine learning technique will be considered, oriented towards improving the reliability of surrogate model construction as well as computational efficiency of the modelling

Figure 2 . 7 Figure 3 .
Figure 2. Examples of microwave design optimization problems.Verbal description of the task (left) is followed by a definition of the target vector F t (middle), and a possible definition of the objective function (right).

Figure 4 .
Figure 4. Characteristic points selection for a microwave coupler: (a) possible feature point choices: open circle-points corresponding to the minima of the matching and isolation characteristics, *-points corresponding to the power split ratio (evaluated at the frequency f 0 being the average of the frequencies of |S 11 | and |S 41 | minima), open square-points corresponding to − 20 dB levels of |S 11 | and |S 41 |; (b) relationshipbetween operating conditions (extracted from the response features, here the center frequency f 0 and power split ratio K P ) and selected geometry parameter of the circuit.The plots are created using a set of randomly-generated designs.Only the points for which the corresponding characteristics allow for extracting the approximated operating parameters, as indicated above, are shown.Clear patterns are visible even though the trial points were not optimized whatsoever.

4 Figure 5 .
Figure 5. Variable-fidelity models: (a) geometry of an exemplary compact branch-line coupler, (b) scattering parameters evaluated using the low-fidelity EM model (gray) and the high-fidelity one (black).At the design shown, the simulation time of the high-fidelity model is about 250 s, whereas the evaluation of the low-fidelity model takes about 90 s.

Figure 6 .
Figure 6.Flow diagram of the proposed ML algorithm for global optimization of microwave passive components.

Figure 10 .
Figure 10.Exemplary runs of the proposed machine-learning framework.Shown are: S-parameters of Circuit I (Case 2) at the designs produced by the proposed technique (top), and the evolution of the objective function value (bottom): (a) run 1, (b) run 2. The iteration counter starts after constructing the initial surrogate model.Target operating frequency, here, 1.2 GHz, marked using vertical lines.

Figure 11 .
Figure 11.Exemplary runs of the proposed machine-learning framework.Shown are: S-parameters of Circuit II (Case 1) at the designs produced by the proposed technique (top), and the evolution of the objective function value (bottom): (a) run 1, (b) run 2. The iteration counter starts after constructing the initial surrogate model.Target operating frequencies, 3.0 GHz and 4.8 GHz, marked using vertical lines.

Figure 12 .
Figure 12.Exemplary runs of the proposed machine-learning framework.Shown are: S-parameters of Circuit II (Case 2) at the designs produced by the proposed technique (top), and the evolution of the objective function value (bottom): (a) run 1, (b) run 2. The iteration counter starts after constructing the initial surrogate model.Target operating frequencies, 2.0 GHz and 3.3 GHz, marked using vertical lines.
of objective function calls.Furthermore, finding the global optimum of U F (x,s (i) (x),F t ) is considerably easier handling the original merit function U(x,F t ) due to inherent regularization that comes with the employment of response features.

Table 6 .
Control parameters of the proposed ML algorithm for global optimization microwave passives.

Table 7 .
EM simulation models for Circuit I and II.