Reduced-cost two-level surrogate antenna modeling using domain confinement and response features

Electromagnetic (EM) simulation tools have become indispensable in the design of contemporary antennas. Still, the major setback of EM-driven design is the associated computational overhead. This is because a single full-wave simulation may take from dozens of seconds up to several hours, thus, the cost of solving design tasks that involve multiple EM analyses may turn unmanageable. This is where faster system representations (surrogates) come into play. Replacing expensive EM-based evaluations by cheap yet accurate metamodels seems to be an attractive solution. Still, in antenna design, application of surrogate models is hindered by the curse of dimensionality. A practical workaround has been offered by the recently reported reference-design-free constrained modeling techniques that restrict the metamodel domain to the parameter space region encompassing high-quality designs. Therein, the domain is established using only a handful of EM-simulations. This paper proposes a novel modeling technique, which incorporates the response feature technology into the constrained modeling framework. Our methodology allows for rendering accurate surrogates using exceptionally small training data sets, at the expense of reducing the generality of the modeling procedure to antennas that exhibit consistent shape of input characteristics. The proposed technique can be employed in other fields that employ costly simulation models (e.g., mechanical or aerospace engineering).

www.nature.com/scientificreports/ • Constructing the surrogate at the level of the response features within a confined domain defined based on random observables, • Demonstrating substantial computational savings as compared to the previously reported constrained modeling techniques, • Demonstrating superiority over conventional data-driven modeling techniques in terms of the CPU cost and surrogate model accuracy.
According to the authors' knowledge, no comparable modeling technique ensuring this level of accuracy at such a low computational cost has not been reported in the context of antenna modeling in the literature thus far.
Two-level constrained modeling with response features. The purpose of this section is to introduce the proposed modeling framework. Our technique capitalizes on the concept of performance-driven modeling, specifically, reference-design-free constrained modeling 49 , as well as the response features technology 50 . "Constrained modeling: Concept and basic definitions" and "Reference-design-free constrained modeling" sections provide a recollection of the constrained modeling technique 49 . "Response features" section outlines the response feature methodology 50 , whose incorporation into the proposed modeling framework is discussed in "Constrained modeling at the level of response features". The formulation of the complete two-level constrained modeling technique, accommodating both the aforementioned technologies, concludes the section.
Constrained modeling: concept and basic definitions. We start by recollecting the performancedriven modeling concept 52 , employed here for a surrogate domain definition purposes. In short, the techniques belonging to this group [47][48][49] aim at identifying the parameter space regions that encompass the designs of highquality from the point of view of the relevant figures of interest. This allows for a significant reduction of the domain volume in comparison to the conventional one, i.e., delimited by the lower and upper bounds on the design variables. As a consequence, substantial savings in terms of training data acquisition cost may be achieved without degrading surrogate model predictive power. This is of paramount importance especially for higherdimensional cases. At the same time, this is achieved without formally narrowing down the ranges of neither antenna geometry parameters nor operating conditions 47,48 . Table 1 gathers the basic objects utilized in constrained modeling 48 : the design variable vector x (typically geometry parameters of the device under study), as well as the two spaces of interest: the parameter space X, and the objective space F. The entries of the design objective vector F may include, e.g., antenna operating frequency/frequencies or bandwidth/bandwidths, but also substrate permittivity the structure is implemented on. The region of validity of the surrogate is supposed to cover the objective space F, delimited by the user-specified ranges of performance figures.
In constrained modeling, the surrogate model domain is to encompass the designs optimal with respect to the assumed figures of merit. The optimal design is understood here as minimizing the scalar objective function U(x,f) that quantifies the design quality 52 The set of designs optimal with respect to all the objective vectors f ∈ F, is denoted as The surrogate is to be set within a domain that constitutes the region of the parameter space adjacent to the manifold U F (F). In nested kriging 48 , this region has been identified with the use of the set of pre-optimized reference designs The pairs {f (j) , x (j) }, j = 1, …, p, constitute a training data set to set up a first-level interpolation surrogate s I (f) : F → X, which served to render an initial approximation of the manifold U F (F).
Needless to say, acquisition of the reference designs is expensive in terms of a required number of full-wave EM-simulations. As a matter of fact, its overall cost has been typically as high as a several hundreds of EM analyses 28 . Furthermore, obtaining these designs required re-designing the antenna at hand over broad ranges of operating conditions. Therefore, the reference design acquisition has been laborious and difficult to automate. Recently, some attempts to make it less dependent on designer's supervision have been reported 53 ).
(1)  49 , the reference designs acquisition is abandoned altogether. Instead, a set of random observables is distributed in the parameter space X. These observables undergo a pre-selection process, in which their acceptance or rejection is based on the information about the design objectives extracted therefrom. The approved observables serve to construct an inverse regression model (a counterpart of the first-level interpolation model of the nested kriging 48 ) for surrogate domain definition. Let {x r (j) , f r (j) }, j = 1, 2, …, be a sequence of pairs containing random vectors x r (j) uniformly distributed in the design space X, as well as the corresponding performance figure vectors f r (j) (extracted from the antenna model responses at x r (j) ). The acceptance/rejection process is carried out as follows: the jth observable is accepted if f r (j) ∈ F; otherwise (i.e., if either of the components of f r (j) is not within the assumed ranges on the performance figures or it is unidentifiable) the observable is rejected. The sample acquisition continues until the required number of observables N r has been acquired (typically, N r should be around ten times higher than the parameter space dimensionality). The said data pairs serve as a training set for setting up the inverse surrogate s r 49 Observe that (2) describes an inverse regression surrogate s r (f) that maps the antenna objective space into its design space. In other words, s r (f) is defined over the objective space F and assumes values in the parameter space X, or s r : F → X. The above inverse model yields an approximation of the optimum design manifold U F . Identification of the surrogate s r requires solving the following nonlinear regression problems where x r.j (k) denotes the jth entry of the observable vector x r (k) , whereas the weighting factors w k = [w max -max{p 1 (x (j) ), …, p N (x (j) )}] 2 , k = 1, …, N r , differentiate "good" observables from the "poor" ones. We have the maximum factor w max = max{k = 1, …, N r , j = 1, …, N : p j (k) }, with p j (k) assuming nonnegative values (a better design is assigned a lower value of p j (k) ). The factors p j (k) are assembled into vectors p r (j) = [p r.1 (j) … p r.N (j) ] T . The above mechanism of the weighted regression allows for ensuring that high-quality observables have more impact on the regression model, but also to take into account information contained in lower-quality ones. The vectors p r (j) are extracted from EM-simulated antenna response, similarly as the vectors f r (j) . As an example, let us consider a dual-band antenna with the operating frequencies being the performance figures of interest. In this case, the vector f r (j) comprises the actual operating frequencies, whereas the vector p r (j) may contain the corresponding reflection levels. The concept of the inverse regression surrogate is visualized in Fig. 1.
In reference-design-free constrained modeling 49 , the surrogate domain definition procedure resembles that of the nested kriging technique 48 : the image of the inverse first-level surrogate s r (F) is extended in order to encompass the majority of the optimum design manifold U F (F). This is because s r (F) provides merely the initial inexact approximation of the location of U F (F). The coefficients of the said extension (towards the vectors normal to s r (F)) are given by · · · a n.0 + a n.1 exp In other words, the domain X S encompasses all the vectors defined by (5) for all f ∈ F, and all the coefficients n (f ) . In reference-design free modeling technique 49 , the values of the extension factors are set individually for each design variable based on the knowledge extracted from the available observable set. For each observable pair {x r (j) ,f r (j) }, a vector P k (x r (j) ) minimizing the distance between the observable and [s r.
Thus, the extension factors T k are given by 49 All the factors are gathered in the extension vector The final, forward surrogate model s(x) is set up in the confined domain X S (defined using (4), (5) and (7)) as a kriging interpolation metamodel 32 the training data samples, whereas R is the EM-simulated antenna response. The training data set is also complemented by the observable set {x r (l) ,R(x r (l) )} l = 1, …, Nr .

Response features.
Modeling of highly-nonlinear antenna characteristics often proves to be a challenging task. In some cases, it is possible to reduce its complexity by employing the response feature technology 50 , where the modeling problem is tackled at the level of suitably defined characteristic points of the antenna at hand. The response feature technology 50 capitalizes on a significantly less nonlinear relationship between the feature coordinates and designable parameters 54 than normally observed for entire antenna responses. In the context of modeling, (but also parameter tuning 54 or yield optimization 55 ) this allows for a notable reduction of the computational overhead, which is of paramount importance, especially for devices described by larger (over ten) numbers of geometry parameters. Clearly, the employment of response feature technology is only realizable when the system outputs are characterized by readily discernible characteristic points. From the practical point of view, the actual selection of response features has to account for the design goals. As an example, let us consider characteristics of a dual-and a triple-band antenna with characteristic points corresponding to antenna resonant frequencies and − 10 dB www.nature.com/scientificreports/ reflection levels, as shown in Fig. 2. These points permit handling design tasks aimed at resonance allocation at the target operating frequencies or bandwidth enhancement. Naturally, the designer has to bear in mind that, in some cases, not all the characteristic points may be distinguished for a specific design, and to handle this issue appropriately during the optimization or modeling process. Technically, the response features are extracted from the EM-simulated antenna responses. Observe, that the feature-based approach permits to directly access information about the performance figures relevant to the assumed design objectives. This is in contrast to the conventional approach, where the entire antenna characteristics are handled, and this knowledge needs to be extracted afterwards. For a more thorough account of the response feature technology see, e.g., Ref. 52 . Constrained modeling at the level of response features. In the proposed modeling technique, the construction of the first-level regression surrogate s r , as well as domain definition procedure follow exactly that Ref 49 , which is recapitulated in "Reference-design-free constrained modeling". Yet, unlike 49 , the secondlevel metamodel is set up using the training data pairs {x B T denotes the response feature vector corresponding to a given design x. The entries of the vector F R are the frequency f j and level coordinates λ j , j = 1, …, p, of p antenna resonances. In other words, the response of the second-level surrogate yields predictions only about the feature point coordinates rather than the entire antenna characteristic at a given design x ∈ X S . Naturally, focusing only on the response features does lead to some unavoidable loss of information. Still, this loss is irrelevant from the point of view of the design goals: as mentioned in "Response features", the features are defined so as to allow for quantifying the design objectives unequivocally. In general, some of the feature points may not be distinguishable (e.g., − 10 dB reflection points do not exist if the antenna resonance level is above that limit). However, this is not an issue for the considered approach, because the very definition of the domain X S ensures that the antenna designs contained therein are of high quality, which ensures the existence of all feature points. Figure 3 shows the conceptual illustration of the proposed modeling procedure. The user needs to define the parameter space and the objective space (by providing the respective lower and upper bounds), and also decide  , utilized to quantify the quality of the observables, are the reflection levels at these frequencies. Observe that the first p entries of the feature vectors F r (j) of "Response features" (i.e., the frequency coordinates) coincide with the entries of f r (j) . At the same time, the remaining p entries of F r (j) (i.e., the level coordinates) are simply the components of the performance vectors p r (j) . The operating flow of the presented modeling procedure has been shown in Fig. 3.

Results
This section provides numerical verification of the proposed modeling technique. The results have been obtained for three antenna structures and benchmarked against the conventional kriging interpolation, as well as the state-of-the-art constrained modeling frameworks: nested kriging, feature-based nested kriging, and referencedesign-free modeling.    Fig. 4a, b and c, respectively. The details concerning the design variables and objectives, as well as simulation models for all the benchmark structures have been gathered in Table 2. For Antenna I and III, the substrate relative permittivity ε r is one of the performance figures, therefore, its ranges are provided under the design objective ranges of Table 2. The computational models are evaluated in CST Microwave Studio and simulated using its time-domain solver.
Modeling results. The surrogate models have been constructed within the respective regions of validity given in Table 2, using the following sizes of the training data sets: 20, 50, 100, 200, 400, and 800 samples. The benchmark techniques include: (i) conventional kriging in an unconstrained domain 32 (Algorithm 1), (ii) basic nested kriging technique 48 (Algorithm 2), (iii) reference-design-free constrained modeling 49 (Algorithm 3), (iv) feature-based nested kriging technique 51 (Algorithm 4). The proposed feature-based reference-design-free constrained modeling framework is referred to as Algorithm 5. The main features of the frameworks considered in this paper are summarized in Table 3, where the surrogate model definition setup and costs are compared, along with the modeling task formulation: conventional (operating on the entire responses) or feature-based. Tables 4, 5 and 6 provide the modeling results: the computational costs of model setup and its accuracy. Observe that in the case of the feature-based performance-driven techniques (i.e., Algorithm 4, and the proposed Algorithm 5), the modeling accuracies of the frequency and level coordinates of the response features are provided. For each antenna structure, a relevant set of characteristic points has been selected: (i) the operating frequency (ring-slot antenna of Fig. 4a), (ii) two operating frequencies (dual-band antenna of Fig. 4b), and (iii) the lower and upper frequencies for which the reflection response assumes − 10 dB levels, as well as the corresponding values of the realized gain characteristic, supplemented by five additional points equally distributed in frequency in between these points (quasi-Yagi antenna of Fig. 4c). The supplementary points are required to adequately reconstruct the gain characteristics within the operating band in order to assess the antenna average in-band gain.    www.nature.com/scientificreports/ The results of Tables 4, 5 and 6 demonstrate that carrying out the modeling process at the level of the response features allows for achieving superior accuracy of representing the characteristic points relevant to the assumed design objectives at a remarkably small computational cost. For all the benchmark antennas, the proposed surrogate allows for achieving accuracy of less than one percent for training data sizes containing merely 50 samples from the constrained domain. Observe that the total cost of training data acquisition includes also the cost of generating 106, 230, and 192 observables for each antenna, respectively, which are necessary to define the surrogate domain. Even when taking into account these additional computational expenses, our approach requires as low as 126, 250 and 212 data samples to assess the operating frequencies of each antenna structure with the Table 4. Ring-slot antenna of Fig. 4a: modeling results and benchmarking. $ The cost includes acquisition of the reference designs, which is 864 EM simulations of the antenna when using feature-based optimization 50 as listed in the table. # The cost includes generation of random observables, here, 106 simulations in total to yield N r = 50 accepted samples.  Table 5. Dual-band antenna of Fig. 4b: modeling results and benchmarking. $ The cost includes acquisition of the reference designs, which is 930 EM simulations of the antenna when using feature-based optimization 50 as listed in the table. Conventional (minimax) optimization required 1201 simulations. # The cost includes generation of random observables, here, 230 simulations in total to yield N r = 50 accepted samples. It should be noted that the accuracy of representing the frequency coordinates of the feature points is exceptionally good even for small data sets, whereas it is not as good for the level coordinates. On the one hand, this is of little practical significance because, for antenna design procedures, it is the frequency allocation that is of primary importance; reflection level, as long as it is below − 10 dB is of secondary relevance. On the other hand, the reason for degraded level rendition are low values of reflection coefficients at the antenna resonances (typically, − 20 dB or less), which implies a considerable amount of numerical noise caused by the EM simulation process itself (related to adaptive meshing techniques as well as terminating the time-domain simulation at relatively high levels of residual energy). The latter is corroborated by considerably better accuracy of representing the levels of the feature-points for Antenna III, which are primarily associated with the antenna gain, less affected by numerical noise due to being integral quantity. www.nature.com/scientificreports/ Another observation is that both reference-design-free constrained modeling approaches (Algorithm 3 and 5) provide similar (Antenna I) or better (Antennas II and III) accuracies than both nested kriging frameworks using the set of reference designs (Algorithm 2 and 4). At the same time, the expenditures required by the proposed approach to determine the surrogate domain are significantly (from five to ten times) lower than that for the nested kriging technique.

Number of training samples
For better visualization, Figs. 5, 6 and 7 shows the scatter plots of the relevant feature points for all the benchmark antennas (the operating frequencies in the case of Antennas I and II, and the frequencies of − 10 dB reflection levels for Antenna III). In all cases, correlation between the surrogate-model-predicted and EM-simulated results is good, even in the case of the model set up using only 20 training data samples. For one hundred data samples, the said correlation is excellent. In addition, Fig. 8 provides also EM-simulated antenna reflection characteristics at the selected test locations, along with the characteristic points yielded by the proposed surrogate. For all antennas, the accuracy of predicting the frequencies that are relevant from the point of view of the assumed design objectives is very good.
In this work, we provided numerical verification of the predictive power of the proposed model. In engineering practice, the purpose of constructing surrogate models for antenna structures is to facilitate design procedures. In particular, the models rendered using the technique presented in this paper can be employed to optimize the antenna structures with respect to performance figures assumed as a part of the objective space. Examples include allocating the operating frequency/bandwidth at their target values, allocating resonant frequencies and improving impedance matching therein, maximizing in-band gain, as well as optimizing dimensions to achieve specific values of the operating frequencies for antenna implemented on the substrate of a specific dielectric permittivity. Application case studies have been provided in our prior works on performance-driven modelling 49,52 .

Conclusion
This work introduced a novel approach to low-cost feature-based surrogate modeling of antenna input characteristics. The proposed surrogate is constructed in the constrained domain, which is determined cost-efficiently using a set of random observables. Our technique enhances the original reference-design-free constrained modeling framework by incorporating the response features technology, thereby allowing for further reduction of the training data acquisition cost. At the same time, it improves the surrogate model predictive power. The proposed modeling procedure has been comprehensively verified using three antenna structures. In all cases, the rendered surrogates are valid for broad ranges of geometry, material and operating parameters. Our approach has been favourably compared to several benchmark techniques: conventional data-driven model, and performance-driven methods operating on the complete antenna responses. The obtained results demonstrate that combining two algorithmic approaches, the reference-design-free model domain definition, and reformulating the modeling task in terms of characteristic points of antenna responses, enables notable computational savings without compromising surrogate model accuracy. The proposed framework may be a viable alternative to conventional data-driven procedures, especially for modeling scenarios that involve multi-parameter spaces and highly nonlinear system outputs. It is particularly suitable for constructing design-ready replacement models valid over broad ranges of operating conditions. Owing to its generic formulation, it can also find applications in various engineering fields that rely on costly simulation models.