Deep Electromagnetic Structure Design Under Limited Evaluation Budgets
Authors: Shijian Zheng
Fangxiao Jin
Shuhai Zhang
Quan Xue
Mingkui Tan
Venue: ICML 2025 poster (2025)
Keywords: Electromagnetic Structure
Surrogate Model
Tree Search
Primary area: applications->chemistry_physics_and_earth_sciences
TL;DR: We introduce a method to efficiently design electromagnetic structures under limited simulation budgets by employing Progressive Quadtree-based Search method and a consistency-based sample selection strategy.

Abstract: Electromagnetic structure (EMS) design plays a critical role in developing advanced antennas and materials, but remains challenging due to high-dimensional design spaces and expensive evaluations. While existing methods commonly employ high-quality predictors or generators to alleviate evaluations, they are often data-intensive and struggle with real-world scale and budget constraints. To address this, we propose a novel method called Progressive Quadtree-based Search (PQS). Rather than exhaustively exploring the high-dimensional space, PQS converts the conventional image-like layout into a quadtree-based hierarchical representation, enabling a progressive search from global patterns to local details. Furthermore, to lessen reliance on highly accurate predictors, we introduce a consistency-driven sample selection mechanism. This mechanism quantifies the reliability of predictions, balancing exploitation and exploration when selecting candidate designs. We evaluate PQS on two real-world engineering tasks, i.e., Dual-layer Frequency Selective Surface and High-gain Antenna. Experimental results show that our method can achieve satisfactory designs under limited computational budgets, outperforming baseline methods. In particular, compared to generative approaches, it cuts evaluation costs by 75∼85%, effectively saving 20.27∼38.80 days of product designing cycle.

Deep Electromagnetic Structure Design Under Limited Evaluation Budgets
Shijian Zheng" Fangxiao Jin
Shuhai Zhang 34
Quan Xue
Mingkui Tan
Abstract
Perform Evaluation
Electromagnetic structure (EMS) design plays critical role in developing advanced antennas and materials, but remains challenging due to high- dimensional design spaces and expensive evalu- ations. While existing methods commonly em- ploy high-quality predictors or generators alle- viate evaluations, they are often data-intensive and struggle with real-world scale and budget constraints. To address this, we propose a novel method called Progressive Quadtree-based Search (PQS). Rather than exhaustively exploring the high-dimensional space, PQS converts the con- ventional image-like layout into a quadtree-based hierarchical representation, enabling progres- sive search from global patterns to local details. Furthermore, to lessen reliance on highly accu- rate predictors, we introduce consistency-dris sample selection mechanism. This mechanism quantifies the reliability of predictions, balancing exploitation and exploration when selecting candi- date designs. We evaluate PQS two real-world engineering tasks, i.e., Dual-layer Frequency Se- lective Surface and High-gain Antenna. Experi- mental results show that our method can achieve satisfactory designs under limited computational budgets, outperforming baseline methods. In par- ticular, compared to generative approaches, it cuts evaluation costs by 75~85%, effectively saving 20.27~38.880 80 days of product designing cycle.
Copper
Null
Candidate Designs
Simulation Software
Output x; (0,1)mxm
Output
y: ((x)
Deliver Feedback
Objective: 0(y))
Optimizer
Performance Metries
Yi = (yn Y.....
Figure 1. Illustration of the EMS design workflow.
domains, ranging from telecommunications to 5G anten- nas, including frequency-selectee surface (Zhu et al., 2022), metamaterials (Chen et al., 2023; Deng et al., 2021), pho- tonic crystals (Peurifoy et al., 2018), and circuit (Cheng et al., 2022; Shahane et al., 2023). Despite broad range of applications, EMS design is challenging primarily for two major reasons.
Firstly, EMS design involves vast design space. For instance, modeling the structure as a 12 x 24 grid with two possible states (metal or empty) per cell, one obtains bd = 2288 x 10 possible configurations. The sheer mag- nitude of this combinatorial space poses a formidable chal- lenge for both domain experts and algorithms, making difficult to locate promising designs in efficient manner.
Secondly, evaluations are costly. Evaluating an EMS design often relies on time-consuming, full-wave electromagnetic simulations, typically governed by complex partial differ- ential equations (Koziel & Ogurtsov, 2014), which cannot be replaced by simpler analytical approximations. Accord- ing to technical report from Inceptra', simulating single design may take anywhere between 660 and 42,780 seconds.
1. Introduction
Electromagnetic structure (EMS) is designed to interact with electromagnetic waves, which crucial for various
Recent works have extended techniques from other high- dimensional, expensive design domains to EMS, typically categorized into two major directions. Predictor-based methods (Koziel et al., 2022; Jing et al., 2022) seek to allevi- ate expensive evaluations by training a deep neural network (DNN) surrogate to approximate costly electromagnetic sim- ulations. Once trained, this surrogate can be integrated into
'School of Future Technologies, South China University of Technology PPngg Cheng Laboratory 'School of Software En- gineering, South China University of Technology 'Pazhou Lab- oratory School of Mieroelectronics, South China University of Technology. Correspondence to: Mingkui Tan <mingkui- tan scut.edu.cn>>, Quan Xue <eeqxue@ www.eesutu.uuuu......
Proceedings of the ndnd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

---

Deep Electromagnetic Structure Design Under Limited Evaluation Budgets
Table I. Comparison of different structure design tasks.
lutions are not overlooked. Together, these two modules accelerate the discovery of high-quality designs, even under real-world large-scale tasks with tight simulation budgets.
NAS
Molccule Design
EMS Design
Large Design Space
v 10 1815
v
10 10
1ligh Evalaaaiion Cost
/f Ramging from minutes days
Public Dataset
V NAS-beach
v ChEMBL
X
Our contributions are summarized as follows:
Pretrained
/ NAS bench- 301
/ Alphafold
x
Predictor Generator
Data Augmentation
Weight Sharing
/ AugLiChem
x
A novel Progressive Quadtree-based Search (PQS) ap- proach for EMS design. We present PQS to tackle the dual challenge of high-dimensional design spaces and limited evaluation budgets. By progressively narrowing down the design space and strategically allocating compu- tational resources, PQS substantially reduces the reliance on massive datasets and repeated full-wave simulations, making it well-suited for industrial scenarios with strict budget constraints.
standard optimizers (e.g., gradient-based or evolutionary algorithms), thus reducing reliance on full-wave solvers. Meanwhile, generative approaches (Brookes et al., 2019; Gao et al., 2023) focus on directly generating promising design candidates under certain performance constraints, of- ten through conditional generative models. By learning end-to-end mapping from performance targets to structural layouts, these methods aim to bypass exhaustive searches over the vast design space. Unfortunately, both predictor- based and generative approaches still exhibit significant drawbacks when scaled to real-world EMS tasks.
A Quadtree-based Search Strategy (QSS) for hierar- chical design exploration. Instead of treating each pixel as an independent variable, introduce hierarchical quadtree representation that partitions the layout into coarser subregions first, and then refines promising ar- eas in finer detail. This strategy effectively mitigates the curse of dimensionality by focusing on higher-po potential regions, thereby accelerating optimization and reducing the overall simulation overhead.
On the one hand, existing methods typically operate directly in a high-dimensional design space, treating every pixel as a free variable to be optimized. As the layout size in- creases, the optimization process can incur rapidly growing evaluation costs, which is commonly known as the curse of dimensionality (Chen et al., 2015). In practical EMS scenarios, where each simulation is time-consuming, such high- gigh-dimensional exploration quickly becomes infcasible.
A Consistency-based Sample Selection (CSS) mecha- nism for adaptive resource allocation. To lessen the reliance on high-fidelity models, CSS measures the reli- ability of surrogate predictions via a consistency metric. When predictions are deemed consistent, more evalua- tions target promising designs; when predictions exhibit high uncertainty, additional true evaluations are triggered to prevent overlooking potentially optimal solutions. This adaptive balance between exploration and exploitation en- hances search efficiency under limited evaluation budgets.
On the other hand, many recent advances heavily rely on high-fieeliyy models, which merely shift the cost from direct full-wave evaluations to extensive data collection. Training such models for EMS tasks often demands tens of thousands of labeled samples, as exemplified by the 10,000-20,00 samples in Wang al. (2023) and the 20,000-2,000,000 in Majorel et al. (2022). Acquiring these large datasets can take months or even years of simulation time, making such solutions impractical for industry applications that must adhere to strict deadlines and budget constraints.
2. Related Work
To overcome the aforementioned challenges, we propose novel Progressive Quadtree-based based Search (PQS) method. Instead of treating every pixel in the layout as free vari- able, PQS introduces a Quadtree- based Search Strategy (QSS). By employing a hierarchical quadtree representation, QSS reduces the intractable complexity of exploring every pixel simultaneously and focuses computational resources on the most promising regions. Moreover, to lessen reliance on high-quality predictors, PQS introduces Consistency- based Sample Selection (CSS) mechanism. CSS leverages consistency metric to gauge the reliability of model out- puts. Based on this metric, CSS adapts its use of the limited evaluation budget. If predictions are consistent, more re- sources are devoted to promising designs. Conversely, the method assigns additional evaluations to designs currently deemed suboptimal, ensuring that potentially superior so-
2.1. Electromagnetic Structure Design
Recent years have seen various data-driven approaches in- troduced to reduce the burden of expensive full-wave simu- lations. For instance, surrogate models coupled with evolu- tionary or gradient-
optimization (Naseri et al., 2022; Jia et al., 2023; Zheng et al., 2023a) have demonstrated effi- ciency in moderateescale EMS scenarios, but constructing accurate surrogates for large-scale spaces remains challeng- ing. Mcanwhile, inverse design with generative models has also gained traction, such cGAN (An et al., 2021) or cVAE (Lin et al., 2022), enabling the direct generation of candidate layouts that meet certain performance criteria. However, most existing approaches still require substantial numbers of samples to train high-fideling models, thereby incurring considerable simulation expenses.
2

---

Deep Electromagnetic Structure Design Under Limited Evaluation Budgets
2.2. Analogous Structure Design
that no single metric falls below an acceptable threshold.
Beyond EMS, data-driven techniques have been extensively explored in domains such neural architecture search (NAS) and molecule design, both of which also share the challenges of vast design spaces and expansive evaluation (see Table 1). For NAS, DARTS (Liu et al.. 2018b) and PNAS (Liu et al., 2018a) iteratively refine candidate neural networks in a vast design space, with benchmarks such as NAS-Bench (Ying et al., 2019; Dong & Yang, 2020) offering public datasets and pretrained predictors (e.g., NAS-Bench- 301 (Siems et al., 2020)). Moreover, augmentation tech- niques such as weight sharing can derive approximate perfor- mance labels from the existing data et al., 2021). These available infrastructures reduce evaluation overhead. Simi- larly, in molecule design, researchers leverage large-scale databases like ChEMBL (Gaulton et al., 2017), pre-trained models (e- (e.g-AlphaFold (Jumper al., 2021); GFlowNets (Madan et al., 2023)), and data augmentation strategies (Ma- gar et al., 2022; Han et al., 2019).
We seek to find a design that maximizes O(S(x)) un- der strict simulation budget Tmxx, yielding the following formulation:
max O(S(x) subject to XE
T S
Taax
(2)
where T is the number of full-wave simulations invoked during the optimization.
Since calling the simulator S is computationally expensive, it is common to introduce an predictor fo to approximate (xx). The details will be presented in Section 4.
4. Proposed Methods
We propose Progressive Quadtree-based Search (PQS), to tackle the EMS design under limited evaluation bud- gets. To this end, PQS integrates two complementary mod- ules: 1) Quadtree-based Search Strategy, which employs a quadtree representation and a progressive refinement pro- cess to reduce the dimensionality of the design space, and 2) Consistency- based Sample Selection, which efficiently allo- cates simulation resources by leveraging imperfect predictor outputs, thus maximizing the benefits of limited evaluations. An overview of our framework is illustrated in Figure 2. Al- gorithm 1 summarizes the entire process. Below, we detail each component of PQS.
In contrast, EMS design is burdened by a larger search space (1086 ~ ) and lacks public datasets, pre-trained models, or augmentation methods (see Table 1). Moreover, EMS are typically developed in a highly customized manner in industrial scenarios, with diverse objectives and boundary conditions. This unique task setting forces researchers to design from scratch, making EMS design fundamentally different from other structure design problems.
4.1. Quadtree-based Search Strategy
3. Preliminary and Problem Statement
Directly searching at the pixel level across an entire EMS layout can be computationally expensive. Therefore, we propose a Quadtree- based Search Strategy (QSS), which leverages a quadtree-basee representation to reduce the scale of the design space and progressive tree-search method to refine only those regions that demand higher resolution. The pseudo-code of QSS is summarized in Algorithm 2.
To formulate EMS design problem, we first give some nec- essary definitions.
Design Space: We consider a binary design space X S {0, 1) 1mmx, where each design x E X' is represented as an m x n matrix. The element Tij indicates a specific material (e.g., metal vs. empty) placed at grid (i.j) in the layout.
Performance Evaluation: To quantify the performance of design x, we employ a high-fidelity simulator S, which merically solves the underlying Maxwell's equations (Bon- deson et al., 2012). The simulator outputs a p-dimensional vector = 5xx)= = [ei..... yp] where each component yk = (x) is performance criterion of interest.
Quadtree- Representation. In conventional pixel- based layouts, every cell is treated uniformly, which inflates the search space. Our idea starts from allowing for varying resolutions across different regions. To this end, we employ quadtree to represent these regions. The quadtree struc- ture recursively subdivides the layout into four child nodes when necessary. Homogeneous areas remain as leaf nodes, requiring minimal variables, while more intricate areas get subdivided into smaller blocks, enabling finer resolution.
Optimization Formulation: In practical tasks, the design must deliver robust performance across multiple criteria simultaneously, rather than excelling in only one while sac- rificing others. We thus define the objective function as the minimum performance among all criteria:
Specifically, each node n in the quadtree Q corresponds to rectangular subregion of the layout matrix x, deter- mined by the row indices [r-ttar ynn and the column in- (1)
((S(x)) = min (x).
SSkSP
dices start sand] Formally, for any pixel (i.j) within n's
subregion, we have
In other words, OsS()) captures the worst-case among the p performance metrics. Maximizing this objective ensures
1 S S
cud
start S j
S c
(3)
R
3

---

Deep Electromagnctic Structure Design Under Limited Evaluation Budgets
Algorithm 1 General scheme of PQS for EMS.
Algorithm 2 EMS Optimization with QSS.
Require:
Initial dataset Do. maximum of simulation runs Timas maximum of tree nodes N
Require: Maximum of leaf nodes Nmxxs predictor fo, size K of Top-K, Maximum iterations M.
Dataset D + Du.
Initialize the root node Ticnot. leaf node set L and Top-K list. for each E [1, MM] do
Current runs of simulation T' + length(Do).
while T < Tux do
while [L] Nuaxx do
Train initial predictor fo-
Randomly select a leaf node n from L.
Conduct optimization under Quadtree-hasee based Search Strat- egy in algorithm 2.
Resample node state &n ~ Uniform([0, ((0, 1}) or split the node based on Eqn. (4).
Conduct Consistency- Sample Selection based on
Reconstruct the design matrix x based on Eqn. (5) and evaluate O ((fx(x)..
Eqn. (10).
Conduct Simulation and obtain feedback ((z,w))
Update the Top-K list.
Add (at, } to dataset D.
end while
Update predictor fe using D.
end for Conduct Depth-wise Importance Assignment based on Eqn. (7),
Update Current runs of simulation T.
end while
Return:The Top-K best designs {xi fxm1
Return: The satisfactory solutions from
If further refinement is needed, node n is split into four child nodes, each covering one of the four quadrants of the original subregion: upper-left (no), upper-right (n1), lower-left (nz), and lower-right (πz). These child nodes recursively divide the region, with their row and column ranges determined by the midpoints of n's intervals:
expands (i.e., maximum of L), we can progressively manage the complexity of the design space in structured manner.
Progressive Tree Search. We propose hierarchical search strategy, which starts from a simple design space and gradually increases the complexity of the design. By partitioning the design matrix into smaller subregions, the search process can adaptively explore finer subregions.
+ y-end
start
snd
'midd
Cmid
'n
(4)
2
2
As shown in Figure 2(a), the tree search process begins with the root node Rooxr which represents sample in the most simplified design space. Subsequently, based on the current set of leaf nodes L, the design matrix x is reconstructed, and its performance ((f((x) (x)) is evaluated by a predictor fo- The tree search process maintains the current Top-K best design matrices x and their corresponding performance values Oj, where O = ((f((x)) represents the k-th best performance found so far.
so that, for example, the upper-left child node no covers rows [r Y'mia] and columns [c% Cmid], with analogous adjustments for the other quadrants.
Otherwise, if no further splitting is performed, node be- comes a leaf node. Each leaf node stores a single binary value s, E {0, 1}. indicating whether its entire subregion is labeled as 0 or 1. This representation reduces redundancy by capturing homogeneous regions with just one bit.
At each iteration, a leaf node is randomly selected from L, and either its state s,, is resampled or is split into four child nodes with randomly initialized states, with both ac- tions following a Bernoulli distribution with parameter 0.5. Resampling explores alternative configurations without ex- panding the design space, while splitting increases search granularity. The design matrix x is reconstructed based on the results of these operations and evaluated by the predictor. If the new performance exceeds the current k-th best perfor- mance O the Top-K list is updated, replacing the k-tth best design matrix x² with x, and recording the corresponding performance value O + (x)())). The iteration repeats until the number of leaf nodes reaches the preset limit N Nmax- at which point the algorithm terminates and returns the final Top-K K best design matrices x
As more nodes subdivide from the root, the quadtree grows to reflect progressively finer resolution. For each pixel æi in the final layout, its value is determined by the unique leaf node n whose subregion contains (i.j). The complete matrix x can thus be reconstructed via:
Zij
Σ Sn, (i, nEL
(5)
where L is the set of leaf nodes, and (i. is an indicator that is if (i, lies in node n'ss subregion and 0 otherwise.
This recursive subdivision effectively compresses the matrix while enabling different resolutions in distinct regions. Let L be the current set of leaf nodes. Since each leaf node has two possible states (0 or 1), the original pixel-wise design space is simplified to
Depth-wise Importance Assignment. In the process of expanding the design space, the initial division of nodes assumes a uniform distribution, treating each newly created leaf node as having equal importance within the EMS matrix.
S = { s = (8n Jnen |
8n E {0, 1))
(6)
whose size is 2| Hence, by controlling how the quadtree
4

---

Deep Electromagnetic Structure Design Under Limited Evaluation Budgets
Expand D alasse with new [x.y]
Increasing
Goal:
Sampling
Yes Deliver
Initial
(x}
(x)
{x.y)
(1) Objective y
dooo
doocoooD
Objective Satisfled Budget exhausted?
(2) Computational B Budget max
%
(a) Quadtree
(b) Consistency based Sample Selection
Simulation Software
{x,y]
Search Strategy
Stage I: Expand Search Tree until
Stage 2: D Depth-wit wise
Importance A ssignment
101%
Tup-k Selection
Fusloianton
0oo0 Step 2
oooo Step
ooooooco
foocoboo Step Depth
foocooo
fa
Step
Step 4
Step Depth =
Feplontioon
(1 1 1) 1209s
fa-- Consistency r Calculation
Candidate Designs
Resources Alocation
(a) Quadtree-based Search Strategy.
Consistency- Sample Selection.
Figure 2. An illustration the proposed PQS. The top portion illustrates the main workflow: (1) the s generates candidate designs in progressively refined design space; predictor Je, then approximates the performance of candidates; Sample Selection mechanism determines which designs proceed to high-fidelity simulations; (4) in the next iteration, the allowable number of leaf nodes N grows enable more complex designs, while fo, is updated with newly evaluated samples. The lower portion provide details of modules: (a) QSS proceeds in two stages, first performing tree search to capture global patterns and then refining local details; (b) CSS uses the prediction consistency r balance between exploitation exploration strategy.
the Consistency-busee
However, certain regions may contribute more significantly overall performance. To address this, we introduce further refinement phase targeting the Top-k designs bet- ter optimize the EMS structure. This further optimization problem is formulated as:
still guide optimization effectively, even with moderate pre- diction errors. Therefore, we adopt Kendall's tau (Kendall, 1938) to measure the consistency between consecutive iter- ations' predicted rankings. Specifically, at iteration 4, we compute Offe, (x)) for candidate designs {x; J and com- pare these values to Offe,,, (x)) from iteration (t-1). The Kendall's ss tau coefficient T is given by:
max O(fo(x:))) ESS
sign(O((s,, (xx)) O(Se..
where Xy' represents the EMS matrix defined by the quadtree structure, and is the search space comprising possible partition parameters in the quadtree:
n(n - 1)
sign (Ofo, O(fe, (x,))),
S' = {(r and start codd
|nE
(8)
where n is the number of data points, and Iy. are the candidate samples. A value of close indicates high consistency, while value is close 0 or even negative indi- cates lower consistency.
4.2. Consistency-base Sample Selection
After the QSS module generates candidate samples, it is of- ten infeasible to simulate all of them due to the high compu- tational cost. Therefore, we must prioritize which samples to evaluate. To achieve this, we propose a Consistency- based Sample Selection (CSS) strategy, which optimizes the evaluation process to enhance search efficiency. The core steps of this strategy are as follows.
Mixed Selection. When the model's predictions become inaccurate, relying solely on the predictor's os' recommended samples may overlook potentially better solutions. To mit- igate such bias, we introduce randomness for additional exploration. Concretely, let 7 be the Kendall's coeffi- cient from the preceding step. A fraction of the total R samples is chosen based on the predictor's highest-ranked candidates, while the remaining randomly. Formally, we define:
Ranking-basse based Prediction Consistency. Since our objec- tive to identify the best design, preserving ranking ac- curacy more critical than achieving precise numerical predictions. model that correctly ranks candidates can
fraction is selected
Rp = T R. R (1 7) R.
(10)
5

---

Deep Electromagnetic Structure Design Under Limited Evaluation Budgets
where Rd is the number of predictor-selected samples and R2 is the number randomly sampled. This mixed-selection strategy balances exploitation (via the predictor) and ex- ploration (via randomness), enhancing robustness under uncertain or less-accurate model predictions.
Considering that structures with similar objective function values can still exhibit differences in quality. Therefore, when the compared methods produce optimal results with closely matched objective function values, we continue to compare the merits of individual criterial (xx)).
Implementation Details. In the predictor-based approach, we employ ResNet50 as the predictor model, initialized with dataset of 300 samples. The total number of sim- ulation runs is limited to 1000 to maintain computational efficiency for Surrogate-GA and Surrogate-RS. For methods like InvGrad and the generative approaches (cGAN, c vVE, and IDN), which require higher model accuracy, we use 6800 and 3800 initial samples for DualFSS and HGA, re- sulting in final sizes of 7000 and 4000, respectively. More implementation details are in Appendix. C.
5. Experiments
We conducted experiments real-world optimization tasks. The experiments answers two key questions: 1) How does PQS compare to state-of-the-art approaches in terms of performance and robustness? 2) How do the individual components of PQS contribute to its overall performance?
Task Settings. Our method is applied to two real-world engineering tasks: 1) Dual-layer Frequency Selective Sur- face (SualFSS), used for electromagnetic noise shielding around chips, and 2) High-gain Antenna (HGA), used in WiFi routers, both involve two optimization objectives. De- tails of these tasks are provided in Table 2 and Appendix. B.
5.1. Comparisons on Real-World Tasks
Table 3 summarizes each method's performance on two real- world tasks. Additionally, Figure 4 presents a comparison of the performance curves across four experimental groups in the HGA task. We highlight three main observations.
Comparison Methods. To highlight the strengths and weaknesses of our approach, we compare it against a diverse set of predictor-bssed, generative, and random sampling methods, which are commonly used in EMS antenna design optimization. The predictor- methods employ a learned surrogate model (or predictor) to approximate the expensive simulation function and guide the optimiza- tion. We evaluate: 1) Surrogate-asistted Genetic Algorithm (Surrogate-GA) (Zhu et al., 2020); 2) Surrogate-asistted Random Search (Surrogate-RS)) RS)); 3) Surrogate- assisted Grey Wolf (Dong & Dong, 2020); 4) varrogate-assisted Gradi- ent Ascent (InvGrad) (Trabucco et al., 2022), which uti- lizes a predictor to acquire the gradient; 5) Two-stage Data- driven Evolutionary Optimization (TS-DDEO) (Zheng et al., 2023b) and 6) Surrogate- assisted hybrid swarm optimiza- tion algorithm (SAHSO) (Li et al., 2022), which represent recent state-of-the-e tfrt---rtt developments in surrogate-assistee evolutionary algorithms. The generative methods learn a direct mapping from target objectives to feasible designs. In our experiments, to ensure fair comparisons, these methods are also updated iteratively in an online fashion. We com- pare: 1) cGAN (Generative Adversarial Network) (An et al., 2021); 2) eVAE (Conditional Variational Autoencoder) (Lin et al., 2022); 3) IDN (Inverse Design Network) (Ma al., 2020), which directly predict designs to fulfill goals; 4)
PQS Achieves Both High Aggregate and Balanced Single-Objective Results. Even with a limited budget of only 1000 simulations, our method attains an aggregate ob- jective of 15.20 dB for DualFSS and 3.66 dB for HGA, markedly surpassing all baselines. Notably, it more than doubles the value of Random Sampling (7.28 dB) under the same budget on DualFSS (a 109% improvement), while also delivering strong individual objectives (15.20 dB for Objl and 31.00 104 dB for Obj2). A similar trend is observed in HGA, where the improvement is more pronounced. This indicates that PQS's superiority in overall design quality.
Predictor-Based Methods Falter in Few-Shot Settings. Despite integrating learned predictors, methods such as Surrogate-RS exhibit lower aggregate objective scores (5.81 dB) than even Random Sampling (7.28 dB) on DualFSS, with SurrogateGGW also underperforming in the HGA (-1.66 dB vs 0.63 dB). This suggests that, under a tight simulation budget (1000 samples), the predictors do not receive enough high-quality training data reliably guide the search. Con- sequently, the search tends become trapped in suboptimal regions, highlighting the vulnerability of purely predictor- driven strategies in data-scarce scenarios.
GenCO (Ferber et al., 2024) which leverages a VAE for generation and a gradient ascent for optimization.
Generative Methods Require Larger Budgets but Un- derperform. Generative models (e.g.. cGAN, CVAE, IDN, GenCO) can be powerful in high-data regimes; however, in this study, we use 7000 simulations, substantially more than the simulations employed by PQS, yet achieve lower aggre- gate objectives. This shortfall underscores the advantage of our progressive quadtree representation and consistency- driven mechanism in effectively allocating a limited budget
Evaluation Metrics. 1). Aggregation Value of Objectives (Agg Obj): To evaluate the search or generation capabilities of different methods, we compare their optimal performance using the O()())) For fair comparison, all methods use the same objective function to guide their optimization or generation process. 2). Single Objective Value (Obj1, Obj2):
6

---

Deep Electromagnetic Structure Design Under Limited Evaluation Budgets
Table 2. Detailed setting of Engineering Tasks.
Problem
x
Design Space Dimension
((c)
Objectives
Objectives2
DualFSS
12*12
10%
S- S-Parameters $2,1
maxx mi 344.5.445 ((x)
maxx pto.sa S(x)(u)
HGA
15*20
10
Realized Gain
maxx millwe (x))w)
maxy mi5 ((x)(u)
Table 3. Comparisons Dual-layer Frequency Selective Surface and High-gain gain Antenna.
Dual-layer Frequency Selective Surface
High-gain Antenna
Method
Age Obj
Obj1* 1
Obj2 f
#Simulations
Agg Obj t
Obj1 f
Obj2 f
#Simulations
RS
7.2824
7.2824
36.6861
1000
0.6314
0.6314
0.7196
1000
Surrogate-RS
5.8116
5.8116
30.2771
1000
3.0857
3.0857
3.1845
1000
Surrogate-GA
4,1946
4.1946
32.1198
10(9)
1,5802
1,5802
4.5598
10(H)
Surrogate-GW
3.1871
3.1871
4.1321
1000
-1.6613
-0,2088
-1.6613
1000
TS-DDEO
5.5627
5.5627
9.0000
1000
0.5201
1.0441
0.5201
1000
SAHSO
4,2066
4,2066
8.0576
1000
0.0589
0.0589
2.2410
1000
cGAN
3.133
8,3658
3.133
7000
-1.0918
-1.0918
-0.6831
4000
<VAE
8,9294
9.5428
8.9294
7000
-1.3657
-1.3657
-0.5326
4000
IDN
4.7335
4.7335
28.8207
7000
-11.9641
-11.9641
-7.1459
4000
InvGrad
2.8941
2.8941
24.6731
7000
3.1783
3.1783
4.4287
4000
GenCO
1.1819
3,9466
1.1819
7000
-5.3032
-5.3032
0.6394
4000
PQS (Ours)
15.1964
15.1964
31.0443
1000
3.6595
3.6595
6.4820
1000
Table 4. Robustmess Comparisons on High-gain Antenna.
of the aggregate and individual objectives over all runs. PQS achieves the highest mean values (4.34, 4.45, and 4.53 for Agg Obj, Obj I, and Obj2. respectively) along with the lowest variance across all objectives, outperforming every baseline in terms of both peak performance and consistency.
Method
Agg 0bj
Obj1* f
Ob2 f
Surrogate-RS RS
-1.14 = 0.82
-0.85 + 1.12
-0.72 = 0.76
Surrgate- GA
-0.93 = 1.16
-0.65 + 1.4
0,36 + 1.12
Surrogate-GWW
-0.96 0.48
0.75
-0.71 0.86
TS-DDEO
0.03 0.42
0.27 0.36
SAHSO
-0.29 = 0.40
0.20 0.75
0,06 0.61
From an engincering perspective, these results suggest that even under challenging, data-scarce conditions, PQS con- sistently discovers effective designs, reducing the risk of suboptimal outcomes due to random initialization or uneven training data. By contrast, other baselines (for example, Surrogate-RS, cGAN, or InvGrad) show higher standard deviations, implying increased sensitivity to training ran- domness and model inaccuracies. Overall, the combination of high average performance and low variance reinforces PQS's ' robustness and reliability in real-world antenna de- sign, where resource- inteeesvee simulations demand stable yet efficient optimization strategies.
cGAN
-2.46 = 1.44
-2.14 1.71
-0.43 = 1.62
eVAE
4.04 = 1.09
-3.97 + 1.07
-1.89 + 2.02
IDN
-17.08 4.23
-17.08 + 4.23
-8.08 + 3.47
InvGrad
-3.73 = 1.72
3.62
-2.11 1.48
GenCO
-3.57 = 1.38
-3.02 + 1.96
-0.34 + 1.58
PQS (Ours)
4.34 1 0.34
4.45 0.30
4.53 + 0.46
toward maximizing multi-objective outcomes.
In short, PQS not only outperforms all baselines in the few- shot regime but also demonstrates robust balance across multiple objectives. These results confirm that our hierarchi- cal representation and sampling strategy can substantially mitigate the data requirements typically faced by predictor- or generator-based solutions.
5.3. Ablation Studies
In this section, we dissect our method's key design choices to understand their individual contributions and overall im- pact on performance. Specifically, we investigate: (1) the effect of varying the maximum dimensionality Nuax in progressive representation, (2) the benefits of the quadtree- based search strategy, and (3) the role of consistency-basee sample selection. All ablation experiments are conducted on the High-gain Antenna (HGA) task for consistency. utilizing identical training and evaluation settings.
5.2. Robustness Comparisons on High-gain Antenna
To assess each algorithm's 's sensitivity to initialization and data variability, we conducted 10 independent runs for both our PQS method and all baselines. Because high-fidelity simulations are computationally expensive, we replaced the simulator with a high-confidence surrogate trained on 37,354 samples from the High-gain Antenna dataset, achiev- ing a KTTau of 0. 0.7969 with respect to real simulations. This ranking correlation is generally sufficient to capture rela- tive performance trends. After fixing the surrogate model's ''s parameters, we applied each method under identical config- urations. Table 5.1 reports the mean and standard deviation
Study on the Number of Variables. As outlined in Section 4.1, the variable Nmax shapes the complexity of the design space and directly influences the challenge of identifying high-quality samples. In this section, we examine the impact of varying Nmax in the context of the HGA task detailed
7

---

Deep Electromagnetic Structure Design Under Limited Evaluation Budgets
Table 5. Effectiveness of Number of Variahles Nmax-
Table 6. Effectiveness of QSS and CSS.
N
Agg Objt
Objl(dB)
Obj2(dB)
Kendall's Taut
QSS
CSS
Agg Objt
Obj2(dByf Obj
16
3.0867
3.0867
6.0295
0,2838 + 0.0527
3.0857
3.0857
3.1845
32
3.6595
3.6595
6,4820
0.2324 1 0.0523
3.2209
3.2209
6,3369
64
3.0233
3.0233
5.1847
0.1255 + 0.0297
V
3.6595
3.6595
6,4820
in the Implementation Details section. We evaluate Nmea values of 16, 32, 64, and as shown in Table 5, our POS achieves satisfactory performance with an objective function value of 3.66 dB when Nmaz 32. indicates that a smaller Nmar is prone to trapping the search locally sub- optimal regions, while larger Nmax makes difficult to construct accurate models given limited budget.
Top-K
CSS(ours)
Random
# Simulations
400
300
200
100
We computed the mean and variance of Kendall's Tau (KTau) for predictors trained on samples generated with varying Naa values, with a fixed total sample size 1000. Samples were split into validation, test, and training sets, and hyperparameter uning was performed separately for each dataset. Using optimal parameters, we conducted 10 tri- als per predictor and calculated KTau. Results showed that Nmaz significantly affects KTau,, with Nmaz 16 yielding the best performance. As Nmaz increased, KTTuu decreased, indicating that predictors map the design space more accu- rately when is smaller. This supports QSS, showing that expanding from low to high-dimensional spaces enhances predictor performance with limited samples.
Sample Selection Strategy
Figure 3. Simulation Costs for Optimal Solution Across Sample Selection Strategy.
3.4
Gain (dB)
RS Surrogate-co GG
Effectiveness of Quadtree-based Search Strategy. We conduct experiments to further demonstrate the effective- ness of our progressive space design. Specifically, we com- pare our methods with a variant which replaces progressive design strategy with random sampling. Our experiments are conducted in High-gain Antenna under the same computa- tional budget and the results are reported in Table 6. From the results, our method outperform the variant without QSS, generating EMS design with higher objective function value (e.g., 3.22dB vs 3..9dBB. These results demonstrate the necessity of the proposed progressive search.
cGAN POStours)
Frequency (GH2)
Figure 4. Simulated Results for High-Gain Antenna Designs ob- tained from Different Methods
labels of the second part were masked. We applied three different sample selection strategies CrCte (ours), Top-K, and Random-to identify the actual optimal solution within the set. In each round, 20 samples were selected and added to the training set to update the predictor, with this pro- cess continuing until the optimal sample was found. Each experiment repeated 20 times. The results in Figure demonstrate that our method outperforms traditional ap- proaches, improving search efficiency by 50%.
Effectiveness Consistency-basee Sample Selection. To verify the effectiveness of the proposed CSS strategy, we compare our methods with variant that simply select the Kop-K structures evaluated by the predictor. Our experi- ments are conducted in High-gain Antenna and we present the objective value in Table 6.It is evident that our method outperforms the variant without CSS, yielding EMS design with higher objective function values (e.g.. 3.22 dB vs. 3.09 dB). These results show that our method could alleviate the bias induced by inaccurate predictions.
6. Conclusion
In this paper, we propose Deep Progressive Search method under Limited Data. Specifically, we devise a Quadtree- based Search Strategy method. By progressively searching in the simplified space, the quality of samples is improved, thus reducing dependence on the number of training sam- ples. In addition, we introduce a Consistency-baat based Sample Selection. With this strategy, the search process can achieve better balance between exploration and exploitation. Ex-
To more clearly demonstrate the advantages of our method, we designed an additional experiments. A dataset of 1000 samples was randomly divided into two equal parts. The first part was used as the initial training for the predictor. The
8

---

Deep Electromagnetic Structure Design Under Limited Evaluation Budgets
tensive experimental results on real-world engineering tasks demonstrate the effectiveness of our method. The results show that our method can achieve satisfactory designs under limited computational budgets and effectively shorten the product designing cycle.
surrogate simulators for artificial electromagnetic materi- als. In Thirty-fifth Conference on Neural Information Pro- cessing Systems Datasets and Benchmarks Track (Round 2). 2021.
Dong. H. and Dong, Z. Surrogate-assisted grey wolf opti- mization for high-dimensional, computationally expen- sive black-box problems. Swarm and Evolutionary Com-
Acknowledgments
This work partially supported by the Joint Funds of the National Natural Science Foundation of China (Grant No.U24A20327).
putation, 57:100715 2020.
Dong, X. and Yang, Y. Nas- mas-bench-201 Extending the scope of reproducible neural architecture search. arXiv preprint SxXi::0001003326 2020.
Impact Statement
Ferber, A. M., Zharmagambetov, A., Huang, T., Dilk- ina, B., and Tian, Y. GenCO: Generating diverse so- lutions design problems with combinatorial nature, 2024. URL https: /openrevieww net/ forum? i id-V7uPprVe10.
This paper presents work whose goal advance the field of Machine Learning. There are many potential societal consequences of our work, none of which feel must be specifically highlighted here.
References
Gao, Z., Tan, C.,, and Li. S. Z. Pifold: Toward effective and efficient protein inverse folding. In The Eleventh Inter- national Conference on Learning Representations, ICLR 2023, Kigali. Rwanda, May 1-5, 2023. OpenReview.net, 2023.
An, S., Zheng, B., Tang, H., Shalaginov, M. Y., Zhou, L., Li, H., Kang, M., Richardson, K. A., Gu, T., Hu, J., et al. Multifunctional metasurface design with a generative ad- versarial network. Advanced Optical Materials, 9(5): 2001433, 2021.
Gaulton, A., Hersey, A., Nowotka, M., Bento, A. P., Cham- bers, J., Mendez, D., Mutowo, P., Atkinson, F., Bellis,
Bondeson, A., Rylander, T., and Ingelström, P. Computa- tional electromagnetics. Springer, 2012.
L. J., AbbrinnUUhale, E., et al. The chembl database in 2017. Nucleic acids research, 45(D1)):995-DD D954, 2017.
Brookes, D., Park, H., and Listgarten, J. Conditioning by adaptive sampling for robust design. In International conference on machine learning, pp. 773-782. PMLR, 2019.
Han, X., Zhang, I.., Zhou, K., and Wang, X. Progan: Pro- tein solubility generative adversarial nets for data aug- mentation in dnn framework. Computers & Chemical Engineering, 131:106533, 2019.
Chen, J., Qian, C., Zhang, J., Jia, Y., and Chen, H. Corre- lating metasurface spectra with a generation-elimintton framework. Nature Communications, 14(1): 14(1):4872, 2023.
Jia, Y., Qian, C., Fan, Z., Cai, T., Li, E.-P., and Chen, H. A knowledge-inherite learning for intelligent metasurface design and assembly. Light: Science & Applications, 12 (1):82, 2023.
Chen, S., Montgomery, J., and Bolufé-Röhler A. Measuring the curse of dimensionality and its effects on particle swarm optimization and differential evolution. Applied Intelligence, 42:514-551 2015.
Jing, G., Wang, P., Wu, H., Ren, J., Xie, 7.,, Liu, J., Ye, H., Li, Y., Fan, D., and Chen, S. Neural network-based surro- gate model for inverse design of metasurfaces. Photonics Research, 1066):1462-14711 2022.
Cheng, R., Lyu, X., Li, Y.. Ye, J., Hao, J.. and Yan, J. The volicy-gradics placement and generative routing neural networks for chip design. Advances Neural Information Processing Systems, 35:26350-2633 26362, 2022.
Jumper, J., Evans, R., Pritzel, A., Green, T., Figurnov, M., Ronneberger, O., Tunyasuvunakool, K., Bates, R., Židek, A., Potapenko, A., et al. Highly accurate protein structure prediction with alphafold. Nature, 596(7873):583-588) 2021.
Daulton, S., Wan, X., Eriksson, D., Balandat, M., Osborne, M. A., and Bakshy, E. Bayesian optimization over dis- crete and mixed spaces via probabilistic reparameteriza- tion. Advances in Neural Information Processing Systems, 35: 15:12760-12771 2022.
Kendall, M. G. A new measure of rank correlation.
Biometrika, 30(1-2):81-93, 1938.
Deng, Y., Dong, J., Ren, S., Khatib, O., Soltani, M., Tarokh, V., Padilla, W. and Malof, Benchmarking data-driven
Koziel, S. and Ogurtsov, S. Antenna design by simulation- driven optimization. Springer, 2014.

---

Deep Electromagnetic Structure Design Under Limited Evaluation Budgets
Koziel, S., Çalik, N., Mahouti, P., and Belen, M. A. Low- cost and highly accurate behavioral modeling of an- tenna structures by means of knowledge-based domain- constrained deep learning surrogates. JEEE Transactions on Antennas and Propagation, 11(1):105-118, 2022.
Ren, P., Xiao, Y., Chang. X., Huang, P.-Y., Li, Z., Chen, X., and Wang. X. A comprehensive survey of neural architec- ture search: Challenges and solutions. ACM Computing Surveys (CSUR), 54(4):1-34, 2021.
Shahane, A. H., Manjiri, S. V. S., Jain, A., and Kumar, S. Graph of circuits with gnn for exploring the optimal design space. In Thirty-seventh Conference Neural Information Processing Systems, 2023.
Li, F., Li, Y., Cai, X., and Gao, L. A surrogate-assister hy- brid swarm optimization algorithm for figh-dimensional computationally expensive problems. Swarm and Evolu- tionary Computation, 72:101096 2022.
Siems, J., Zimmer, L., Zela, A., Lukasik, J., Keuper, M., and Hutter, F. Nas-bench-301 and the case for surrogate benchmarks for neural architecture search, arXiv preprint arXiv:2008 2007v:2008.09777, 4:14, 2020.
Lin, H., Hou, J., Wang, Y., Tang, R.,. Shi, X., Tian, Y., Xu,
W., et al. Machine-leammin-asiiee inverse design of scattering enhanced metasurface. Optics Express, 30(2): 3076-3088, 2022.
Trabucco, B., Geng, X., Kumar, A., and Levine. Design- bench: Benchmarks for data-driven offline model-based optimization. In International Conference Machine PMLR, 2022.
Liu, C., Zoph, B., Neumann, M., Shlens, J., Hua, W., Li, L.-J., Fei-Fei, L., Yuille, A., Huang, J., and Murphy, K. Progressive neural architecture search. In Proceedings of the European conference on computer vision (ECCV), pp. 19-34, 2018a.
Learning, pp.
Wang. Y., Yang. Z., Hu, P., Hossain, S., Liu, Z., Ou, T.- H., Ye, J., and Wu, W. End-to-end diverse metasurface design and evaluation using an invertible neural network. Nanomaterials, 13(18)22561, 2023.
Liu, H., Simonyan, K., and Yang, Y. Darts: Differentiable architecture search. arXiv preprint arXiv: 118800009555 2018b.
Ying, C., Klein, A., Christiansen, E., Real, E., Murphy, K., and Hutter, Nas-bench-10k 101: Towards reproducible neural architecture search. In International conference machine learning, pp. 7 7105-7114 14. PMLR, 2019.
Ma, J., Huang, Y., Pu, M., Xu, D., Luo, J., Guo, Y., and Luo, X. Inverse design broadband metasurface absorber based on convolutional autoencoder network and inverse design network. Journal of Physics D: Applied Physics, 53(46)):64002,, 2020.
Zheng, L., Karapiperis, K., Kumar, S., and Kochmann, D. M. Unifying the design space and optimizing linear and non- linear truss metamaterials by generative modeling. Nature Communications, 14(1):7563, 2023a.
Madan, K., Rector-Broode Brooks, J., Korablyov, M., Bengio, E., Jain, M. Nica, A. C., Bosc, T., Bengio, Y., and Malkin, N. Learning gflownets from partial episodes for improved convergence and stability. In International Conference PMLR, 2023.
Zheng, L., Shi, J., and Yang. Y. two-stage surrogate- assisted meta-heuristic algorithm for high-dimensional expensive problems. Soft Computing, 27(10):4465-4488 2023b.
on Machine Learning, pp.
Magar, R., Wang, Y., Lorsung, C., Liang, C., Ramasubra- manian, H., Li, P., and Farimani, A. B. Auglichem: data augmentation library of chemical structures for machine learning. Machine Learning: Science and Technology, (4):045015, 2022.
Zhu, F.,, Li, F.,, Wei, Z., and Yin, W.-Y. Adversarial-networrk regularized inverse design of frequency-selective surface with frequency-temporal deep learning. IEEE Transac- tions on Antennas and Propagation, 70(10):9400-9469, 2022.
Majorel, C., Girard, C., Arbouet, A., Muskens, O. I.., and Wiecha, P. R. Deep learning enabled strategies for mod-
eling of complex aperiodic plasmonic metasurfaces of arbitrary size. ACS photonicx, 92):575-588, 2022.
Zhu, R., Qiu, T.. Wang. J., Sui, S., Li, Y., Feng, M., Ma, H., and Qu, S. Multiplexing the aperture of a metasurface: in- verse design via deep-learning forward genetic algorithm. Journal of Physics D: Applied Physics, 53545)4455002, 2020.
Naseri, P., Goussetis, G., Fonseca, N. J., and Hum, S. V. Synthesis of multi-band reflective polarizing metasur- faces using a generative adversarial network. Scientific Reports, 17016, 2022.
Peurifoy, J., Shen, Y., Jing. I.,, Yang, Y., Cano-Renteria, F., DeLacy, B. G., Joannopoulos, J. D., Tegmark, M., and Soljačić, M. Nanophotonic particle simulation and inverse design using artificial neural networks. Science advances, 4(6):eaar4206, 2018.
10

---

Deep Electromagnetic Structure Design Under Limited Evaluation Budgets
A. Supplementary Materials
In the supplementary, we provide more implementation details and more experimental results of our PQS. We organize our supplementary as follows.
In Section B, we provide more details of the considered two real-world challenging electromagnetic structures, dual-layer frequency selective surface and high-gain antenna
In Section C, we depict more implementation details of our PQS and the compared methods.
In Section D, we give more experimental results demonstrate the effectiveness of our PQS.
B. More Details on Electromagnetic Structures
Dual-layer Frequency Selective Surface. The dual-layer Frequency Selective Surface (DualFSS) is an electromagnetic structure specifically designed for selectively filtering electromagnetic waves. It consists of two layers of conductive elements, each containing a grid or array of metallic elements exhibiting specific resonant behavior at particular frequencies. This configuration endows the DualFSS with rrqueeny--depeneen transmission and reflection characteristics. Despite its more intricate structure compared to a single-layer FSS, the DualFSS provides higher degrees of freedom, allowing for more flexible performance adjustments across different frequencies. In practical applications, the DualFSS finds common usage in scenarios demanding enhanced performance and broader frequency coverage, such RF communication, radar systems, and engineering designs within the radio frequency spectrum. In our experiments, we utilized the proposed methodology to design the structure of the DualFSS. focusing on the two layers of metallic grids. The designed structure aims to meet specific performance criteria, with detailed parameters and expected metrics outlined below.
Specifically, the DualFSS under investigation is depicted in Figure The FSS is composed of three layers of boards, each having a thickness of 0.035mm, and features cylindrical elements with radius of 0. 1mm. It is noteworthy that the bottommost layer, representing the grounded metallic surface, remains unaltered throughout the optimization process. In contrast, the upper two layers, initially comprising entirely of air, constitute the optimization space. The objective of the optimization is to strategically convert certain regions within the air layers into metallic elements, adhering to the constraint that each designed metallic block must have a minimum size of 0.2 0.2m0..mm. The optimization encompasses determining the specific configuration of metallic blocks within the upper layers to achieve desired electromagnetic properties. Importantly, the maximum extent of the air region available for optimization corresponds to the footprint of the bottommost layer.
In terms of optimization objectives, this scenario aims to eliminate electromagnetic noise caused by ultra-high-frequene circuits, preventing such noise from interfering with the operation of mobile phone cameras. The high-frequency noise primarily occurs in two frequency bands: 10.5-15.5 GHz and 31.5-34.5 GHz. We evaluate the suppression capability using the S-Parameter 1, which is a parameter that takes only negative values. A smaller magnitude of this parameter indicates stronger suppression performance. We aim to achieve broad absorption capability in the 31.5-34.5 GHz range, while maximizing suppression in the 10.5-15.5 GHz range. To enhance generalization, it is necessary to minimize the maximum value within the 31.5-34.5 GHz band, while minimizing the minimum value within the 10.5-15.5 GHz band to strengthen absorption performance. Accordingly, we define the first objective as minimizing the maximum value of S-Parameters 12,1 over the 31.5-34.5 GHz band, represented by the formula minx max 5.34.. ((x)(u). By taking the inverse, we can transform it into maximization problem and define it using a new mathematical expression maxx [ 5,,445 - (u)) and denote this as Objl. Similarly, the second objective is to minimize the minimum value of S-Parameters 12 over the 10.5-15.5 GHz band, defined by the formula miny e (1)) We can also transform into a maximization problem and represented by the formula max, max 15.5] - (u)). and referred to as Obj2. In this formulation, x represents the vector of structural design parameters, while u denotes the frequency in GHz, serving as the independent variable across both frequency bands. The term ((x)(u) refers to the S-Parameters 12 of the FSS for a given design x at a specific frequency t.
An aggressive objective function is set to maximize the worst-case performance of the single objective to achieve balanced shielding capabilities. This is mathematically expressed as maxx (min (Objn (x), Objz(x))), where Obj((x) represents min.g(311.534.55 (u) while Objz(x) represents max (s))
High-gain Antenna. The high-gait gain antenna is a specialized electromagnetic structure designed to achieve significant directional amplification of radio frequency signals. This type of antenna is characterized by its ability to focus transmitted

---

Deep Electromagnetic Structure Design Under Limited Evaluation Budgets
Dual- Dual-layer Frequency Selective Surface Size Parameters
Optimization Area
0.20
229 mm
0.60 min
0.30
0.50
Optim ze ion Area
Modeling Diagram of Dual-layer Frequency Selective Surface
1. Load design parameters (Area 1)
2. Create metal unit
3. Rotate 90 degrees around the axis times
Optimization Area
E
E
4. Load design parameters (Area 2)
Create metal unit
degrees around axis three times
Optimization
Figure 5. The detailed settings of the Dual-layer Frequency Seleetive Surface.
received signals in specific direction, resulting in a concentrated radiation pattern. In the design of it, the integration of metal array plays a crucial role in shaping the antenna's 's radiation pattern and achieving enhanced performance. The metal array structure involves carefully arranged grid array of metallic elements, such as reflectors and directors, strategically positioned to optimize the antenna's is gain and directional characteristics. High-gain antennas find extensive applications in scenarios requiring long-range communication, satellite communication, and situations where a concentrated signal strength essential.
Specifically, our structure a rectangular prism with dimensions of 60 mm in length, 40 mm in width, and 4.6 mm in thickness in Figure 6. The prism is then divided into halves along the midpoint of its length, parallel the width. One half is designated the design region, and after the design completed, it is mirrored across the vertical plane of symmetry.
The target of this scenario is to design a dual-band router antenna operating at both 2.4 GHz and GHz, ensuring optimal communication performance in both frequency bands simultaneously, rather than having one band perform well while the other lags. Consequently, our two objective functions represent the minimum Realized Gain for the 2.4 GHz and GHz bands, respectively. An Aggregation Value of Objective set to maximize the worst-case performance across both objectives, aiming to achieve effective dual-band communication.
Specifically, since the minimum value within given range dictates the weakest communication capability of that band, to meet the communication requirements of the 2.4 GHz band, the first objective is defined as maximizing the minimum Realized Gain in the 2.55-2.55 GHz frequency range. This is mathematically expressed as maxx miinue/2.45.2.00 Su)) and denoted as Obj1. In this formulation, x represents the vector of structure design, while u denotes the frequency in GHz, acting as the independent variable across both frequency bands. The term (1x) denotes the Realized Gain the antenna for given design a specific frequency u. Similarly, to ensure robust communication in the 5 GHz band, the
12

---

Deep Electromagnetic Structure Design Under Limited Evaluation Budgets
High-gain Antenna Size Parameters
90.00
40 0,000 mm
Design Area
Modeling Diagram of High-gain Antenna
Load design parameters (Area 1)
Create metal unit
Building symmetrical structures
Figure Designs of the High-gain Antenna.
second objective is defined maximizing the minimum Realized Gain the 5.0-6.0 GHz frequency range, expressed as maxx [5s.g (u) and denoted Obj2.
To achieve strong communication performance across both bands, the smaller of the two objective values is chosen as the Aggregation Value of Objective. This is mathematically expressed as:
((x)(u), min S(x)(w) wE[5.6]
max min
min [5522.45,2.58)
This formulation ensures that the EMS design is optimized for both frequency bands by focusing improving the worst-case communication performance across the two bands, thereby achieving balanced and robust Aual-band communication.
C. More Implementation Details
PQS (ours). For the proposed PQS, maintain consistent setup across both experimental scenarios. In both cases, the initial dataset comprises 300 samples, derived from progressive design strategy, which allows for more adaptive and efficient sampling. We utilize ResNet50) as the predictor, ensuring consistency in model structure across different approaches and facilitating a fair comparison of performance. The design variable Nmaz is set to 32. Additionally, when implementing CSS, we estimated the initial r (at = 0) by training multiple predictors with independent random initializations. We set maximum iteration
10000000 and K 10. The total number samples R in CSS is 10.
Compared Methods. We re-implement the following state-of the-art electromagnetic structures design methods in our two challenging task, dual-layer frequency selective surface and high-gain antenna. More details of the baseline design methods provided in the subsequent discussion.
Random Sampling (RS). In this approach, we randomly generate 1000 electromagnetic structures, evaluating them with the simulation software.
Bayesian Optimization (Daulton et al., 2022). After attempting, Bayesian optimization failed to generate non-empty structures, and the updates of new samples did not alter the algorithm's 's behavior. We believe that Bayesian methods are
13

---

Deep Electromagnetic Structure Design Under Limited Evaluation Budgets
not effectively applicable our scenario due to the curse of dimensionality. Therefore, we did not include them in the comparative experiments.
Surrogate-assisted Random Search (Surrogate-RS). This approach adopts the ResNet50 architecture as the surrogate model in both scenarios, consistent with the setup in PQS, to ensure a fair comparison between the methods. The surrogate model is designed to provide prediction of the objective function given the input of the electromagnetic structures and is ultilized to guide the random search. Specifically, we begin by randomly sampling 300 electromagnetic structures to train an initial surrogate model. In each subsequent iteration, the surrogate model guides the selection of the Top-K samples from M randomly sampled candidates (where M > K), which are then evaluated through simulation. The surrogate model is updated accordingly, and this process is repeated until the total number of simulated samples reaches the predefined limit of 1000. Finally, we select the best one as optimized result. In practice, we set M = 200000 and K 10 for our experiments in both two real-world tasks.
Surrogate- GA (Zhu et al., 2020). This method exploit surrogate model to accelerate the evolutionary algorithm. Specifically, this method fit a DNN-based surrogate model with a simulated dataset to assist fitness evaluation during the evolution process. In particular, we have adapted the mutation operator to suit our electromagnetic structural scenario by modifying it to perform transformations between and I in the matrix elements. In our experiments, we use the ResNet50 model as the surrogate model. The model's batch size is set to 256, trained for 200 epochs, with learning rate of 0.01. First, 300 samples were obtained through random sampling, and simulations were performed on these samples to train the surrogate model. Based on the surrogate model, we set K=10 for our experiments, meaning that in each generation, the top 10 samples are selected using a Top-K strategy for simulation verification. The surrogate model is then updated with these results, and a new population is generated. This process continues until the total number of simulated samples reaches 1000.
Srroogatt-GG(Donngg & Dong, 2020). The authors introduced method based on the Grey Wolf Optimizer (GWO) for designing electromagnetic structures that satisfy specific electromagnetic performance requirements. GWO is a metaheuristic optimization algorithm inspired by the hunting behavior of grey wolves in nature. By simulating the hierarchical social structure of wolf pack (a, B, 8, w) and the dynamic processes of encircling, chasing, and attacking prey, GWO demonstrates excellent global search and local exploitation capabilities in complex high-dimenssonal optimization problems. In this study, the initial population consists of 500 randomly sampled simulation samples for DualFSS and 300 for HGA. Subsequently, GWO simulates the cooperative hunting mechanism of wolf pack by continuously adjusting the positions of the leader wolves (a, B, 8) and the remaining wolves, guiding them toward the current optimal individual while maintaining population diversity to prevent premature convergence to local optima. Regarding the experimental setup, the parameter which controls the balance between exploration and exploitation during the search process, is set to 3. As the same as Surrogate-GA, we use the ResNet50 model as the surrogate model. The model's batch size is set to 256, trained for 200 epochs, with a learning rate of 0.01. Based on the surrogate model, we set K=10 for our experiments, meaning that in each generation, the top 10 samples are selected using a Top-K strategy for simulation verification. The surrogate model is then updated with these results, and a new population is generated. This process continues until the total number of simulated samples reaches 1000.
InyGrad (Trabucco et al., 2022). Trabucco et al. introduces a simple baseline method based on gradient ascent. In this approach, a ResNet50- baasee surrogate model is initially trained on a dataset of electromagnetic structures, which is designed to establish an accurate mapping between the electromagnetic structure and the objective function. Specifically, the dataset consists of 6800 randomly sampled simulation samples for DualFSS and 3800 for HGA, which represent the minimum number of samples required to ensure the stability and reliability of the model without compromising its performance. Subsequently, the method performs multiple gradient updates the input electromagnetic structure based on the surrogate model output, ultimately yielding an satisfactory electromagnetic structure that satisfies the specified criteria. The gradient update could be formulated as I + 2z + a (I),, where t represents the update step and a denotes the learning rate. In pratice, we set T = 1000, a = 0.01 for two design tasks. We randomly sampled 10 electromagnetic structures, input them into the surrogate model for optimization, and forwarded the optimized results to the simulation software for evaluation. The surrogate model is then updated with these results, and this process is repeated until the total number of simulated samples reaches the predefined limit of 7000 for Du and 4000 for HGA.
IDD (Ma et al., 2020). Ma et al. introduces a baseline method for inverse design based on Convolutional Autoencoder Network (CAN) and Inverse Design Network (IDN). In this approach, the authors utilized CAN to compresses input spectrums with the dimension of 1 1*1000 into dow-dimensional spectrums with the dimensionof $00 Subsequently, the
14

---

Deep Electromagnetic Structure Design Under Limited Evaluation Budgets
compressed latent space values were fed into the IDN with the expectation of generating structures that conform the input spectrum. In our experiments, we set the target values as either the maximum or minimum values within specific frequency range, eliminating the involvement of high-dimensional inputs. Consequently, we exclusively adopted the IDN component of the method for our purposes,In In terms of network architecture, we introduced an additional fully connected layer before the first convolutional layer of the Inverse Design Network. This layer elevates the input target values to a 50-dimensional space to align with subsequent dimensions. The model's batch size is set to 128, trained for 200 epochs, with a learning rate of 0.0001. Adam optimizer is employed, and MAB(Mean Absolute Error) is used for loss computation. Initially, we performed random sampling to generate 6800 samples for DualFSS and 3800 samples for HGA. These samples were simulated to calculate their respective objective values, which were subsequently used to train the initial models. In the next stage, the models were utilized to generate 10 additional samples, which underwent simulation-based validation. The validated samples were then added to the dataset, and the models were retrained iteratively. This process was repeated until the simulation budget of 7000 and 4000 was reached for DualFSS and HGA, respectively. Ultimately, the sample with the best simulation performance during this process was selected as the final result.
cGAN (An et al., 2021) An et al. presents a generative adversarial network that can generate metasurface designs to meet design goals. Generative adversarial nets can be extended to a conditional model if both the generator and discriminator are conditioned on some extra information. It could be any kind of auxiliary information, such as class labels or data from other modalities. We can perform the conditioning by feeding extra information into the both the discriminator and generator as additional input layer. Consequently, cGAN introduces extra information as conditions in both the encoder and decoder inputs to confer the ability to generate pecific structures based on varying conditions. model's batch size is set to 64, trained for 200 epochs, with a discriminator learning rate of 0.00005 and generator learning rate of 0.0002. In addition, the latent dimension is set to 100, Adam optimizer is employed. We began by randomly sampling 6800 instances for DualFSS and 3800 instances for HGA, followed by simulations to derive their objective values for training the initial models. Using these models, 10 new samples were generated and validated through simulations. These validated samples were incorporated into the dataset, and the models were updated iteratively. This iterative procedure continued until the simulation budgets-7000 for DualFSS and 4000 for HGA- were exhausted. The final result was determined by selecting the sample exhibiting the optimal performance during simulations.
vVE (Lin et al., 2022). Lin et al. introduces an approach utilizing Conditional Variational Autoencoder (cVAE) to generate metasurface retroreflectors (MRF) structures satisfying specified performance criteria. cVAE represents a variant incorporating both Variational Autoencoder (VAE) and Autoencoder (AE) principles. While VAE extends the training paradigm of AE by transforming it from encoding inputs into a single point in latent space encoding inputs into a distribution in latent space, endowing it with generative capabilities, the generated content is inherently uncontrollable. Consequently, eVAE introduces conditions in both the encoder and decoder inputs to confer the ability to generate specific structures based on varying conditions. The model's 's batch size is set to 128, trained for 200 epochs, with a learning rate of 0.0005. In addition, the latent dimension is set to 20, Adam optimizer is employed, and the loss function is obtained through linear summation of Mean Squared Error (MSF) and 0.00000001 times the Kullback-Leibe (KL) divergence. An initial random sampling of 6800 samples for DualFSS and 3800 samples for HGA was conducted, with simulations performed to compute the corresponding objective values for initial model training. The trained models then produced 10 new samples, which were subjected to simulation validation. These validated samples were appended to the dataset, and the models were retrained iteratively until the simulation budgets of 7000 for DualFSS and 4000 for HGA were fully utilized. The final output was chosen as the sample demonstrating the highest simulation performance during the process.
GenCO (Ferber et al., 2024) GenCO utilizes VAE (Variational Autoencoders) to generate a variety of designs that account for specific constraints, such as those encountered in nanophotonic materials. Following the approach outlined in the original paper, we integrate electromagnetic structure performance as a constraint objective into the model's training loss function. GenCO requires computing the gradient of the objective function with respect to the design variables. However, in our case, obtaining such gradient information directly through simulation is unavailable. To overcome this challenge, use a surrogate model to approximate the gradient. The surrogate model, implemented using ResNet50, serves to predict the objective function's 's gradient efficiently. The training parameters of the surrogate model, such as the network architecture and optimization procedure, are consistent with those used in similar works. Since the original paper does not provide detailed hyperparameter settings for VAE, we made reasonable choices based on standard practices for training generative models. We use a four-layer convolutional and transposed convolutional network architecture for the VAE model. The specific training parameters for our implementation are as follows: latent dimension = 255,, number of
15

---

Deep Electromagnetic Structure Design Under Limited Evaluation Budgets
Table Effecc of Importance Assignment.
Method
Objective Function Valuet
Objectivel
Objective2(dB))
PQS w/o IA PQS
2.32
2.32
5.57
3.66
3.66
6.48
embeddings 512, learning rate 1e-3, and the number of epochs 100.
TS-DDEO (Zheng et al., 2023b). Zhen et al. introduces a two-stage data-driven evolutionary optimization algorithm for solving computationally expensive in high dimensions optimization problems. In the first stage, a surrogate-assisted hierarchical particle swarm optimization method (SHPSO) is used to identify a promising area in the entire search space, In the second stage, a best data-driven optimization method (BDDO) is used to improve the convergence speed in the promising area. In our electromagnetic structure scenario, used the ResNet50 model the surrogate model of TS-DDEO and randomly generated and evaluated 500 simulation samples of DualFSS (300 for HGA) to train the initial surrogate model. The batch size of this model was set to 256, and it was trained for 200 epochs, with a learning rate 0.01. Consistent with the parameters set in the original paper, we set the population size of DE sampling in SHPSO and BDDO to 50, and the probability r of FC strategy in the BDDO phase was 0.2. In addition, out of a total budget of 1000 simulation evaluations, we allocated 300 evaluation budgets for the first phase, and then switched to the second phase. After searching for candidate populations in each stage, we select the top 10 candidate solutions for simulation verification. Then, we use the simulated results to update the historical database and retrain the ResNet50 model.
SAHSO (Li et al., 2022). Li et al. propose a surrogate-assisted hybrid swarm optimization algorithm (SAHSO) for high-dimensional computationally expensive problems. In the first stage, the Teaching-leaminn--asse optimization (TLBO) learner phase is employed to enhance global exploration through random interactions among individuals. In the second stage, the algorithm switches to standard partiele swarm optimization (SPSO) to accelerate local exploitation, with the transition controlled by To- In our scenario, we adapted SAHSO with modifications to address high dimensionality. To enhance exploration, we increase the duration of the first stage by setting To to 60% of the maximum evaluation budgets. During the second stage, to ensure both computational efficiency and fair comparison with other methods, the number of pre-screened candidate solutions was limited to 10-15 per iteration. The original surrogate model RBF was replaced with ResNet50. In each iteration, we randomly sample 30,000 candidates, evaluate them using ResNet50, and select the one with the highest predicted performance the surrogate optimum for updating the global best. The ResNet50 model was trained using 500 initial simulation samples for DualFSS and 300 for HGA, with batch size of 256, 200 training epochs, and learning rate of 0.01.
Time-saving Calculation. We present detailed breakdown of the average computational time savings achieved by method compared to generative approaches. Based on our statistics, the average simulation time per sample is 583.7 seconds on Dual-layer Frequency Selective Surface and 558.8 seconds on the High-gain Antenna. The time-saving calculation is as follows:
Dual-layer Frequency Selective Surface: Compared to generative approaches, our method reduces the number of required simulations by 3,000. The total time saved is 583.7x3.000 ~ 20.27 days.
3,600 24
High-gain Antenna: Compared to generative approaches, our method reduces the number of required simulations by 6,000. The total time saved is 3,600x21
558. 8,0000 xv 38.80 days.
These results clearly demonstrate that our method achieves significant efficiency gains (75~85% cost reduction) compared to generative approaches.
D. More Experimental Results
Effect of the Number of Variables Nmar. To further illustrate the effect of the parameter Nmax: we present kernel density estimation (KDE) plots of the sample performance distribution under different parameter settings in Figure 7. We sampled and simulated 1000 samples for each parameter setting and random sampling. The experimental results demonstrate that our method is more likely to sample higher- perbbrrnefoominggggg instances across various parameter configurations, whereas most the samples generated through random sampling tend to cluster in the lower performance range.
16

---

Deep Electromagnetic Structure Design Under Limited Evaluation Budgets
30
SandonSSmpiing
GursiNi-15)
OursiN-32)
OursIN-64)
Density
65-
Objective Function Value
Figure Comparison of Sample Performance Distribution under Different Parameter Settings.
Laver3
Layer2
Layeri
Layor2
Layerd
Layer2
Syrrl
RS
Surtogate-GA
<VAE
InvGrad
Layeri
Layer2
Surrogate-RS
Layer2
Layorl
Layer2
Layorl
Layer2
Surrogate
TS-DDEO
SAJISO
PQS(ours)
Figure 8. Optimized Electromagnetic Structures of Different Methods on Dual-layer Frequency Selective Surface.
Effect of Dipt Importance Assignment. We investigate the effect of the depth-wise importance assignment. For fair comparison, we conduct this experiment under the same simulation budget in high-gain antenna design task. From Table without importance assignment, the PQS tends to find sub-optimal electromagnetic structure. When equipped with the proposed importance assignment, the searched structure consistently outperforms that without importance assignment (e.g., 3.66dB 2.32dB). These findings illustrate the essential nature and efficacy of the introduced depth-wise importance assignment.
Comparisons of electromagnetic structures on Dual-layer Frequency Selective Surface. In Figure 8, we visualize the optimized electromagnetic structures searched by our proposed method and the baseline methods in the task of dual-layer frequency selective surface. can observed that, compared to the baseline methods, the structures designed by our approach exhibit better adherence to physical priors, showcasing more regular and manufacturable design.
Comparisons of Simulated Results Dual-layer Frequency Selective Surface. In Figure 9, We present the simulated results of the optimized electromagnetic structures for our methods and all baseline methods. From the results, it is evident that the optimized electromagnetic structures obtained through our proposed method exhibit lower values in the frequency ranges [31.5, 34.5] and [10.5, 15.5]. This indicates that our dual-layer frequency-selectiv surface performs better, providing empirical evidence for the effectiveness of our optimization algorithm.
Comparisons of electromagnetic structures on High-gain Antenna. Figure 10, we visualize the optimized electro-
17

---

Deep Electromagnetic Structure Design Under Limited Evaluation Budgets
-10
-22
PV (0h1
RS
Surrogate-RS
oVAE
InvGrad
-31
Surrogate-GA
cGAN
GenCO
-10
-10
12
-25
1-2
18
20
26
-00
20
Surrogate-GWW
TS-DDEO
SAHSO
PQS(ours)
Pigure 9. Simulated Results of Optimized Electromagnctic Structures on Dual-layer Frequency Selective Surface.
magnetic structures designed by our proposed method and the baseline methods in the task of high-gain antenna. be observed that, compared to the baseline methods, the structures designed by our approach also exhibit better adherence to physical priors, showcasing more regular and manufacturable design. This further demonstrates the strong generalization ability and robustness of our proposed method, proving its effectiveness across multiple real-world tasks.
Comparisons of Simulated Results on High-gain Antenna. In Figure 11, We further present the simulated results the optimized electromagnetic structures for our methods and all baseline methods. The experiments further demonstrate that in the frequency ranges [2.45, 2.55] and [5.00, 6.00), the structures designed by our method significantly outperform all baseline methods, achieving substantial performance improvements in the target frequency bands.
Visualizations of Optimized Dual-layer Frequency Selective Surface. We illustrate the optimized electromagnetic struc- tures obtained through our proposed methodology and the reference methods in the High-gain Antenna task. In Figure 12, in contrast to the reference methods, the structures formulated by our approach demonstrate a heightened conformity to physical priors, presenting a more regular and manufacturable design.
Visualizations of Optimized High-gain Antenna. In Figure 13, the results indicate that the optimized electromagnetic structures obtained through our proposed method exhibit higher values in the frequency ranges [2.45, 2.55] and [5.00. 6.00). This implies that high-gain antenna demonstrates superior performance, providing validation for the effectiveness of our algorithm.
18

---

Deep Electromagnetic Structure Design Under Limited Evaluation Budgets
Surrogate-GA
eVAE
InvGrad
SurrogateRRS
eGAN
IDN
GenCO
Surregate-GW
TSDD-EO
SAHSO
PQStours)
Figure 10. Optimized Electromagnetic Structures of Different Methods on High-gain Antenna.
12
Geocili
Concet
Claeciii
FroqueseyyGHe))
PregengeeGll
Proprnetlil
PropireyGGla
RS
Surrogate-GA
cVAE
InvGrad
Churedit
Prepencytiiln
ProperestGGli IDN
Surrogate-R5
GenCO
-13
Propanec/OHaa
Propuney(GGly
SurrogateGGW
TSDD-EO
SAHSO
PQS(ours)
Figure I1. Simulated Results of Optimized Electromagnetic Structures High-gain Antenna.
19

---

Deep Electromagnetic Structure Design Under Limited Evaluation Budgets
0
-5
-10
S2.1/abs(dB)
15
-2
-25
-30
5
10
15
20
25
30
35
40
Frequency (GHz)
a
(
Figure 12. (a) Optimized Electromagnetic Structures of Different Methods on Dual-layer Frequency Selective Surface, (b) Simulated Results of Optimized Electromagnetic Structures on Dual-layer Frequency Selective Surface.
bb)
Figure 133. (a) Optimized Electromagnetic Structures of Different Methods on the High-gain Antenna. (b) Simulated Results of optimized Electromagnetic Structures on High-gain Antenna.
20


Reviews:
[{"review_id": "Mft5zUVdcE", "reviewer": "Reviewer_Kmfh", "summary": "# I have no experience in this field so I am not capable of completing a sound review. I acknowledge that some of the comment are generated with the assistance of LLM, and some parts are left blank.\nThe paper proposes Progressive Quadtree-based Search (PQS) for electromagnetic structure (EMS) design under limited budgets, tackling high-dimensional spaces (e.g., 10^86 configurations) and costly simulations (660-42,780 seconds each). PQS uses Quadtree-based Search Strategy (QSS) for hierarchical exploration and Consistency-based Sample Selection (CSS) to balance exploitation/exploration. Tested on DualFSS and HGA, PQS outperforms baselines (e.g., 15.20 dB vs. 7.28 dB for DualFSS) with 1000 simulations, cutting costs by 75-85% and saving 20-48 days. Contributions include PQS, QSS, and CSS for efficient EMS design.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely appreciate your time and valuable feedback. We acknowledge your concerns and would like to clarify our motivations and technical contributions.\n\n**1. Significance of the Task & Motivation**\n\nElectromagnetic structure (EMS) design is fundamental to modern antenna/material development, yet faces two critical challenges:\n+ **Vast Design Space**: Conventional grid-based representation creates exponentially growing search spaces (e.g., 16×16 layout can contain over $10^{77}$ potential design).\n+ **Expensive Evaluations**: Each design must be evaluated via time-consuming electromagnetic simulations (often taking minutes to hours), making large-scale evaluations impractical for industrial timelines.\n\nExisting approaches primarily revolve around two strategies:\n* **Predictor-Based Methods:** Use predictors to approximate the costly electromagnetic simulator. However, such predictors need **large amounts of high-quality data** to achieve sufficient accuracy, which is infeasible when budgets are tight.\n* **Generative Approaches:** Use generative models to produce promising designs directly. They also require **tens of thousands of labeled samples** for training.\n\nIn short, **our motivation** is to **reduce the reliance on large evaluation costs** by **introducing a more efficient search mechanism** that narrows down the design space in a structured manner.\n\n**2. Methodological Innovation**\n\nOur proposed **Progressive Quadtree-based Search (PQS)** addresses these challenges through:\n* **Quadtree-based Search Strategy:**\n    * **Quadtree-based Representation:**\nTo reduce the dimensionality, we encode the layout in a **quadtree** that recursively subdivides the structure from a coarse resolution to finer details.\n    * **Progressive Tree Search:**\nTo focus the limited budget on the most impactful design decisions, we decompose the high-dimensional search into **a tree search from global pattern to local refinement**.\n* **Consistency-based Sample Selection:**\nRather than requiring a highly accurate predictor, we adopt an adaptive sample selection. Specifically:\n    * **Ranking-based Reliability:**\nKendall's Tau is used to measure how consistently the predictors ranks candidate designs across consecutive iterations, which reveals the model's reliability to guide the search effectively.\n    * **Dynamic Exploitation & Exploration:**\nBased on the consistency, our method dynamically allocates evaluations: high consistency leads to evaluate top-ranked designs, while low consistency leads to an additional exploration to uncover overlooked candidates.\n\n**3. Practical Validation**\n\nOur proposed approach has already been successfully validated in real-world engineering tasks:\n* **High-Quality Outcomes:** Compared to baseline methods, PQS yields designs with approximately 70% greater improvement in DualFSS design objectives.\n* **Cost Savings:** Achieved **75%-85% cost reduction** vs. generative approaches.\n* **Robust Performance:** Repeated experiments showed that PQS exhibited **29%-92%** lower variance compared to baseline methods. \n\n**4. Broader Impact to ICML Community**\n\nBy bridging the gap between theoretical optimization and industrial constraints, PQS provides:\n* **A scalable framework** for high-dimensional and expensive structural design problems.\n* **A low-budget solution** for EMS design with limited computational resources.\n\nWe will revise the manuscript to better emphasize these aspects and would be happy to provide supplementary materials detailing industrial implementation.\n\n> Q1. Time-saving estimate (20-48 days) lacks detailed derivations.\n\n**A1.** Thank you for your valuable feedback. We acknowledge that the original time-saving estimate (20–48 days) lacked detailed derivation. Upon re-evaluating the calculations, we identified an error in rounding and task-specific simulation reductions. Here is the correct derivation:\n+ **DualFSS Task**: The average simulation time is 583.7 seconds; The reduction in simulations is 3,000 times; The total time saved is $\\frac{583.7 \\times 3,000}{3,600 \\times 24} \\approx 20.27$ days.\n+ **HGA Task**: The average simulation time is 558.8 seconds; The reduction in simulations is 6,000 times; The total time saved is $\\frac{558.8 \\times 6,000}{3,600 \\times 24} \\approx 38.80$ days.\n\nThus, the refined time-saving range is **20.27–38.80 days**. We apologize for the oversight and have updated the manuscript to reflect this correction. Importantly, the revised values still robustly demonstrate that our method achieves significant efficiency gains compared to generative approaches.", "claims_and_evidence": "Claims (PQS outperforms baselines, QSS reduces dimensionality, CSS boosts efficiency, 75-85% cost reduction) are supported by experiments (Tables 3, 6; Figures 3, 4) and metrics (Agg Obj, Kendall’s tau). Time-saving estimate (20-48 days) lacks detailed derivation but aligns with simulation reductions. Evidence is clear and convincing overall.", "methods_and_evaluation_criteria": "N/A (not familiar)", "theoretical_claims": "No formal proofs", "experimental_designs_or_analyses": "Experiments (Section 5, Appendix D) compare PQS vs. baselines (1000 vs. 7000/4000 simulations) and ablate QSS/CSS (Tables 5, 6; Figures 3, 7). \n\nlimited to two tasks, but conclusions hold.", "relation_to_broader_scientific_literature": "Not familiar with this field", "essential_references_not_discussed": "Not familiar with this field", "other_strengths_and_weaknesses": "Strengths: Original QSS+CSS combo, significant cost savings, clear presentation (e.g., Figure 2).", "other_comments_or_suggestions": "N/A", "questions_for_authors": "N/A", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "ePqOrtLKQy", "reviewer": "Reviewer_63Rq", "summary": "This paper proposes a novel method for electromagnetic structure (EMS) design under limited computational budgets, called Progressive Quadtree-based Search (PQS). PQS employs a Quadtree-based Search Strategy (QSS) to progressively explore the high-dimensional EMS design space and incorporates a Consistency-based Sample Selection (CSS) mechanism to optimize search efficiency under constrained evaluation resources. The method is evaluated on two real-world engineering tasks: Dual-layer Frequency Selective Surface (DualFSS) and High-gain Antenna (HGA). Experimental results demonstrate that PQS can efficiently identify high-performance designs while reducing computational costs by 75%–85% compared to baseline methods, significantly shortening the product design cycle.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "> Q1. PQS should be compared with SOTA surrogate-assisted evolutionary algorithms (SAEAs). \n\n**A1.** Thank you for your constructive suggestion. We compared our method with TS-DDEO[1] and SAHSO[2] in Tables 1-3. PQS outperforms them by 3.14-3.60 and 9.63-10.99 and achieves significantly higher robustness in the aggregation value of objectives, with 15%-19% lower variance. This stems from PQS’s ability to hierarchically explore the design space and adaptively allocate the tight evaluation budget via CSS. This efficiency could significantly benefit real-world applications where evaluation budgets are tight.\n\n**Table 1:** Comparisons on High-gain Antenna (HGA).\n|Method|Agg Obj↑|Obj1↑|Obj2↑|\n|-|-|-|-|\n|TS-DDEO[1]|0.5201|1.0441|0.5201|\n|SAHSO[2]|0.0589|0.0589|2.2410|\n|**PQS (Ours)**|**3.6595**|**3.6595**|**6.4820**|\n\n**Table 2:** Comparisons on Dual-layer Frequency Selective Surface (DualFSS).\n|Method|Agg Obj↑|Obj1↑|Obj2↑|\n|-|-|-|-|\n|TS-DDEO[1]|5.5627|5.5627|9.0000|\n|SAHSO[2]|4.2066|4.2066|8.0576|\n|**PQS (Ours)**|**15.1964**|**15.1964**|**31.0443**|\n\n**Table 3:** Robustness comparison results on High-gain Antenna (HGA).\n|Method|Agg Obj↑|Obj1↑|Obj2↑|\n|-|-|-|-|\n|TS-DDEO[1]|0.03±0.42|0.62±0.87|0.27±0.36|\n|SAHSO[2]|-0.29±0.40|0.20±0.75|0.06±0.61|\n|**PQS (Ours)**|**4.34**±**0.34**|**4.45**±**0.30**|**4.53**±**0.46**|\n\n[1] A two-stage surrogate-assisted metaheuristic algorithm..., Soft Comput. 2023.\n\n[2] A surrogate-assisted hybrid swarm optimization..., Swarm Evol. Comput. 2022.\n>Q2. Only 300 samples may hinder generalization. Authors should examine dataset sizes and classifiers' impact on performance.\n\n**A2.** We acknowledge dataset size and model choice's importance. However, in EMS design, simulation takes hours to days (e.g., 559-583 seconds per evaluation). To address this, we begin with a small initial dataset (300 samples) and use the remaining evaluation budget iteratively during optimization. This ensures that computational resources are prioritized for optimization rather than data collection.\n\nIn Table 4, we evaluated Kendall's Tau (KTau) between predicted and ground truth across dataset sizes (300-1,100) on a fixed 5,800-sample test set (10 trials). We clarify two key observations:\n* Expanding the dataset from 300 to 1,100 samples maintains weak KTau correlation (<0.3), showing larger datasets don't proportionally enhance accuracy in our task.\n* All models perform poorly, implying that model capacity is not the limiting factor.\n\nThese observations suggest that while gathering more data can be helpful, it is not always cost-effective in EMS design. Our PQS is designed to achieve data-efficient optimization, attaining superior performance even with a limited budget. \n\n**Table 4:** Impact of training sample size and predictor on model performance.\n|Model|KTau @300 Samples|KTau @700 Samples|KTau @1100 Samples|\n|-|-|-|-|\n|GoogLeNet|0.2148±0.0088|0.2610±0.0084|0.2718±0.0071|\n|ResNet50|0.2175±0.0065|0.2600±0.0126|0.2896±0.0084|\n|ResNet101|0.2146±0.0135|0.2605±0.0092|0.2882±0.0065|\n\n> Q3. The observation that RS and Surrogate-RS outperform Surrogate-GA and Surrogate-GW is unexpected.\n\n**A3.** One possible factor is **each method's reliance on the predictor for solutions**. Specifically:\n* Surrogate-GA and Surrogate-GW **rely on the predictor to guide the evolution of new populations**. With a limited training set (300 samples), the predictor's accuracy suffers, misleading GA and GW optimizers to suboptimal regions.\n* By contrast, RS and Surrogate-RS avoid predictor dependence: they generate candidates via uniform random sampling. **This is like our QSS, which also avoids heavy predictor reliance** by hierarchical random exploration.\n\n>Q4. Why QSS is superior to other dimensionality reduction approaches, such as PCA or AutoML-based search space simplification?\n\n**A4.** Our QSS excels in **search space reduction** and **dynamic adaptation**:\n* **Explicit space shrinking**: Unlike PCA that projects the design space into a continuous latent space, our QSS preserves the discrete, grid-like nature of EMS through its quadtree-based hierarchical representation, ensuring clear shrinking.\n* **Dynamic adaptation during search**: PCA or feature selection methods perform static, global dimensionality reduction. In contrast, our QSS dynamically adjusts the search granularity during optimization. Initially, it explores coarse global patterns and later focuses on locally promising regions, enabling efficient exploration under tight budgets.\n\n>Q5. It does not cover recent advances such as deep reinforcement learning or Bayesian optimization.\n\n**A5.** We will carefully discuss and cite these works in our revised paper.\n\n>Q6. Important training details are missing.\n\n**A6.** We clarify training details: Batch Size: 256, Epochs: 200, Cosine Annealing LR (lr=0.01, T_max=200, η_min=1e-6), Gradient Clipping (Global L2 norm ≤1.0/batch), Optimizer: Adam, resnet_lr=0.01, fc_lr=0.01. These will be added to the manuscript.", "claims_and_evidence": "* The paper employs Surrogate-GA, which is no longer representative of state-of-the-art (SOTA) surrogate-assisted evolutionary algorithms (SAEAs). Recent survey [1] indicate that SAEAs have become the dominant approach for high-dimensional expensive optimization problems. The paper should compare PQS with SOTA SAEAs rather than a 2020-era Surrogate-GA.\n\n* The paper utilizes ResNet50 as the predictor but lacks details on training data sources and training methodology. The dataset size is limited to 300 samples, which may be insufficient for complex EMS tasks. A larger dataset should be used, and additional experiments should evaluate the impact of dataset size on performance.\n\n* The observation that random sampling (RS) and Surrogate-RS outperform Surrogate-GA and Surrogate-GW is unexpected, as it contradicts typical evolutionary algorithm performance trends. The paper does not provide a satisfactory explanation for this result.\n\n[1] M. Zhou, M. Cui, D. Xu, S. Zhu, Z. Zhao and A. Abusorrah, “Evolutionary Optimization Methods for High-Dimensional Expensive Problems: A Survey,” in IEEE/CAA Journal of Automatica Sinica, vol. 11, no. 5, pp. 1092-1105, May 2024, doi: 10.1109/JAS.2024.124320.", "methods_and_evaluation_criteria": "* The dataset contains only 300 samples, which may hinder generalization. The study should explore the impact of dataset size by conducting experiments with larger datasets.\n\n* The baseline methods do not include the most recent SAEA algorithms, affecting the fairness of the evaluation.\n\n* Only ResNet50 is used as the predictor. The study should compare different classifiers to assess their impact on performance.", "theoretical_claims": "The paper does not provide theoretical justification for why QSS is superior to other dimensionality reduction approaches, such as PCA or AutoML-based search space simplification.", "experimental_designs_or_analyses": "* The paper relies on older methods (e.g., Surrogate-GA, Surrogate-RS), omitting SOTA SAEAs, which undermines the validity of the comparisons.\n\n* The predictor is trained with only 300 samples, which may lead to poor generalization. The study should analyze the predictor’s accuracy across different dataset sizes.\n\n* The superior performance of RS and Surrogate-RS over Surrogate-GA contradicts standard evolutionary optimization trends, and the paper does not provide a satisfactory explanation.", "relation_to_broader_scientific_literature": "The paper references some relevant works but fails to cite the latest SAEA methods. Additionally, it does not cover recent advances in EMS optimization, such as deep reinforcement learning (DRL) or Bayesian optimization (BO).", "essential_references_not_discussed": "I do not have more suggestions.", "other_strengths_and_weaknesses": "* The paper does not compare with SOTA surrogate-assisted evolutionary algorithms.\n\n* Only 300 samples are used, which may not be sufficient to train a robust predictor.\n\n* The better performance of RS over Surrogate-GA is counterintuitive and requires further justification.", "other_comments_or_suggestions": "* Expand the dataset size and analyze its impact on the predictor’s accuracy.\n\n* Include SOTA SAEAs for fair comparisons.\n\n* Explain the unexpected experimental results (why RS outperforms Surrogate-GA).", "questions_for_authors": "I have no more questions.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "ZY7z4owT5a", "reviewer": "Reviewer_55Dq", "summary": "The authors present Progressive Quadtree-based Search (PQS), a novel method for electromagnetic structure (EMS) design under limited evaluation budgets. PQS leverages a quadtree-based hierarchical representation to mitigate the curse of dimensionality inherent in conventional pixel-wise optimization. In addition, it introduces a Consistency-based Sample Selection (CSS) module that leverages model prediction consistency to dynamically allocate scarce evaluation resources. PQS is benchmarked on two real-world tasks, where it substantially outperforms baseline methods in both efficiency and robustness. Comprehensive ablation studies further validate the important role of the hierarchical representation and the adaptive sample selection mechanism.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": ">Q1. The CSS mechanism relies on assessing the consistency of historical predictors. However, there is only one predictor available at t=0.\n\n**A1.**  Thank you for your insightful comment. We actually train multiple predictors at t=0 rather than relying on a single one. These predictors are then used to compute and assess consistency. We will clarify this detail in the revised manuscript.  \n\n>Q2. It would be interesting to discuss whether using different predictor architectures could lead to significant differences in prediction accuracy.\n>\n\n**A2.** We have conducted further experiments to validate this. Please refer to A2 in our response to Reviewer 63Rq for more details.\n\n>Q3. I recommend that the authors release the annotated EMS design data.\n\n**A3.** We will release the annotated EMS design dataset upon acceptance. \n\n>Q4. The stopping condition in Algorithm 1 is inconsistent with that in Figure 2 ... Furthermore, Algorithm 1 should output the optimal solution found by the search, not a satisfactory solution.\n\n**A4.** Thank you for your suggestion. We will correct this in our revised version.\n\n>Q5. For example, it would be beneficial to explore more thoroughly how much the quadtree-based representation reduces the search space size. What is the specific relationship between the search space size and the parameter $N_{max}$?\n\n**A5.** The quadtree representation is used to reduce the complexity of the design space. Each leaf node in the quadtree represents a binary decision (either 0 or 1) for its corresponding subregion. Thus, if the current set of leaf nodes is $L$, the total size of the design space is $2^{|L|}$. The parameter $N_{{max}}$ limits the maximum number of leaf nodes, i.e., $|L| \\leq N_{{max}}$, so the upper bound of the search space is $2^{N_{{max}}}$. In other words, a smaller $N_{{max}}$ effectively reduces the search space size.\n\n>Q6. It would still be beneficial to provide a more detailed computational time including simulations and model-updating—especially since PQS continuously retrains its predictor.\n\n**A6.** Below, we provide a detailed breakdown of the average computational time for simulation in **Table 2** and model-updating in **Table 3**. Specifically, we measured the model-updating time under both initial and final dataset conditions for each task:\n* HGA: 300 samples → 1,000 samples.\n* DualFSS: 500 samples → 1,000 samples.\n\nAll model updates were consistently performed for 200 epochs.\n\n **Table 2:** Simulation cost per iteration (seconds).\n|Task|Avg.Time|\n|-|-|\n|HGA|5588|\n|DualFSS|5837|\n\n **Table 3:** Model updating cost per iteration (seconds).\n|Task|Updating Samples|Avg.Time±Std|\n|-|-|-|\n|HGA|300|18.8716±0.8278|\n|HGA|1000|43.8452±0.2518|\n|DualFSS|500|26.9258±0.5701|\n|DualFSS|1000|44.5284±0.2120|\n\nIn both tasks, simulation is the dominant cost. Model updating remains lightweight and efficient throughout the search.\n\n>Q7. Since Bayesian optimization is commonly used for optimizing expensive evaluations, I recommend that the authors include a more thorough discussion on this topic.\n\n**A7.** We will include a more comprehensive discussion of Bayesian optimization in our revised manuscript.", "claims_and_evidence": "This work provides a successful demonstration of deep learning’s application in electromagnetic structure design. By directly addressing the often-overlooked challenge of high data collection costs in real-world scenarios, it shows the potential to shorten product design cycles in industrial settings, making it well-suited for practical, large-scale applications. The claims of the paper are well-supported by evidence from experiments. \n1.\tExperiments on two real-world tasks demonstrate that PQS yields designs with enhanced performance, with limited evaluation budgets.\n2.\tAblation studies confirm the importance of the QSS and CSS strategy.\n3.\tRobustness tests indicate that PQS consistently delivers reliable performance.", "methods_and_evaluation_criteria": "The paper presents effective methods for addressing EMS design challenges. Its innovative quadtree-based representation reduces the problem’s dimensionality, and the CSS mechanism ensures efficient utilization of computational resources under limited evaluation budgets. Additionally, the evaluation tasks (DualFSS and HGA) are highly relevant to real-world electromagnetic structure design applications. However, some detailed issues still remain.\n1.\tRegarding the CSS mechanism, it relies on assessing the consistency of historical predictors, but at t=0 there is only one predictor available, so it is unclear how consistency is computed. The authors should provide additional details on this aspect.\n2.\tAlthough the predictor itself is not the main contribution of the methods, I noticed that the authors consistently used ResNet-50 as the architecture. Therefore, it would be interesting to discuss whether using different predictor architectures could lead to significant differences in prediction accuracy.\n3.\tAs noted in the paper, there is currently a lack of publicly available real-world datasets and benchmarks in the EMS design field. Therefore, I recommend that the authors release the annotated EMS design data, as such a resource would greatly assist future researchers in improving and advancing related methodologies.\n4.\tThe stopping condition in Algorithm 1 is inconsistent with that in Figure 2. In the former, the algorithm stops once the simulation budget is exhausted, whereas in the latter, it stops when a satisfactory solution is reached. Furthermore, Algorithm 1 should output the optimal solution found by the search, not a satisfactory solution.", "theoretical_claims": "This paper demonstrates strong practical applicability and innovation, especially with the introduction of the PQS and the CSS mechanism, both of which have been experimentally validated for their effectiveness. However, the theoretical background could be expanded further. For example, it would be beneficial to explore more thoroughly how much the quadtree-based representation reduces the search space size. What is the specific relationship between the search space size and the parameter $N_{max}$?", "experimental_designs_or_analyses": "The paper compares the proposed method with various baseline approaches and supports its claims with comparative evaluations, robustness tests, and ablation studies, providing solid evidence of its effectiveness. Nevertheless, some improvements are possible:\n1.\tWhile the authors note that simulation time represents the primary cost within the overall optimization, it would still be beneficial to provide a more detailed computational time including simulations and model-updating—especially since PQS continuously retrains its predictor.\n2.\tAlthough the authors explain in the appendix why Bayesian optimization cannot be successfully applied, the paper lacks a more detailed analysis. Since Bayesian optimization is commonly used for optimizing expensive evaluations, I recommend that the authors include a more thorough discussion on this topic.", "relation_to_broader_scientific_literature": "The key contributions of this paper are well-grounded in the existing literature, addressing key gaps.", "essential_references_not_discussed": "The authors have done a thorough job of addressing the essential references relevant to their contributions. \n1.\tThe paper clearly discusses key methodologies in the field of electromagnetic structure design, including both surrogate-based optimization and generative approaches. \n2.\tThe authors have made meaningful comparisons with analogous fields, highlighting the unique challenges of this task.", "other_strengths_and_weaknesses": "1.\tThe images in the appendix are excessively large. Please adjust their sizes to improve readability.\n2.\tThe citation for the Genco baseline is missing the publication year.", "other_comments_or_suggestions": "1.\tFigure 1 can benefit from a longer caption.\n2.\tIn the \"Depth-wise Importance Assignment\" section, there appear to be repetitive sentences.\n3.\tIn the \"Effectiveness of Consistency-based Sample Selection\" section, \"Top-M\" should be corrected to \"Top-K\".\n4.\tIn the \"Quadtree-based Representation\" section, \"so taht\" should be corrected to \"so that\".", "questions_for_authors": "n/a", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: The paper proposes a method named Progressive Quadtree‑based Search (PQS) that tackles electromagnetic‑structure (EMS) design when every simulator call is painfully slow and budgets are tight. By encoding the 2‑D layout as a quadtree, the method searches from coarse global patterns down to fine local details. The paper introduces a practically motivated strategy for budget‑constrained EMS optimisation. Empirical gains seem large (3–10× objective improvement, 75–85 % cost cut) and now benchmarked against two recent SAEA algorithms after the rebuttal. While theoretical framing is light and some baselines (e.g. Bayesian optimisation, DRL) are still absent, the added experiments strengthened the results in the paper.

Given the current scores and the rebuttal, I will recommend the paper for acceptance.
Decision: Accept (poster)