Machine intelligence continues to rise in popularity as an aid to the design and discovery of novel metamaterials. The properties of metamaterials are essentially controllable via their architectures and until recently, the design process has relied on a combination of trial-and-error and physics-based methods for optimization. These processes can be time-consuming and challenging, especially if the design space for metamaterial optimization is explored thoroughly. Artificial intelligence (AI) and machine learning (ML) can be used to overcome challenges like these as pre-processed massive metamaterial datasets can be used to very accurately train appropriate models. The models can be broad, describing properties, structure, and function at numerous levels of hierarchy, using relevant inputted knowledge. Here, we present a comprehensive review of the literature where state-of-the-art machine intelligence is used for the design, discovery and development of metamaterials. In this review, individual approaches are categorized based on methodology and application. We further present machine intelligence trends over a wide range of metamaterial design problems including: acoustics, photonics, plasmonics, mechanics, and more. Finally, we identify and discuss recent research directions and highlight current gaps in knowledge. Keywords Metamaterials ·Artificial intelligence ·Machine learning ·Neural Networks ·Evolutionary algorithms · Surrogate modelling ·Optimization APREPRINT - SEPTEMBER 30, 2023 Contents 1 1 Introduction 4 2 1.1 Introducing Metamaterials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3 1.2 Machine intelligence and metamaterials: a growth in interest and practice . . . . . . . . . . . . . . . 6 4 1.3 Highlights from recent reviews on the AI/ML based design of metamaterials . . . . . . . . . . . . . . 8 5 1.4 Virtue of this current review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 6 2 Machine intelligence: a brief overview of concepts 8 7 2.1 Machine Learning (ML) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 8 2.1.1 Classification of ML Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 9 2.1.2 ML Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 10 2.2 Nature Inspired AI Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 11 2.2.1 Genetic Algorithms (GA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 12 2.2.2 Swarm Intelligence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 13 3 Machine Intelligence in Metamaterial Design 16 14 3.1 Machine Learning in Metamaterials Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 15 3.1.1 Gradient-Based Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 16 3.1.2 Dimensionality Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 17 3.1.3 Other ML Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 18 3.2 Artificial Neural Networks in Metamaterials Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 19 3.2.1 Surrogate Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 20 3.2.2 Inverse Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 21 3.2.3 Other ANN Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 22 3.3 Nature-Inspired Algorithms in Metamaterials Design . . . . . . . . . . . . . . . . . . . . . . . . . . 27 23 3.3.1 Genetic Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 24 3.3.2 Particle Swarm Intelligence Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 25 4 Discussion 29 26 4.1 General Trends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 27 4.2 Opportunities and Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 28 4.2.1 Efficient Exploration of Design Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 29 4.2.2 Multi-Objective Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 30 4.2.3 Materials Discovery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 31 4.2.4 Data Availability and Quality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 32 4.2.5 Computational Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 33 4.2.6 Interdisciplinary Expertise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 34 4.2.7 Robustness and Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 35 4.2.8 The Causal Relationship Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 36 4.2.9 Hyperparameter Optimization and Hybrid ANN & GA . . . . . . . . . . . . . . . . . . . . . 33 37 2 APREPRINT - SEPTEMBER 30, 2023 4.3 Future Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 38 3 APREPRINT - SEPTEMBER 30, 2023 1 Introduction 39 1.1 Introducing Metamaterials 40 The properties of metamaterials are a function of material 41 architecture and as such, metamaterial properties are not 42 solely related to the properties of material constituents. 43 Properties can therefore be controlled and manipulated by 44 changing metamaterial ‘unit cell’ topology and while the 45 bulk properties of the constituent materials have influence, 46 these properties are often not the sole dominating variable 47 used in designing metamaterials. When designing with 48 respect to topology, the properties of metamaterials can 49 be customized within a very large design space, and this 50 is one of the reasons for the exponential growth in meta- 51 materials R&D across the breadth of modern engineering. 52 Metamaterials can be categorized according to four main 53 objective based areas: mechanical, acoustic, optical and 54 electromagnetic, [1]. 55 The mechanical properties of metamaterials are of inter- 56 est as the manipulation of architectures has been found 57 to enable a very wide range of properties and behaviors. 58 Design by computation can improve the consistency of 59 metamaterial functionality and fundamentally desirable 60 characteristics, such as improved material longevity, high 61 mechanical energy absorption, and unique modes of load- 62 controlled deformation, can be made possible by initially 63 computing the outcome of architectural iterations to the 64 metamaterial structure. The process is two-fold as both 65 structure and composition are best if conjointly optimized, 66 typically with a specified design objective. This bilateral 67 approach is an enabler for the development of intentionally 68 distinct mechanical properties and behaviors with signif- 69 icance in many generic and niche areas of application 70 [2]–[5]. Mechanical metamaterials are often analyzed via 71 normalized properties as a function of their relative densi- 72 ties through the generic relationship shown in Equation 1, 73 where P∗is an apparent mechanical property of the meta- 74 material including any extant air space (e.g. E(apparent 75 elastic modulus), σy(apparent yield strength), G(apparent 76 shear modulus), etc.), Psis the mechanical property of the 77 solid constituent material only (i.e. the bulk property of 78 the material assuming no air space), ρ∗is the apparent den- 79 sity of the metamaterial (i.e. the density per unit volume 80 including air space), ρsis the density of the constituent 81 solid material only, and Canddare adjustable constants 82 (mathematical correction factors). The presence of the 83 mathematical correction factors Canddclarifies that the 84 relationship between density and properties in mechani- 85 cal metamaterials is not always linear. Here, ρ∗ ρs =ρ, 86 andρis the relative density of the metamaterial, while the 87 relative mechanical property Pis equal toP∗ Ps. 88 P∗ Ps=Cρ∗ ρsd (1) Mechanical metamaterials are often characterized by the 89 type of basis structure they exhibit, Figure 1. As seen 90in this figure, structures can be beam based (also strut 91 based), plate based, or minimal surface based. Beam based 92 structures are essentially beams or struts that connect at 93 common nodes (ends). The mechanical efficiency of a 94 metamaterial is understood as P∝ρd. As discussed by 95 [6], the mechanical efficiency of beam based mechanical 96 metamaterials is related to the nodal connections between 97 beams (Maxwell’s rigidity), as b−2j+ 3 = s−mand 98 b−3j+ 6 = s−m, for 2D and 3D respectively, where 99 bis the number of beams (or struts), jis the number of 100 nodes in connection, sis the number of self-stress states 101 andmis the number of mechanisms. According to these 102 forms, in 2D b≥4and in 3D b≥6for the case of 103 stretching domination. When stretch dominated, beams 104 are sufficiently ‘thin’ (with a low ρ).P∝ρ1, for stretch 105 dominated and P∝ρ)1.5→2for bend dominated. When ρ 106 is high (i.e. the beams are sufficiently thick) this scaling 107 law no longer applies as the geometrical features of the 108 nodes play a more significant role in controlling P. While 109 beam based structures are always essentially of the open 110 cell form, plate based structures can be either open cell 111 or closed cell structures. The plate based metamaterials 112 are built up by connected thin walled sections that connect 113 at vertices. Unlike beam based structures, which bear 114 load more effectively along the beam axes, thin walled 115 structures have a multi-axial resistance to loading, and 116 when ρis sufficiently low, P∝ρ1as they are effectively 117 stretch dominated at low relative density. Finally, minimal 118 surface based metamaterials are comprised of curved shells 119 where the mean curvatures are zero (κ1+κ2 2= 0) and the 120 Gaussian curvatures are less than zero ( κ1·κ2<0). Since 121 minimal surface structures have no connections at nodes or 122 vertices, like beam or plate based structures, respectively, 123 they benefit from reduced stress concentration points when 124 loaded. 125 While beam based, plate based and minimal surface based 126 classifications are based on geometry type , mechanical 127 metamaterials are also commonly classified based on ge- 128 ometrical behavior . A simple set of examples include 129 pentamode, auxetic and negative stiffness, which are each 130 characterized by the Poisson’s ratio [7], ν, a ratio of trans- 131 verse strain ( εt) to axial strain ( εa) such that ν=εt εa. A 132 pentamode material exhibits the behavior of an ideal fluid 133 and phenomenologically, a solid pentamode structure re- 134 tains a constant volume under loading, which means that 135 the Poisson’s ratio remains at a constant value, similar to 136 that of an incompressible fluid ( ν= 0.5). Recognizing 137 this it can be noted from Lamé’s relation [8], Equation 2, 138 where Eis elastic modulus and Kis the bulk modulus, 139 thatE 6Kdiminishes →0. Auxetic metamaterials are those 140 where the Poisson’s ratio is negative, i.e. ν <0. Here, an 141 auxetic metamaterial is stretched, it becomes wider, while 142 when compressed it becomes thinner, behaviors that are 143 atypical to most materials. Negative stiffness metamate- 144 rials are those that deform in opposition to the direction 145 of the applied force, resulting in material instability when 146 unconstrained. When constrained, the same material can 147 be stable [9] since within the allowable Poisson limits 148 4 APREPRINT - SEPTEMBER 30, 2023 −1< ν < 0.5, stronger ellipticity from constraint results 149 in reduced conditions for stability, Equation 3, where Gis 150 shear modulus. 151 ν=1 2−E 6K(2) −∞< E < ∞or−4G 3< K < ∞ (3) A third method of classification that is commonly used 152 for mechanical metamaterials, beyond those just discussed 153 is based on mechanical performance , where performance 154 criteria based on relationships similar to those discussed in 155 Equation 1 form the the basis for objective oriented design. 156 Common examples of coupled relationships include maxi- 157 mized strength, stiffness, and energy absorption, alongside 158 minimized mass. An example form for coupled stiffness 159 maximization with mass minimization is shown in Equa- 160 tion 4, where Vis the total volume of the structure and σ 161 is the engineering stress, mis mass, and Ueis the elastic 162 strain energy, which is itself determined through Equation 163 5. 164 E ρ=σ2V 2Uem V−1 (4) Ue=Zε 0σdε (5) Acoustic, electromagnetic and optical metamaterials are 165 each controlled through the refractive index, n=c v, where 166 cis the speed of light and vis the phase velocity of light. In 167 each case, the properties of metamaterials can be controlled 168 by rationally designing the structure to have variability in 169 the refractive index, which in turn results in a specific 170 property outcome. As waves travel through a material, 171 the refractive index nis a function of the wave frequency, 172 ω. Acoustic metamaterials are able to direct and redirect 173 sound waves in ways that are ordinarily unachievable in 174 conventional materials. The two main inherent material 175 properties that control sound wave propagation in materials 176 are density ( ρ) and bulk modulus ( K). Metamaterials func- 177 tion on the basis of structure and as such, by altering the 178 structure and architecture of a material, both ρandKcan 179 be manipulated, such that an engineer can purpose-design 180 a metamaterial to control the directing/redirecting of sound 181 waves. Acoustic pressure is connected to ρaccording to 182 Equation 6, where vdefines the acoustic particle perturba- 183 tion, and tis time. Additionally, pandKare related as the 184 motion of a non-viscous fluid is directly connected to K, 185 which acts as a scaling constant for material compressibil- 186 ity. The conservation of mass is assumed in the continuity 187 Equation 7, where the propagation of acoustic waves is 188 taken to be isentropic. Combining therefore, Equations 6 189 and 7, yields the scalar wave equation Equation 8. As such, 190 changes in the direction of waves at interfaces are con- 191 trolled by the acoustic wave velocity which is described as 192c=q K p, while the reflection and transmission amplitudes 193 of waves at interfaces is controlled by the acoustic wave 194 impedance, which is described as Z=p v=√Kρ, [14]. 195 As such, both Kandρare evidently important parameters 196 in the control of sound waves through solids, and are anal- 197 ogous to the electromagnetic parameters of permittivity 198 (the electric component of light), ϵ, and magnetic perme- 199 ability (the magnetic component of light as a function of 200 wave frequency, ω),µ, such that in the electromagnetic 201 sense, c=√ϵµandZ=pµ ϵ[15]. Many such electro- 202 magnetic parameters are effectively algebraically derived 203 from the following two relations, ε(ω) =ε1(ω) +iε2(ω),204 and,µ(ω) =µ1(ω) +iµ2(ω), where 1and2indicate 205 properties relative to the different phases of the metama- 206 terial (whether they be matter and matter, or matter and 207 air). Electromagnetic metamaterials have many potential 208 areas of application, including as cloaking devices, in lens 209 technologies, for antennas, and more. One particularly in- 210 teresting property of certain electromagnetic metamaterials 211 is the ability to exhibit a negative refractive index. Conven- 212 tionally, the refractive index is always positive, indicating 213 that light, or electromagnetic waves, bends towards the 214 normal when it enters the material. Metamaterials with 215 a negative refractive index will not bend light, or elec- 216 tromagnetic waves, in this way and phenomena such as 217 negative refraction are possible specifically by achieving 218 ε(ω)<0concurrent with µ(ω)<0. Since the refractive 219 index is n(ω) =p ε(ω)µ(ω), this would result in an index 220 n(ω)<0, [16]. 221 ρ∂v ∂t=∇p (6) ∂p ∂t+K∇ ·v= 0 (7) ∂2p ∂t2=K ρ∇2p (8) Many acoustic metamaterials contain air, and this should 222 be factored into our overall understanding of sound trans- 223 mission. While the effective density ρeffof a two compo- 224 nent composite can be linearly volume averaged, Equation 225 9, where ρ1andρ2are the material densities of compo- 226 nents 1 and 2 and V1is the volume fraction of material 227 1, there is the underlying implication that both composite 228 components can move together under wave motion. This 229 is nevertheless generally not the case, as once the wave 230 frequency surpasses the natural frequency of one of the 231 two components, motion becomes out of phase and thence, 232 relative [17]. A mathematical consideration to this issue 233 has been discussed by Pierre and co-workers [18] who 234 modeled the concept using a cylindrical pipe and spring 235 connected materials separated by air, a toy model that 236 could also be used to evidence negative effective density, 237 Equation 10, where ris the radius of the material parts in 238 the cylinder, eis the thickness of the same parts, Meff 239 is the effective mass, ρ0is density, Sis total section area 240 5 APREPRINT - SEPTEMBER 30, 2023 Figure 1: Examples of mechanical metamaterials based on geometry type. Illustration of 3D (a) beam-based, (b) plate-based, and (c) minimal surface-based topologies. Reproduced from [6], [10] with permission under the CC-BY licence. (S=S1+S2), and dis the distance between material 241 layers. 242 ρeff=V1ρ1+ (1−V1)ρ2 (9) ρeff= 1−e d +Meff Sd(10) Metamaterials are an emerging class of cellular materials 243 that has attracted interest since their unique properties can 244 be rationally designed. The design process is exacerbated 245 and enhanced through the use of computer-aided tools for 246 optimization, coupled with experimental measurements for 247 validation [19]-[20]. The complexity of metamaterials cre- 248 ates opportunities for vastly extended design spaces, which 249 in turn encourages more refined time-saving methods such 250 as machine intelligence [21]-[22]. 251 1.2 Machine intelligence and metamaterials: a 252 growth in interest and practice 253 Over recent decades we note a peak in interest in the use 254 of Artificial Intelligence (AI) and Machine Learning (ML) 255 techniques for metamaterials engineering and design, fol- 256 lowing their successful implementation in other branches 257 of engineering [23]–[25]. Examples include neural net- 258 works and evolutionary algorithms, which offer speed- 259 ier solutions to problems with lower computational costs. 260 High-complexity problems with a large number of design 261variables, such as is evident in the design of metamaterial 262 architectures, take advantage of the fast and robust nature 263 of AI algorithms. Metamaterial design has been shown 264 to be accelerated through the use of ML algorithms [26] 265 when compared against more traditional methods such as 266 trial-and-error, and analytical physics-based approaches. 267 New technologies that have been introduced to the field 268 allow not only quicker solutions, but also the identifica- 269 tion of unseen trends between the design variables and 270 metamaterial performance [27]. There is an ever-growing 271 desire for novel metamaterials with unique behaviors and 272 this encourages designers to come up with new methods to 273 further speed up the development and discovery of meta- 274 material architectures. As reported in [28], this requires a 275 framework that automates the design process while con- 276 currently increasing the rate at which the new structures 277 are evaluated. Such frameworks can be created using a 278 variety of AI techniques. In fact, it is now evident that AI 279 can contribute significantly towards the future success of 280 new metamaterials by automating labor-intensive discov- 281 ery processes, and finding patterns that were previously 282 unrecognized [29]–[31]. There are several diverse ML arti- 283 cles and reviews that have been published in recent times in 284 the area of materials science and engineering, for example 285 on glasses [32], energy materials [33], polymers [34], com- 286 posites [35], additive manufacturing [36]-[37], continuum 287 materials mechanics [38], and bio-inspired materials [39]. 288 6 APREPRINT - SEPTEMBER 30, 2023 (a) (b) (c) Figure 2: Examples of mechanical metamaterials defined by geometrical behavior (a) pentamode [11], (b) auxetic [12], and (c) negative stiffness [13]. Reproduced with permission under the CC-BY licence. 7 APREPRINT - SEPTEMBER 30, 2023 020004000600080001000012000 2018 2019 2020 2021 2022 2023Number of Publications Figure 3: Distribution of publications by year which shows the increasing research trend of using ML, AI and nature- inspired optimization solutions for metamaterial applica- tions. 1.3 Highlights from recent reviews on the AI/ML 289 based design of metamaterials 290 While there are numerous reviews focusing on various 291 classes of metamaterials and their design processes, there 292 are only a few that consider the use of AI and ML tech- 293 nologies. Emergent trends in the literature inform us that 294 the majority of current work focuses on surrogate and in- 295 verse design models, with the aim of replacing numerical 296 analysis techniques. Nevertheless, there is seemingly still 297 no consensus as regards the techniques suited for specific 298 design tasks and generally, the literature lacks guidance 299 on the de novo design of lattice metamaterials [40]. The 300 increase of community interest in AI and its application to 301 metamaterials science and engineering is evidenced by an 302 exponential increase in research publications since 2018. 303 Figure 3 illustrates this rise and considers published scien- 304 tific work where AI and ML have been used in conjunction 305 with metamaterials engineering. 306 The first review detailing metamaterials design by AI was 307 by Choudhury et al. [41] in 2015. Here, they consid- 308 ered the metamaterial-based design of broadband antennas. 309 Their comprehensive review discusses the applications of 310 evolutionary algorithms that in broadband antenna design, 311 are primarily used for solving optimization problems. Most 312 other review papers are more recent, with the majority pub- 313 lished from 2020 and beyond. Wang et al. [42] reviewed 314 the use AI in the context of additive manufacturing. They 315 summarize recent trends in the design of topologically 316 optimized metamaterials and structures that are made by 317 additive manufacturing. More specifically, they discuss 318 the use of AI and ML techniques as methods for the de- 319 velopment of high performance materials and customized 320 mechanical parts. Jia et al. [43] summarized AI applica- 321 tions as a means to designing smart metamaterials. In their 322 review, they emphasize the challenges that AI-enabled 323structures would face in terms of design, analysis and 324 fabrication. Their review furthermore discussed the future 325 trends and limitations of ML-based data-driven approaches. 326 Bonfanti et al. [44] reviewed data-driven development in 327 the context of metamaterials manufacturing techniques. 328 They also considered other architected structures as well 329 as biomimetic materials. Their review focused fundamen- 330 tally on the mechanical applications of such structures. A 331 few other reviews available in the literature concentrate on 332 analyzing the AI techniques themselves and on how they 333 are specifically exploited to generate optical metamateri- 334 als. So et al. [45] review on deep learning approaches 335 for designing nanophotonics provided a detailed analysis 336 of deep neural networks that were suitable for solving in- 337 verse design problems. Another 2020 review by Zhang et 338 al.[46] summarizes machine learning and evolutionary 339 algorithm based design and optimization techniques for 340 the intelligent development of photonic devices. A more 341 recent review by Picinotti et al. [47] discussed the use of 342 AI in photonics while also considering nanophotonics and 343 plasmonics in significant detail. 344 1.4 Virtue of this current review 345 Although the aforementioned reviews are highly relevant 346 and important, each review tends to have a narrow scope 347 or application focus. In contrast, this review uses a sys- 348 tematic approach to provide a broad understanding of how 349 AI and ML have been considered across the full spectrum 350 of metamaterials (mechanical, acoustic, electromagnetic 351 and optical). After consideration of the currently available 352 reviews on AI/ML and metamaterials, it is our opinion that 353 the current body of literature is missing an overall, broad 354 comparison of AI and ML in metamaterials science and 355 engineering. Yet, the obvious rapid development of new AI 356 technologies indicates that a review such as this is needed, 357 as it will summarize advancements in the field, whilst con- 358 currently providing useful insights based on previously 359 published knowledge. We believe therefore, that this sys- 360 tematic review is both timely and essential as it will help 361 the metamaterials community identify gaps in knowledge 362 while highlighting both current and future trends. 363 2 Machine intelligence: a brief overview of 364 concepts 365 There is no doubt that Artificial Intelligence, Machine 366 Learning and Artificial Neural Networks (ANN) have be- 367 come buzzwords in a number of areas ranging from science 368 and engineering to marketing. These terms are now com- 369 monly being used interchangeably to generically describe 370 data-driven approaches and by doing so increase interest 371 and encourage further discussion related to data analysis. 372 While there are many definitions of AI, it may be best 373 defined as a branch of computer science that aims to au- 374 tomate the smart or learned behavior of algorithms [48]. 375 Unfortunately, to an inexperienced user, this might make 376 it more challenging to understand the benefits these tech- 377 8 APREPRINT - SEPTEMBER 30, 2023 nologies can offer. Therefore, the following section aims 378 to provide a general background knowledge that would 379 allow the reader to distinguish and understand the basics 380 of AI and associated methods. To keep this review relevant 381 to metamaterials, we focus primarily on the approaches 382 analyzed and discussed in the reviewed literature that have 383 been utilized in metamaterials design processes. Figure 384 4 categorizes different AI approaches, breaking AI into 385 the subsets of Nature Inspired Algorithms and Machine 386 Learning, the latter of which contains the subset of Deep 387 Learning. The following subsections of this review briefly 388 introduce and discuss different types of machine intelli- 389 gence and their bases. 390 2.1 Machine Learning (ML) 391 ML is a subclass of AI focused on the learning and adjust- 392 ment of algorithms during the data processing stage. These 393 adjustments are based on the sets of data received and the 394 technique is commonly associated with systems that dis- 395 cover trends and patterns within the data under analysis 396 [49]. Within this subclass are artificial neural networks 397 (ANN). ANN is a statistical machine learning technique 398 that will process data in similitude to that of the human 399 brain. The brain consists of interconnected neurons, or 400 nodes, in a layered pattern and the task of ANN is to mimic 401 the neural network pattern. ANN designs an adaptive sys- 402 tem to continuously upgrade ‘the machine’ by learning 403 from errors or miscalculations. An ANN is essentially 404 inspired by the functionality of a neuron, which is a cell 405 that receives electro-chemical signals from other neurons, 406 processes the information received, and then transmits the 407 information electro-chemically to other neurons, Figure 408 5. ANN is programmed to be similarly constructed such 409 that an input layer aggregates input information from con- 410 nected ‘neurons’, which are processed in a hidden layer 411 that is responsible for training, and an output layer, which 412 provides output, A, based on Equation, 11, where an ANN 413 node takes the input value, x, multiplied by weight, W, and 414 added to the bias, b, and this is fed through the activation 415 function, ϕ. The abstraction of layered ANN from a neu- 416 ron and its alignment to the output equation is summarized 417 in Figure 5, from [50]. ANNs solve complex problems, 418 such as identification, classification, or recognition with 419 high accuracy and at high speed. The ANN analyzes input 420 data to make accurate decisions [51]. 421 A=ϕ(Wx+b) (11) Deep Learning (DL) is also a subset of ML. DL mimics the 422 learning process of the human brain, using artificial neural 423 networks to achieve this goal. While machine learning 424 algorithms require human correction, DL algorithms are 425 self-correcting through repetition and as such there is no 426 requirement of human involvement. While ML algorithmic 427 training needs relatively small datasets, DL algorithms use 428 large datasets in the training process, and include both 429 unstructured and diverse data [52]. 4302.1.1 Classification of ML Techniques 431 ML can be subdivided into three main categories: super- 432 vised learning, unsupervised learning, and reinforced learn- 433 ing. Supervised learning is a task-oriented technique that is 434 trained on labeled data (considered the ground fact) to map 435 both inputs to outputs. Although unsupervised learning is 436 a data-driven approach, it does not require labeled data to 437 make ground predictions during training. In reinforcement 438 learning, certain labels are present that distinguish it from 439 supervised and/or unsupervised learning. A reinforcement 440 learning approach is based on the interactions between 441 agents and their environment and it is primarily concerned 442 with the maximization of a cumulative reward. Algorithm 443 performance in both supervised learning and unsupervised 444 learning, is determined by the decreases in both the loss 445 functions and objective functions. In addition to the three 446 categories of learning mentioned, there is the sub-category 447 of semi-supervised learning (or, weak supervised learning) 448 which falls between supervised and unsupervised learning. 449 This is a broad category that is trained on both (a) labeled 450 data (for ground predictions) and (b) unlabelled data (to 451 determine the larger distribution of data and its shape). 452 Supervised learning methods are utilized regularly in meta- 453 materials design as they are considered more mature and 454 accurate than other ML techniques [49]. 455 2.1.2 ML Algorithms 456 Classical ML algorithms are the simplest type comprising 457 multilayer models. This section defines some of the more 458 commonly used ML algorithms that have been used in the 459 design of metamaterials. 460 Linear Regression (LIR) is an elementary method for 461 calculating a linear relationship between input and output 462 variables. LIR does this by relating input features with a 463 continuous output, to predict the continuous properties of 464 metamaterials, such as refractive index, conductivity, or 465 other mechanical parameters like the strength and modu- 466 lus of mechanical metamaterials [53]. The equation for a 467 regression model used in metamaterial design depends on 468 the specific goals and parameters of the design problem. 469 However, in general, regression models for metamaterial 470 design aim to predict specific material properties or per- 471 formance metrics based on the structural characteristics 472 or parameters of the metamaterial [54]. Equation 12 is a 473 generic form of a regression equation that can be used for 474 the design of metamaterials: 475 A=f(X1, X2, X3, X4, ....Xn ) +ϵ (12) where Arepresents the target output to predict, fis the 476 regression function that relates input features to the output, 477 andX1,X2,X3,X4,...Xnare the input features or 478 parameters that describe the metamaterial in terms of its 479 design. These include features and parameters such as 480 geometric dimensions, material properties, and any other 481 relevant parameters. 482 9 APREPRINT - SEPTEMBER 30, 2023 Agents and environmentGNN, RBF NetworksK-Means Clustering, Autoencoders, GAN, PCA, LDA, Feature selection, Mean shiftLinear regression, logistic regression, K -Nearest neighbours, Naïve Bayes, Decision Trees, SVR, MLP , CNN, RNN, LSTM, GRU, ResNets , FFNN NATURE INSPIRED ALGORITHMS Swarm intelligence, Evolutionary algorithms, Genetic algorithms, etc FEEDFORWARD (FFNN) BACKPROPAGATION (BP)ARTIFICAL INTELLIGENCE The imitation of human processes of thinking MACHINE LEARNING Learning based on large, structured data DEEP LEARNING Learning through neural networks without human instructio n Figure 4: Categorisation of AI techniques used in the field of metamaterials. Here the following abbreviations are used: ANN - Artificial Neural Networks, CNN - Convolution Neural Networks, GNN - Graph Neural Networks, GAN - Generative Adversarial Networks, GRL - Graph Representation Learning, RNN - Recurrent Neural Networks, RBF - Radial Basis Functions, PCA - Principle Component Analysis, LDA - Linear Discriminant Analysis, SVR - Support Vector Regression, MLP - Multilayer Perceptron, LSTM - Long Short-Term Memory, GRU - Gated Recurrent Unit, FFNN - Feed Forward Neural Network 10 APREPRINT - SEPTEMBER 30, 2023 Figure 5: Overview of artificial neural network. (A) In a biological neuron, the nucleus transforms the chemical inputs from the dendrites into electric signals. The signals are transmitted to the next neurons through the axon terminals. (B) In a node (perceptron), the input values are transformed by the weights, biases, and activation functions. The output values are transmitted to the next perceptron. (C) Multilayer perceptron consists of multiple perceptrons. Reproduced with permission from [50] under the Creative Commons CC-BY licence. 11 APREPRINT - SEPTEMBER 30, 2023 min β0,β1,...,β n  1 2NNX i=1(Ai−β0−nX j=1βjXij)2+λnX j=1|βj|  (13) Least Absolute Shrinkage and Selection Operator 483 (LASSO) is a modification of the LIR algorithm. Here, an 484 absolute penalization value is added to the loss function 485 [55]. LASSO is used to select the most important features 486 (parameters) when designing a metamaterial. LASSO is 487 expressed by Equation 13, where Nis the number of data 488 points or observations in the dataset, Aiis the target vari- 489 able (e.g. a material property or performance metric) for 490 theithobservation to predict or optimize., Xijis the jth491 input feature (parameter) for the ithobservation that de- 492 scribes the metamaterial, β0is the intercept term, β1,β2,..., 493 βnare the regression coefficients associated with the input 494 features, and λis the regularization parameter. This pa- 495 rameter controls the extent of regularization applied to the 496 coefficients. Higher values of λlead to greater shrinkage 497 and hence feature selection. 498 Polynomial Regression (PR) is another modification of 499 LIR. The PR method uses terms within a polynomial to 500 find linear solutions to predict polynomial relationships 501 between input and output variables [56]. Regression al- 502 gorithms supporting nonlinear models, such as Random 503 Forest, include the construction of multiple decision trees 504 for the purpose of both prediction and classification. Ran- 505 dom forests are versatile, and able to handle complex re- 506 lationships between both input features and metamaterial 507 properties by providing feature importance rankings [57]. 508 Support Vector Regression (SVR) divides high- 509 dimensional data space with one or more sets of hyper- 510 planes. SVR is useful when dealing with nonlinear re- 511 lationships between input parameters and metamaterial 512 properties. A Support Vector Machine (SVM) uses the 513 SVR algorithm for both classification and regression prob- 514 lems [58]. Classification tasks include identifying specific 515 properties of a material and deciding the category within 516 which they belong based on Equation 14, 517 Y=sign nX i=1αiAi⟨Xi, X⟩+b! (14) where Ais the predicted class label for a new input X(e.g., 518 "+1" for positive class and "-1" for negative class), αiare 519 the Lagrange multipliers obtained during the SVM training 520 process, Aiare the corresponding labels for the training 521 data,Xiare the support vectors, which are a subset of the 522 training data that influence the decision boundary, and bis 523 the bias term. The key concept behind SVM classification 524 is to determine the sign of the expression inside the ‘sign’ 525 function to classify a new input into one of the two classes. 526 This expression represents a hyperplane decision bound- 527 ary that separates the classes, and the support vectors are 528the data points closest to this decision boundary. SVM 529 is especially useful in dealing with non-linear separation 530 boundaries. Such nonlinear models are very accurate, and 531 can handle outliers better than linear regression models. 532 Apart from regression tasks, classification is another impor- 533 tant category in ML algorithms as it enables the classifying 534 of metamaterials into predefined groups. 535 Logistic Regression (LOR) is a classification algorithm 536 that includes a logistic loss function. In the context of 537 materials design, it can be used to predict categorical out- 538 comes, e.g. will a particular material exhibit a specific 539 property or behavior [57]. Based on Equation 15, logistic 540 regression for materials design as is, 541 P(A= 1) =1 1 +e−(β0+β1X1+β2X2+...+βnXn)(15) where P(A= 1) represents the probability that the cate- 542 gorical outcome Ais equal to 1, eis the base of the natural 543 logarithm (ca. 2.71828), β0,β1,β2...βnare the regres- 544 sion coefficients associated with the input features that 545 allow the prediction of the probability of a positive out- 546 come based on the material characteristics, X1,X2,X3... 547 Xnare the input features, or parameters, that describe the 548 characteristics of a material. 549 Numerous ML algorithms deal with both classification and 550 regression solutions such as Gradient Boosting [59], De- 551 cision Tree (DT) [60], and K-Nearest Neighbors (KNN) 552 [61]. These algorithms can be used for data representa- 553 tion, feature engineering, and hyperparameter selection in 554 metamaterial design problems. In classical ML algorithms, 555 Artificial Neural Networks (ANN) are the most vital basic 556 building blocks based on perceptrons as a means of devel- 557 oping network structures by generating a stack of multiple 558 layers of artificial neurons to establish nonlinear relation- 559 ships between input and output variables, and the distri- 560 bution of data [62]. As the number and depth of network 561 layers increase, DL models showcase their capacity to 562 provide tremendous results in numerous interdisciplinary 563 areas including in computer science. 564 Multilayer Perceptron (MLP) or Feedforward Neural 565 Networks (FFNN) are simple and perfect DL models. In- 566 formation in FFNN models travels in one-way, as the name 567 implies. Every layer in FFNN contains multiple neurons 568 that generate output to succeeding layers, depending on 569 the input from the preceding layers. The trainable parame- 570 ters/weights are used to calculate respective neuron values 571 and enhanced to decrease the value of the loss function 572 [62]. 573 12 APREPRINT - SEPTEMBER 30, 2023 Back Propagation (BP) is another general ANN method, 574 implemented with the Gradient Descent (GD) method, and 575 aimed at gaining the lowest value of a loss function during 576 the training process [63]. BP functions are similar to the 577 calculation of derivatives, while GD methods decide how 578 to transfer the value of the loss function to a lower value. 579 The process repeats until the value of the loss function 580 reaches a minimum. The mathematical form of backpropa- 581 gation for a feedforward neural network for metamaterial 582 design is provided in Equations 16-25. For a given input 583 sample, a forward pass computes the network’s output 584 A(L), where Lis the number of layers. It calculates the ac- 585 tivations and weighted sums in the activation function A(l)586 for each layer lfrom 1 to Las in Equation 16, where f(l)587 is the activation function for the layer l, Equation 17. For a 588 Loss computation, loss is calculated between the network’s 589 output A(L), and the true target values Yusing a suitable 590 loss function, typically a mean squared error (MSE) for 591 regression tasks, or, cross-entropy for classification tasks 592 as shown in Equation 18. 593 Forward Pass: 594 Z(l)=A(l−1)·W(l)+b(l)(16) A(l)=f(l)(Z(l)) (17) L=Loss(A(L), Y) (18) Backward Pass: 595 In a Backward pass, gradients of the loss with respect to 596 the weights and biases are computed to minimize the loss, 597 Equation 19, with respect to the output of the last layer. 598 Gradients are then computed for each layer lfrom L−1 599 to 1. Equation 20, and Equation 21 show the Gradient 600 of Weighted Sum and Gradient of Weights respectively. 601 The Gradient of Biases with respect to axis = 0 shown in 602 Equation 22 and Equation 23 shows the activation for the 603 previous layer. 604 dA(L)=∇L (19) dZ(l)=dA(l)·f(l)′(Z(l)) (20) dW(l)=A(l−1)T·dZ(l)(21) db(l)=X (dZ(l),axis= 0) (22)dA(l−1)=dZ(l)·W(l)T(23) Update Weights and Biases: 605 Using the gradients computed in the backward pass, 606 weights based on Equation 24 are updated as are biases as 607 shown in Equation 25 within each layer using a gradient 608 descent optimization algorithm, such as stochastic gradient 609 descent (SGD): 610 W(l)=W(l)−α·dW(l)(24) b(l)=b(l)−α·db(l)(25) where αis the learning rate, a hyperparameter that controls 611 the step size during weight and bias updates. This process 612 is iterated until the neural network’s weights and biases 613 converge to values that minimize the loss function, effec- 614 tively training the network to make accurate predictions 615 for material design. The specific architecture and hyperpa- 616 rameters of the network will depend on the problem and 617 dataset in the context of materials design. 618 Convolutional Neural Networks (CNN) : With the ex- 619 ception of general FFNN and BP algorithms, CNN are 620 the most well known DL architectures, gaining extensive 621 attention due to their numerous applications in Natural 622 Language Processing (NLP), and computer vision. Over 623 recent decades, CNN have become one of the most pow- 624 erful tools as they have an immense capacity for handling 625 significant image data. 626 CNN are image-based DL models where an artificial neu- 627 ron receives input images, assigns learnable weights and 628 biases to various objects or aspects within the images, 629 processes them, and subsequently processes the result as 630 output. The input layer takes input image pixels in a ma- 631 trix form and sends them to multiple hidden layers, where 632 feature extraction is performed by a mathematical linear 633 operation called ‘Convolution’. CNN differ from simple 634 neural networks in that the data is convolutional, which is 635 beneficial when processing images. A CNN can be math- 636 ematically represented as a series of mathematical opera- 637 tions, including convolution, pooling, and fully connected 638 layers. Equation 26 shows the convolution operation of a 639 simple CNN architecture, where the convolution operation 640 is∗and involves applying a filter (also called a kernel), 641 W, to a 2D or 3D input image X, resulting in an output 642 feature map Y.iandjrepresent the spatial coordinates 643 of the output feature map, mandnrepresent the spatial 644 coordinates of the filter, and ·defines element-wise multi- 645 plication [64]. The architecture and the number of layers 646 in a CNN can vary significantly depending on the prob- 647 lem at hand, and more complex architectures often involve 648 additional components like skip connections, batch normal- 649 ization, and dropout to improve training and generalization 650 performance. 651 13 APREPRINT - SEPTEMBER 30, 2023 Y(i, j) = (X∗W)(i, j) =X mX nX(i+m, j+n)·W(m, n) (26) CNN require far less pre-processing as compared to other 652 ML classification techniques. While traditional techniques 653 use hand-engineered filters, with abundant training, CNNs 654 can learn these filters/characteristics [65]. In materials 655 design problems, CNN are good for describing material 656 properties through their ability to extract features at vari- 657 ous hierarchical levels [64]. It is not solely materials that 658 have various features at innate hierarchical levels, but this 659 concept is universal e.g. sound, language, matter, etc [66]. 660 Recurrent Neural Networks (RNN) have also gained 661 popularity along with CNN through their capacity to deal 662 with sequential information. There is the main difference 663 between CNN and RNN is that in CNN, input and output 664 are considered to be independent of each other, while RNN 665 analyze the sequential data, and the output depends on the 666 latter and previous sequence, using memory to decide the 667 output of each layer. CNN might not deal with sequen- 668 tial tasks, for example, it would be difficult for CNN to 669 emphasize the sequence of information to predict the next 670 word in an incomplete sentence if the sequence of sen- 671 tences is absent. RNN deals precisely with these types of 672 sequential problems. The gradient manipulated by BP can 673 be vanished or expanded in RNN [67]. To meet this chal- 674 lenge, various types of mechanisms have been developed 675 such as Gated Recurrent Units (GRU), Long Short-Term 676 Memory (LSTM) [68], Attention [69], and ResNet [70], 677 growing the power of RNN in different natural language 678 processing problems for instance speech processing and 679 language translation. RNN are also gaining more routine 680 use in scientific tasks such as de novo protein design and 681 protein folding. 682 Generative Adversarial Networks (GAN) are based on 683 generative algorithms and are used to create new data 684 points based on real data distributions. GAN are a success- 685 ful architecture and includes two separate neural networks: 686 the ‘Generator’ and the ‘Discriminator’ [71]. The task of 687 the generator is to create new ‘fake’ data points, while the 688 discriminator matches these ‘fake’ data points with the ex- 689 isting data. During the training process, both the generator 690 and the discriminator contend as the generator produces 691 new genuine but ‘not real’ data points as a part of its aim 692 to fool the discriminator. Concurrently, the discriminator 693 discriminates the real data from the fake data as effectively 694 as possible. The mathematical form of GANs in the con- 695 text of material design involves two main components: the 696 generator and the discriminator as shown in Equation 27 697 and 28 respectively. The generator Gtakes random noise 698 zas input and produces fake data Mfakeas output: 699 Mfake=G(z;θg) (27)Mfakerepresents the generated material structure or prop- 700 erty,zis the input random noise vector, and θgrepresents 701 the parameters of the generator neural network. The dis- 702 criminator Devaluates whether an input data point Mis 703 real or fake: 704 D(M;θd) (28) D(M;θd)represents the probability that Mis a real ma- 705 terial structure or property, θdrepresents the parameters 706 of the discriminator neural network. GAN achieve conver- 707 gence at the moment when the Nash equilibrium is held 708 between the generator and the discriminator. The method 709 of balancing the performance of both networks is similar to 710 the equilibrium process in a physical system that includes 711 both forces of attraction and repulsion, which shows GAN 712 can likely also describe physical phenomena. Conditional 713 GAN (CGAN) is a type of GAN, that generates data under 714 controlled characteristics or conditions, using labels as a 715 control variable [72]. The image-to-image translation is an 716 example of CGAN that use an original image as a control 717 variable in the generator [73]. 718 Variational Autoencoders (V AE) include a single neural 719 network to encode the input information into a latent code, 720 after which the output data is reconstructed by decoding the 721 latent code [73]. It is another type of generative model used 722 for tasks such as feature extraction, data generation, and 723 dimensionality reduction. While V AE have not typically 724 been used in metamaterial design, they can play a role in 725 assisting with certain aspects of the design process, such as 726 in generating new material designs, exploring latent spaces, 727 or in optimizing the material properties [74]. 728 Principal Component Analysis (PCA) andLinear Dis- 729 criminant Analysis (LDA) are dimensionality reduction 730 techniques commonly used in metamaterial design. While 731 neither are direct design tools for metamaterials, they can 732 be a valuable pre-processing step or analysis tool in the 733 context of metamaterials research. Metamaterial design 734 deals with high-dimensional feature spaces representing 735 material properties, geometries, or other characteristics. 736 PCA and LDA can be used to reduce the dimensionality of 737 the feature space by identifying and retaining the most in- 738 formative features (principal components) while discarding 739 less important ones [75][76]. They can potentially simplify 740 the design space and thus make it more manageable. 741 Reinforcement Learning (RL) is a class of ML algo- 742 rithms based on environmental variations and aimed at 743 maximizing long-term achievements. The training process 744 focus is in finding an equilibrium between the exploitation 745 of existing knowledge and the exploration of new territory 746 [77]. 747 14 APREPRINT - SEPTEMBER 30, 2023 Unlike normal neural networks, Graph Neural Networks 748 (GNN) are another type of ML technique, operating on 749 Euclidean information and acting on non-Euclidean data 750 graphs with nodes connected by corners in unnatural or- 751 ders [78]. Graph Convolutional Networks (GCN) are 752 current breakthroughs in GNN and have been demonstrat- 753 ing the competency of GNN to learn graph embedding 754 through the passing of messages between nodes. GCN 755 performed outstandingly in semi-supervised classification 756 problems that are equally applicable to both mechanics 757 and materials design problems, including graphical struc- 758 tures [79]. Radial Basis Function (RBF) networks are 759 a type of artificial neural network that can be applied to 760 various machine learning tasks, including regression and 761 classification. While RBF networks are not as commonly 762 used as other neural network architectures such as CNN or 763 RNN in metamaterial design, they still show potential as 764 a valuable tool in data representation and evaluation and 765 optimization tasks [80]. 766 Transfer Learning is a recent breakthrough in machine 767 learning and is a powerful tool as it recycles previous 768 training, which can later be used for both the basic and 769 advanced stages of machine learning. Transfer learning 770 is a beneficial approach when the user starts from scratch. 771 When the resources to build a model from the ground up are 772 unavailable, using Transfer Learning, the user can benefit 773 from a pre-existing model. The user can apply the same 774 principles that have been used for training on similar large 775 quantities of data, and can then modify it for a specific use 776 case. By adjusting model parameters and feeding it new 777 data, the model rapidly adapts and learns to classify new 778 objects, saving computational resources and a significant 779 amount of time [81] [82]. 780 2.2 Nature Inspired AI Algorithms 781 Nature inspired AI algorithms draw inspiration from natu- 782 ral processes, behaviors, or phenomena in the biological, 783 physical, or ecological world to solve complex optimiza- 784 tion and design problems. These algorithms are often used 785 in metamaterials design to efficiently explore vast design 786 spaces, and to discover novel structures with desirable 787 properties. This section will explore some commonly used 788 nature-inspired AI algorithms in metamaterials design. 789 2.2.1 Genetic Algorithms (GA) 790 Evolutionary algorithms are a nature-inspired optimization 791 algorithm that mimics the process of natural selection and 792 evolution to search for optimal or near-optimal solutions 793 in a design space. Genetic algorithms (GA), are amongst 794 the most well-known evolutionary algorithms used in the 795 design of metamaterials. GA have found extensive applica- 796 tions in the optimization of metamaterial lattice structures, 797 especially in combination with machine learning formu- 798 lations. These approaches leverage the power of GA in 799 exploring the vast design space of metamaterials, and by 800 combining them with machine learning techniques, the 801user can enhance the optimization process to discover in- 802 novative lattice structures [83]. 803 Machine learning models, such as surrogate models or neu- 804 ral networks, are used to approximate an objective function 805 that evaluates the performance of a given lattice structure. 806 GA can then be used to optimize the surrogate model, 807 thereby reducing the computational cost that would nor- 808 mally be required if evaluating each design individually 809 [84][3]. Mathematically representing a GA for metamate- 810 rial design requires definitions of the key components that 811 govern the optimization process. Equation 29 illustrates 812 design as a chromosome or genome, where Xrepresents a 813 chromosome, which is a vector of the design parameters, 814 and each xicorresponds to a design parameter, such as 815 materials properties or geometrical characteristics. 816 X= [x1, x2, . . . , x n] (29) In Equation 30, Ptis an initial population of Nindividu- 817 als (chromosomes) obtained randomly or using heuristic 818 methods. Equation 31 shows objective function f(X)that 819 quantifies the performance of a metamaterial design based 820 on the design parameters X. This function evaluates how 821 well the metamaterial meets specific design objectives or 822 constraints. 823 Pt= [X1, X2, . . . , X N] (30) f(X) :X→R (31) Equation 32 is used to evaluate the fitness of each individ- 824 ual in the population using an objective function, where 825 F(Xi)is the fitness of individual Xi. 826 F(Xi) =f(Xi),fori= 1,2, . . . , N (32) Selection mechanisms such as roulette wheel selection or 827 tournament selection can be used to choose individuals 828 from the current population to become parents for the next 829 generation. The probability of selection for an individual, 830 Xi, is proportional to its fitness, F(Xi). After selection, it 831 applies crossover (recombination) to pairs of parent chro- 832 mosomes to create offspring chromosomes as shown in 833 Equation 33. A common crossover operator is a single- 834 point crossover. The specific crossover method depends 835 on the problem and on chromosome representation. 836 Xoffspring =Crossover (Xparent1, Xparent2) (33) Mutations introduce small random changes to offspring 837 chromosomes to add diversity to the population. Mutations 838 help the user explore new regions of the design space as 839 understood in Equation 34. 840 15 APREPRINT - SEPTEMBER 30, 2023 Xmutated =Mutation (Xoffspring ) (34) A New Population is formed by selecting the best individ- 841 uals from the current population (elitism) and adding the 842 offspring and mutated individuals as shown in Equation 843 35. 844 Pt+1= [Xelite1, Xelite2, . . . , X mutated 1, Xmutated 2, . . .] (35) GA specifies termination criteria, such as a maximum num- 845 ber of generations or a convergence threshold based on 846 fitness values. GA continues until the termination criteria 847 are met. The final population consists of individuals that 848 represent optimal or near-optimal metamaterial designs. 849 The choice of operators, chromosome representation, and 850 objective function, will depend on the problem at hand and 851 the characteristics of the metamaterial design task [85]. 852 GAs are versatile and can be used in various aspects of 853 metamaterial design, including optimizing electromagnetic 854 properties (e.g., achieving desired frequency responses 855 or refractive indices) [86], mechanical properties (e.g., 856 stiffness or flexibility) [87], etc. They are particularly 857 useful when the design space is large, complex, and non- 858 linear, enabling the discovery of innovative solutions that 859 might be challenging to find through manual design or 860 through other optimization methods. 861 2.2.2 Swarm Intelligence 862 Swarm intelligence algorithms can be beneficial in opti- 863 mizing the structure and properties of metamaterials for 864 various applications. These algorithms emulate collective, 865 decentralized behaviors observed in natural swarms such 866 as bird flocks, fish schools, and ant colonies, to efficiently 867 search across a design space, and subsequently discover 868 novel metamaterial architectures. These systems exhibit 869 emergent intelligence and adaptability, allowing them to 870 find solutions to complex problems with no centralized 871 control. Swarm intelligence algorithms emulate these be- 872 haviors to solve optimization, search, and decision-making 873 tasks [88]. Researchers and practitioners often choose the 874 most suitable algorithm based on the characteristics of the 875 problem, the optimization goals, and the available domain 876 knowledge. Ant Colony Optimization (ACO) is one of 877 the most popular swarm intelligence algorithms, inspired 878 by the foraging behavior of ants. It is used to solve combi- 879 natorial optimization problems. Ants deposit pheromones 880 on paths, and other ants use these pheromone trails to 881 make decisions. Over time, paths with higher pheromone 882 levels are favored [89]. Particle Swarm Optimization 883 (PSO) is another popular swarm intelligence algorithm 884 used in metamaterial design. It is inspired by the social 885 behavior of birds when flocking or fish when schooling. 886 It optimizes a problem by iteratively adjusting a popula- 887 tion of particles within the solution space. Particles move 888towards the best-known solution (individual best) and the 889 best-known solution among all particles (global best) [90]. 890 Bee Colony Optimization (BCO) algorithms are based 891 on the foraging behavior of honeybees. Bees explore the 892 solution space and communicate information about promis- 893 ing solutions to their hive mates. It is commonly used for 894 solving optimization problems and is particularly effec- 895 tive in continuous optimization [91]. Firefly Algorithms 896 (FA) are based on the flashing behavior of fireflies, where 897 fireflies attract each other with their light. In optimization 898 problems, fireflies represent potential solutions, and their 899 brightness corresponds to the quality of the solution. Fire- 900 flies move towards brighter neighbors, and the algorithm 901 converges to an optimal solution [92]. The Bat Algorithm 902 (BA) is inspired by the echolocation behavior of bats used 903 for continuous optimization and employs a population of 904 virtual bats that search for solutions in the search space. 905 Bats emit sound waves, and the algorithm adjusts the fre- 906 quency and loudness of these waves to explore and exploit 907 the solution space. Cuckoo Search (CS) algorithms are 908 inspired by the breeding behavior of cuckoo birds, this 909 algorithm is used for global optimization problems. It in- 910 volves a population of virtual cuckoos that lay eggs in host 911 nests. The quality of a nest corresponds to the fitness of 912 a solution and the algorithm mimics the natural selection 913 process [93]. The Krill Herd Algorithm (KHA) is based 914 on the herding behavior of krill and involves a population 915 of virtual krill that move collectively toward improved 916 solutions. Krill update their positions based on both per- 917 sonal experience and social interactions [94]. Each of the 918 swarm intelligence algorithms mentioned has been used 919 for the design optimization of metamaterials, which will 920 be discussed in Section 3. The methods are versatile and 921 applicable to various optimization and search problems 922 across different domains. 923 3 Machine Intelligence in Metamaterial 924 Design 925 This section prioritizes publications where machine intel- 926 ligence has been applied to solve metamaterial problems. 927 The section begins (Section 3.1) with machine learning 928 applications focusing on: gradient-based methods, dimen- 929 sionality reduction, and other ML methods, including pre- 930 dominantly regression. This is followed by consideration 931 of the application of ANN to metamaterials design (Section 932 3.2). ANN has a very specific set of application methods 933 including surrogate modeling and inverse design, and as 934 such we focus our discussion on these. Finally, we high- 935 light papers applying nature-inspired methods (Section 936 3.3), which are commonly used in the design of metamate- 937 rials. 938 16 APREPRINT - SEPTEMBER 30, 2023 3.1 Machine Learning in Metamaterials Design 939 3.1.1 Gradient-Based Methods 940 Gradient-based methods are efficient approaches for de- 941 sign optimization problems comprising multiple variables. 942 Gradient-based optimization techniques iteratively update 943 structural parameters or materials properties to minimize 944 (or maximize) an objective function, typically related to de- 945 sired performance criteria. Algorithms, such as conjugate 946 gradient and steepest descent, perform searches to find opti- 947 mal design arrangements taking into account the prescribed 948 constraints while minimizing an objective function [95]. In 949 various fields of engineering, gradient-based optimization 950 is employed as the basis for topology optimization, which 951 aims to maximize structural performance by optimizing the 952 distribution of material. This application includes but is not 953 limited to automotive, aircraft, and structural engineering 954 [4], [96]. In the metamaterials context, gradient-based non- 955 linear topology optimization has been shown as effective 956 for the microscale design of elastic structures [97]. Backer 957 and co-workers [98] use the steepest descent algorithm 958 to optimize metasurfaces. Their approach made use of 959 analytical simulations in the context of Fourier optics, for 960 which gradients were evaluated using an adjoint-gradient 961 approach. In another paper [99] metasurface designs were 962 optimized using a level-set method to define the refrac- 963 tive index of a meta-atom cross-section. The objective 964 function was then described as the difference between the 965 desired output and the optical response of a meta-atom, 966 and gradient descent optimization was used to seek out 967 and find the most suitable meta-geometries. Other papers 968 in the field employ algorithms such as gradient descent to 969 optimize problems related to elastic metamaterial-based 970 vibration absorbers [100], electromagnetic devices [101], 971 photonic band gap structures [102] and acoustic metamate- 972 rials [103]. Parameter optimization is also possible from 973 the calculation of analytical gradients. This approach is 974 commonly used in ANN training, and gradient-based al- 975 gorithms employed in back-propagation [104] are also a 976 means by which model parameters can be adjusted [105]– 977 [111]. 978 Gradient-based methods have several limitations which 979 become more apparent when applied to both metamate- 980 rial discovery, and design optimization problems. Firstly, 981 the calculation of analytical gradients is often impractical 982 or unfeasible. Unlike in topology optimization problems 983 where the analytical gradients can be found using adjoint 984 methods [101], the equations governing the behavior of 985 metamaterials and the exact solution to these equations are 986 usually unknown. Therefore, most metamaterial design 987 processes rely on computationally expensive FE-based ap- 988 proaches that approximate the numerical gradients. Hence, 989 for more complex problems with high numbers of vari- 990 ables, the evaluation of the numerical solution becomes a 991 primary limiting factor in the optimization process. Sec- 992 ondly, gradient-based methods tend to get caught at local 993 minima or saddle points, resulting in the manifestation of 994 sub-optimal solutions. They also struggle to handle non- 995smooth functions, where gradients cannot be evaluated at 996 the points of discontinuity. Consequently, the performance 997 of gradient-based optimization may vary drastically de- 998 pending on the initial estimates defined as a starting point 999 of the algorithm. To overcome these limitations, some 1000 researchers combine gradient-based methods with other 1001 optimization techniques. For example, Boddeti et al. [112] 1002 reports a hybrid inverse design framework using gradient 1003 descent and gradient-free (GA) algorithms to find an op- 1004 timal metamaterial structure for a thermal-photovoltaic 1005 emitter coating application. The gradient descent algo- 1006 rithm optimizes the geometrical properties of the structure 1007 while the genetic algorithm searches for the most suitable 1008 materials from a given database, resulting in the optimiza- 1009 tion of the arrangement of inclusion particles. 1010 3.1.2 Dimensionality Reduction 1011 Dimensionality reduction (DR) refers to ML approaches 1012 that are designed to reduce the number of features de- 1013 scribing a given data set. This is achieved by either the 1014 generation of new features based on the existing ones or 1015 through the selection of key features within the set that 1016 best represents the data. A few applications that benefit 1017 from such approaches include but are not limited to, data 1018 visualization, storage, and tasks involving heavy compu- 1019 tations. The latter is the most frequently applied in the 1020 development of metamaterials as they help reduce com- 1021 putational expenses during the simulation, prediction or 1022 generation of new metamaterial architectures. 1023 A method invented and first introduced by Reuter et al. in 1024 2005 [113] called ‘ShapeDNA’ is a technique for shape 1025 characterization based on spectral shape descriptors. It ap- 1026 proximates the geometrical shape of any 2D or 3D object 1027 by taking the eigenvalues of its Laplace-Beltrami operator 1028 [114]. The method is simple, does not rely on previous 1029 knowledge, and is highly suitable for the retrieval of non- 1030 rigid shapes [115]. This was used as a DR technique 1031 in a data-driven mechanical metamaterial design frame- 1032 work as presented by Wang et al. [116]. The researchers 1033 showed that by approximating a complex unit cell shape 1034 with a low number of descriptors, they were able to sig- 1035 nificantly reduce computational requirements. A similar 1036 dimensionality reduction approach has been reported in the 1037 two publications by Bostanabad et al. [117], [118] where 1038 these researchers used spectral shape descriptors based 1039 on Laplace-Beltrami operators to reduce the metamaterial 1040 input dimensions from 2500 pixels to 16 scalar descriptors. 1041 Another approach is to employ a principal component anal- 1042 ysis (PCA) which is a linear DR technique that constructs a 1043 best fitting lower dimensionality subspace within the initial 1044 data space. It aims to position the linear subspace so that 1045 the data approximation error can be minimized. PCA has 1046 been used as means to reduce 5 design variables of pho- 1047 tonic devices to no more than 2, allowing them, therefore, 1048 to reduce the optimization problem size and perform an 1049 in-depth exploration of the design space [119]. Wang et 1050 al.[111] further demonstrated that this technique could 1051 17 APREPRINT - SEPTEMBER 30, 2023 be used to reduce dimensionality by applying it to the la- 1052 tent space generated by a variational autoencoder, hence 1053 enabling visualization of the results. In the same paper, 1054 PCA was applied as a tool to cluster data points within the 1055 latent shape according to their representative metamaterial 1056 shapes and structures. A more in-depth analysis of differ- 1057 ent DR techniques has been summarized by Kiarashinejad 1058 et al. [120] who compare a traditional PCA with KPCA 1059 (kernel principal component analysis), a nonlinear PCA 1060 technique, and an autoencoder. They suggested that al- 1061 though autoencoders are more computationally intensive 1062 to train and use, the DR capabilities and performance of the 1063 approach surpass PCA and KPCA. Other less commonly 1064 applied ML algorithms for DR problems in metamaterial 1065 design include t-distributed stochastic neighbor embed- 1066 ding (t-SNE) [121] and the Isomap [122] algorithm and its 1067 extensions. 1068 3.1.3 Other ML Algorithms 1069 This section summarizes some of the less commonly ap- 1070 plied machine learning algorithms in metamaterials design, 1071 primarily regression algorithms, which are used to deter- 1072 mine regression trends within the available data. Kurniwati 1073 et al. [123] employed random forest regression algorithms 1074 to build models that were able to predict antenna parame- 1075 ters, such as gain, V oltage Standing Wave Ratio (VSWR), 1076 bandwidth, and return loss, based on the metamaterial 1077 structures and their distinct performance parameters as 1078 applied to training in the algorithm. Their work suggests 1079 that a predictive approach such as was used, enabled a 1080 reduction in computational time, when compared against 1081 conventional optimization techniques. Liu et al. [124] 1082 presented a similar justification for the use of machine 1083 learning algorithms. The research team reported that sim- 1084 ple regression algorithms were more accurate and efficient 1085 for predicting the performance of graphene-based meta- 1086 materials, as compared to ANN. Their work compared 1087 Decision Tree, Extra Tree, KNN, Random Forest algo- 1088 rithms and ANN performance for the forward and inverse 1089 design of photonic devices [125]. Mukherjee et al. [126] 1090 considered ANN, complex tree, weighted KNN. These are 1091 referred to as soft computing techniques and the group 1092 used these to classify metamaterial-based defected pho- 1093 tonic crystal structures. Clustering methods were used by 1094 Moon et al. [121], specifically K-Means clustering, to de- 1095 sign metaplasmonic biosensors with negative index meta- 1096 materials. Further, nonlinear programming has been shown 1097 by Bacigalupo et al. [127] to be an effective method for 1098 designing acoustic metamaterials. Other forms of less com- 1099 mon forms of metamaterial optimization besides Cluster- 1100 ing techniques include Nelder-Mead and Bayesian, [128], 1101 which has been used to optimize structure and also to opti- 1102 mize wave band gaps in mechanical metamaterials, [129]. 1103 Bayesian machine learning has been reported as a method 1104 for the design of super-compressible micro scale mechan- 1105 ical meta-structures, [130], while Gaussian Radial Basis 1106 Function (RBF) networks used together with Quasi-Monte 1107 Carlo sequences [131] are a form of multiple optimiza- 1108tions, which is also noted in [132] where CNN (Resnet50) 1109 and Monte Carlo methods were used in parallel for the 1110 optimization of actuators. 1111 3.2 Artificial Neural Networks in Metamaterials 1112 Design 1113 In this section, papers that focus on the use of artificial neu- 1114 ral networks (ANN) in metamaterials design are discussed. 1115 The section is broadly divided into: surrogate modeling 1116 (Section 3.2.1), inverse design (Section 3.2.2), and a pre- 1117 sentation of less commonly used methods (Section 3.2.3). 1118 3.2.1 Surrogate Models 1119 Artificial neural networks (ANN) are commonly used for 1120 building surrogate models. ANN are highly suitable for 1121 this application as they have the ability to learn and predict 1122 outputs based on the training data supplied. According to 1123 [133] artificial neural networks can approximate any con- 1124 tinuous function, making them suitable for use alongside 1125 numerical methods such as finite element analysis (FEA). 1126 Further, designers can benefit from the abundance of avail- 1127 able network architectures that can further be customized 1128 to meet predictive modeling needs. As FEA is commonly 1129 used for structure-function optimization in metamaterial 1130 design, multiple publications can be found discussing their 1131 research in terms of replacing this numerical method with 1132 ANN-based solutions. 1133 Some researchers use ANN as a tool for the analysis and 1134 prediction of mechanical properties. Gu et al. [134], for 1135 example, have shown that this unconventional approach to 1136 optimization enables the design of metamaterials without 1137 the need for complete micro-structural data, also referred 1138 to as coarse-grained modeling. Using a CNN architec- 1139 ture to speed up the process of predicting mechanical re- 1140 sponse, they report that while FEA methods required a 1141 5 day compute time, a trained ANN-enabled framework 1142 required only 1 minute to compute 100,000 microstruc- 1143 tures. Importantly, Gu and co-workers also showed that the 1144 final characteristics for material placement in an optimized 1145 tough-structure, are those that localize in the areas of crack 1146 opening, as shown in Figure 6. This corresponds well to 1147 the placement of energy absorbing material at the site of 1148 splits and openings in natural structures such as in insect 1149 wings, [135]. 1150 In an aligned paper by Xue et al. [136], similar findings 1151 are reported. Here, they showed that a surrogate ANN 1152 model reduces computational time by two orders of mag- 1153 nitude when compared to finite element simulations. Their 1154 work employed a single hidden layer architecture to predict 1155 the nonlinear mechanical behavior of hyperelastic meta- 1156 materials. Constitutive models can be used to analyze 1157 hyperelastic lattice structures [137]. Here, the authors de- 1158 veloped models based on a feedforward ANN architecture, 1159 and inform that in some cases, higher accuracy results 1160 were achieved as compared to the analytical model. The 1161 work of [137] provides clarity on the distinctness of the 1162 18 APREPRINT - SEPTEMBER 30, 2023 Figure 6: Machine learning generated designs. (a) Strength and toughness ratios of designs computed from training data and ML output designs. The strength ratio is the strength normalized by the highest training data strength value. The toughness ratio is the toughness normalized by the highest training data toughness value. The ML output designs are shown from training loops of 1000 and 1000000. Envelopes show that ML material properties exceed those of training data. (b) Effects of learning time on ML models for minimum, mean, and maximum toughness ratio start to converge as training loops increase. (c) Microstructures from partitions A (lowest toughness designs in training data) and B (highest toughness designs from ML) in part (a) of the figure with corresponding colors for unit cell blocks (blue = U1, orange = U2, yellow = U3). Also shown in the right-most columns for designs A and B are the strain distributions, which show lower strain concentration at the crack tip for the ML-generated designs. Reprinted with the permission of [134] under a Creative Commons CC-BY licence. 19 APREPRINT - SEPTEMBER 30, 2023 deformational, F, and 1st Piola-Kirchhoff stress Pstates 1163 of a baseline structure described as an X-cell, for uniaxial, 1164 biaxial, planar, volumetric and shear cases, Figure 7. The 1165 work highlights the importance of algorithm training with 1166 respect to deformational behavior, as the authors note that 1167 equal stress and deformation for each strut of an X-cell 1168 is only true for a volumetric loading case, and as such, 1169 structure organization has to be considered in the context 1170 of its utility. Here, the plots correspond to 3×3matrices 1171 denoted by the colors in the legends. 1172 Similar surrogate models have been used for the analysis 1173 and simulation of high frequency electronic products such 1174 as antennas and antenna arrays. In a paper by [138], Saha 1175 et al. discuss a data driven approach in analyzing split 1176 ring resonator metamaterials and reveal that high accuracy 1177 can be achieved using a surrogate model. Another team 1178 [139], [140] demonstrated ANN predictions of permeabil- 1179 ity and permittivity in split ring resonator type (SRR) meta- 1180 material designs. ANN-based models are also shown to 1181 appropriately replace commercially available FE packages 1182 and other analytical models as a design tool for electro- 1183 magnetic metamaterials [141], meta-optics components, 1184 and optical devices [142]–[145]. Wu et al. [5] in addition 1185 considered the design of 1D acoustic metamaterials using 1186 an artificial neural network-based surrogate model. In their 1187 work, they found the method was useful even for aperi- 1188 odic metamaterials, and noted that their approach neither 1189 requires a predefined number of unit cells to function ef- 1190 fectively nor is there a need for the user to understand the 1191 physical model well for it to work. 1192 Convolutional networks are an attractive basis for surro- 1193 gate modeling as they have the ability to recognize pat- 1194 terns within a design space, irrespective of pattern location. 1195 Geometrical features such as chamfers or fillets can be 1196 easily distinguished and optimized to reduce stress con- 1197 centrations. In a paper by Garland et al. [40], machine 1198 learning methods were developed to be Pareto Optimal, 1199 an approach that took on two objectives: wave propaga- 1200 tion through a material, and the elastic modulus. They 1201 found the best possible trade-off between the two without 1202 stipulating any particular claim about which combination 1203 is best. Essentially, a Pareto Optimal approach does not 1204 allow the improvement of one objective, without leading 1205 to the worsening of the other objective. In Figure 8 (a) 1206 their initial results from 1000 random designs can be seen 1207 and in (b) both numerical and experimental validations 1208 of three of their final generated designs are shown. As 1209 ANN-based surrogate models significantly reduce com- 1210 putational time in comparison to traditional FE analysis, 1211 they are commonly incorporated in optimization frame- 1212 works. These models are routinely combined with genetic 1213 algorithms [40], gradient-based approaches [95] and other 1214 optimization techniques. 1215 Deep physical informed neural networks (DPINN) have 1216 been used by Fang et al. [141] in the design of electro- 1217 magnetic metamaterials who generated their designs by 1218 DPINN, focusing on improvements to the activation func- 1219tion as a route to circumventing high wave number prob- 1220 lems. Deep neural networks as used to model all-dielectric 1221 metasurfaces can also be trained to predict performance 1222 at the atomic scale [144], which the researchers validated 1223 by comparing direct results against full-wave simulations 1224 using a commercial electromagnetic simulator. Magnusson 1225 and co-workers [143] use deep neural networks focused 1226 on improving the functionality of optical devices. In their 1227 work, they related the device outputs to the four parameters 1228 of the Stokes vector using a linear 4×4matrix transfor- 1229 mation. Deep neural networks were found to increase the 1230 number of outputs substantially over traditional algebraic 1231 methods that establish a correlation between input and 1232 output data. Chiroptical responses in 2D chiral metamate- 1233 rials have been shown to benefit from deep learning (DL) 1234 approaches. When compared with traditional, rigorous 1235 coupled wave analysis (RCWA) methods for the study of 1236 circular dichroism (CD) in higher-order diffraction beams, 1237 DL is found to accelerate the design of hypersensitive pho- 1238 tonic devices. Akashi et al. [142] compared Mie theory 1239 against neural network (NN) computations to solve scat- 1240 tering problems over the cross-section of a multi-layered 1241 cylinder (forward calculation). In their work, they report 1242 that the NN calculations were 3500 times faster than those 1243 based on Mie theory. 1244 3.2.2 Inverse Design 1245 The solving of inverse design problems is another major 1246 application area for ANN. It is well understood that the tra- 1247 ditional forward design process can be time-consuming. In 1248 inverse design, a metamaterial structure needs to initially 1249 be suggested using intuition and insight, and in alignment 1250 with the performance requirements, as illustrated in Figure 1251 9. The metamaterial structure is then analyzed, and fur- 1252 ther iterations or optimization steps are typically needed 1253 to further customize the structure. The design process is 1254 made more efficient when the relationship between the per- 1255 formance and design parameters is well established. The 1256 inverse design framework essentially analyzes the perfor- 1257 mance requirements and directly matches a metamaterial 1258 structure that perfectly suits the required specifications, by 1259 solving the problem inversely. 1260 To build and train an ANN-based inverse framework, there 1261 has to be a solution for both forward design and the inverse 1262 problem. Some researchers choose to implement two sepa- 1263 rate ANN to deal with this task. In a publication by Kumar 1264 et al. [146] a multi-layer perceptron (MLP) architecture is 1265 used, with the first ANN mapping the design parameters 1266 onto the stiffness response of a spinodoid metamaterial 1267 in a forward fashion. This part of the framework serves 1268 as a surrogate model to replace FE analysis as described 1269 previously. The second, inverse ANN, is then trained using 1270 knowledge obtained from the surrogate model, to solve 1271 the inverse problem and relate stiffness requirements to 1272 the specific design parameters defining the metamaterial. 1273 As the inverse design approach significantly reduces both 1274 time and effort in the design process, it has gained popu- 1275 larity, for example, a similar approach employed by Ma 1276 20 APREPRINT - SEPTEMBER 30, 2023 Figure 7: Training dataset of the X cell, showing all components of FF (left column of plots) and PP in Pa (right column of plots) for uniaxial, biaxial, planar, volumetric, and simple shear cases (from top to bottom). Reprinted with the permission of [137] under a Creative Commons CC-BY licence. 21 APREPRINT - SEPTEMBER 30, 2023 (a) (b) Figure 8: (a) Pareto front for the initial set of 1000 random designs, identified as generation 1. The axis values are normalized by the maximum value associated with 60 %dense vertical columns and (b) normalised experimental results for the three generated designs compared to FEA results. Each design is given a different color. FEA data points are shown as squares, and VeroTMMagenta are shown as circles. Reprinted with the permission of [40] under a Creative Commons CC-BY licence. 22 APREPRINT - SEPTEMBER 30, 2023 A FORWARD PROBLE M AN INVERSE PROBLE MPerformance measuresDesign parameters Figure 9: Forward and inverse problems in metamaterial design et al. [105] focused on choosing 2D cellular structure 1277 patterns to match targeted force-displacement responses. 1278 The framework was able to predict the entire nonlinear 1279 response history, including large deformations within the 1280 plasticity region, rather than just a single mechanical prop- 1281 erty of the whole cell. Importantly, Ma and co-workers 1282 [105] elucidate that structure predictions made after only 1283 20%of training reflect those at 100 %, Figure 10. Since 1284 the inverse design approach has the ability to match the 1285 metamaterial structure to its full range of response, inverse 1286 design frameworks are gaining popularity for the predic- 1287 tion of full range spectral responses in photonic and plas- 1288 monic metamaterials. Hou et al. [147] demonstrated the 1289 use of ANN for on-demand design. A desired reflectance 1290 spectrum was provided as input to their framework, which 1291 consequently adjusted the parameters of a split ring res- 1292 onator based nano-optic metasurface. Other research out- 1293 puts show that similarly implemented frameworks used 1294 for the design of metamaterials match the required absorp- 1295 tion [148], [149], circular dichroism (CD) spectra [150], 1296 [151], and scattering spectra [152]. Similar deep learn- 1297 ing techniques have also been applied by some groups 1298 in the prediction of plasmonic [153] and photonic [109] 1299 properties in graphene-based metamaterials. A paper by 1300 Nanda et al. demonstrates a similar technique employing 1301 inverse ANN to design an electromagnetic split ring res- 1302 onator based on metamaterial structures, relating design 1303 parameters with desired frequency and permeability values 1304 [108]. 1305 To ensure design suitability in some inverse problems, the 1306 range of admissible solutions may require the imposition 1307 of limits. In such cases, physics-informed neural networks 1308 (PINN) are useful as they solve learning tasks that need 1309 to respect the laws of physics, which are fundamentally 1310 solved through nonlinear partial differential equations. A 1311paper by Fang et al. [141] and Chen et al. [154] demon- 1312 strated the usefulness of PINN in this respect, by using the 1313 method for the inverse design of cloaking metamaterials. 1314 A number of researchers discuss the use of convolutional 1315 neural networks (CNN) in terms of their high suitability 1316 in image processing and analysis. CNN are mostly ap- 1317 plied to metamaterial design processes when the structural 1318 or atomic arrangement is defined by a patterned image. 1319 Wilt et al. [155] used CNN to analyze pseudo-randomized 1320 Rorschach (ink blot) images and developed varying aux- 1321 eticity metamaterials based on these patterns. Another 1322 example is from Liu et al. [156], who employed CNN 1323 architecture in meta-surface reflection phase predictions 1324 by analyzing planar patterns containing 16×16square 1325 sub-blocks. Similarly, architectured networks have been 1326 utilized for mapping the geometrical properties of 2D fre- 1327 quency selective meta-surfaces and their corresponding 1328 reflectance and transmission spectrums [157]. In a paper 1329 by He et al. [158] CNN architectures were shown to be 1330 demonstrably useful for the analysis of spectral images 1331 when compared against processing the same type of infor- 1332 mation in a conventional data table format [148], [149]. 1333 The work elucidated inverse design processes of plasmonic 1334 metamaterial structures as also achievable by mapping their 1335 geometrical properties and optical responses. 1336 With regards to autoencoder type architectures commonly 1337 used in dimensionality reduction tasks [120], [159], [160], 1338 a few papers report the exploitation of similar approaches 1339 for inverse design purposes. Harper et al. [161] ex- 1340 plain many-to-one problems, discuss the limitations of 1341 traditional autoencoders, and propose a new pseudo- 1342 autodecoder (decoder-encoder architecture) that achieves 1343 high prediction accuracy. Their further work utilized the 1344 technique on broadband highly reflective meta-surfaces 1345 [110]. 1346 Generative networks (GN) are another class of the neural 1347 network framework that can be used in the discovery of 1348 new metamaterial structures. As the name suggests, GN 1349 are developed with the aim of generating new data, as 1350 opposed to the mapping of input-output relationships. In 1351 metamaterial design, applying GN enables a very broad ex- 1352 ploration of the design space, much more so than in other 1353 ML techniques. This is because these frameworks are able 1354 to produce new structures that are not limited by the train- 1355 ing set. The GN approach is therefore very suitable for the 1356 types of problems where the number of possible metamate- 1357 rial structural arrangements is infinite, and the training data, 1358 no matter how large, is unable to capture all the possible 1359 variations. The two most commonly used in metamaterial 1360 related applications are variational autoencoders (V AE) 1361 and generative adversarial networks (GAN). V AE is a gen- 1362 erative type neural network used in the inverse design of 1363 metamaterials. Similarly to traditional autoencoders, the 1364 network has an encoder part compressing the number of 1365 input variables into latent space, and another part with the 1366 objective of decoding the information in the most accurate 1367 way. V AE have an additional regularization term intro- 1368 23 APREPRINT - SEPTEMBER 30, 2023 Figure 10: (a–c) Top four patterns and their reference curves for a targeted response with one, two and three load- dropping points only using the predicted responses by 20 %training ratio. (d) Top four feasible patterns and their reference curves for the same target response in (c) using the reference responses of all available patterns. Reprinted with the permission of [105] under a Creative Commons CC-BY licence. 24 APREPRINT - SEPTEMBER 30, 2023 duced into the loss function of the autoencoder that ensures 1369 regularity within the latent space during training and helps 1370 to avoid data over-fitting. As a result, this ensures that the 1371 data within the latent space will have a sensible meaning 1372 once decoded, thus enabling a generative process using this 1373 network architecture. Liu et al. [162] use this technique 1374 for the inverse design of photonic nanostructures. They 1375 trained a V AE to reconstruct the input geometrical data of 1376 the metamaterial structures and then separate the decoder 1377 part to use it as a data generator. Randomly sampled vector 1378 values matching the distribution (check) of the latent space 1379 data were fed to the generator, which transformed the data 1380 to their corresponding structures. A similar approach was 1381 employed by Wang et al. [111] who used a V AE architec- 1382 ture to generate new 2D metamaterial structural assemblies 1383 that deform in a pre-programmed manner. The framework 1384 discussed in the publication is able to generate functionally 1385 graded metamaterials as well as heterogeneous structures 1386 that follow desired distortional behaviors. Distortional be- 1387 havior has also recently been designed using combinations 1388 of biological structures [163], specifically leaf venation 1389 and trabecular bone. In their work, they used an unsu- 1390 pervised GAN to design novel metamaterial architectures 1391 with differing levels of structural hierarchy, showing thus 1392 that there is near infinite design potential for future meta- 1393 material structures Figure 11. Lastly, Ma et al. [164] 1394 report on how V AE can be incorporated into a deep gener- 1395 ative model for designing photonic metamaterials. A GAN 1396 usually has two separate neural networks that are trained 1397 simultaneously in an adversarial process. A generative part 1398 of the model produces a new metamaterial sample which is 1399 similar to those used in network training, while a discrim- 1400 inative part estimates the probability of the new sample 1401 coming from the generator rather than from the training set 1402 [165]. Throughout this competitive process, a GAN pro- 1403 duces sets of completely new metamaterial structures that 1404 fall within the statistical distribution of the training set. A 1405 method of employing GAN for topology optimization has 1406 also been demonstrated by Kudyshev et al. [166] in their 1407 paper on meta-surface based thermophotovoltaics. The 1408 researchers created a GAN-assisted topology optimization 1409 framework by training the neural network using a set of 1410 topology-optimized designs. The design framework was 1411 shown to significantly reduce optimization time in com- 1412 parison with direct optimization methods while generating 1413 high efficiency designs. Generative networks have also 1414 been shown as highly suitable for the discovery of new 1415 metasurfaces while solving inverse design problems [167]– 1416 [171]. In one of the earliest publications in the field [167], 1417 Liu and co-workers use a GAN architecture to concep- 1418 tualize metasurface designs for optical applications that 1419 match user-defined optical spectra. After training was com- 1420 plete, the network was able to produce sets of previously 1421 unknown arbitrary patterns. This is in contrast to other 1422 ML practices, which fundamentally aim to optimize the 1423 geometrical parameters of the metasurface structure. The 1424 approach also ensures that local minima are avoided while 1425 the design space is explored, as multiple structures are 1426 generated for a single optical spectrum input. 14273.2.3 Other ANN Applications 1428 Optimization frameworks are usually parametric or topo- 1429 logical when used to design metamaterial structures. Meta- 1430 heuristic or gradient-based approaches are amongst the 1431 most commonly used for optimizations, however, many re- 1432 searchers also discuss approaches when optimization relies 1433 primarily on ANN. 1434 Chen and co-workers [95] discuss a methodology using 1435 two ANNs in combination for the design of metamateri- 1436 als. The first one called the ‘predictor’ acts as a surrogate 1437 model while the second one, the ‘designer’ is an optimiza- 1438 tion network that searches for the global minimum. It 1439 uses a backpropagation technique, which computes an- 1440 alytical gradients to find a solution to the optimization 1441 problem. In their findings, they note that although this 1442 gradient-based method might still get caught at a local 1443 minima, it is much faster than conventional numerical 1444 gradient-based approaches, such as finite-difference meth- 1445 ods (FDM). Sarmah et al. [172] proposed the use of an 1446 ANN-based optimization framework, trained using several 1447 gradient descent algorithms. Their findings suggest that 1448 ANN is highly suitable for the optimization of design pa- 1449 rameters and in their case for defining the performance of 1450 a split ring resonator. ANN have also been considered an 1451 alternative method for topology optimization. Kollmann 1452 et al. [173] used a CNN architecture to design 2D me- 1453 chanical metamaterials by implementing a non-iterative 1454 topology optimization framework. In general, ANN is a 1455 data ‘hungry’ approach and sufficient data might be hard 1456 to obtain. This is especially true for more complex 3D 1457 meta-structures. Manufacturing or other design constraints 1458 might also limit the data that can be collected. Active learn- 1459 ing may at times be a solution to this problem as the ANN 1460 trains while the other optimization cycle is running. Pre- 1461 dictions at the beginning of this process are less accurate, 1462 however, they improve with each design iteration. Exam- 1463 ples being active learning with GA [40] and learning with 1464 two ANN: a ‘designer’ and a ‘predictor’ [95] as discussed 1465 earlier in this paragraph. Calik et al. [174] discusses the 1466 optimization of ANN architecture and model parameters 1467 in more detail. The researchers use Bayesian optimiza- 1468 tion to enhance the performance of a deep learning based 1469 surrogate model replacing full-wave electromagnetic sim- 1470 ulations. Their results suggest that a model developed 1471 using such methodology demonstrates a higher prediction 1472 accuracy and lower sensitivity to the variation in training 1473 samples in comparison to conventional and state-of-the-art 1474 regression methods. Harper et al. [175] also address the 1475 subject of optimal ANN design in metamaterial develop- 1476 ment. It employs Bayesian optimization with Gaussian 1477 processes to adjust the architecture and then the hyperpa- 1478 rameters of the network mapping geometrical properties 1479 of all-dielectric metasurface structures to their reflective 1480 spectra. Their research focuses on the optimization of 1481 fully connected ANNs with one or two hidden layers and 1482 states the performance improvements. The findings sug- 1483 gest that similar optimization techniques can be applied to 1484 deeper networks such as CNN or GAN. There are also hy- 1485 25 APREPRINT - SEPTEMBER 30, 2023 Figure 11: Unsupervised GAN for the combination of trabecular bone and leaf venation to create novel metamaterial structures. Reproduced with the permission of [163] under a Creative Commons (CC-BY) licence. 26 APREPRINT - SEPTEMBER 30, 2023 brid approaches where genetic algorithms (GA) and ANN 1486 are used in combination. [176] use this approach to de- 1487 velop a model predicting the chiroptical response of 2D 1488 diffractive chiral metamaterials. GA is used for finding 1489 the optimum number of neurons in each of the four hidden 1490 layers of the ANN. Similarly, Dong et al. [128] used GA 1491 in this hybrid approach to increase the predictive power 1492 of an ANN-based surrogate model for elastic metamateri- 1493 als. Their results show that by adjusting the architecture 1494 and the hyperparameters of the network, the normalized 1495 mean absolute error (NMAE) was reduced from 1.1109 to 1496 0.0278 in comparison to the results obtained from a finite 1497 element model. The final architecture contained 5 hidden 1498 layers with 19 neurons in each layer. 1499 3.3 Nature-Inspired Algorithms in Metamaterials 1500 Design 1501 Many nature inspired algorithms used in metamaterials 1502 design, are meta-heuristic approaches inspired by evolu- 1503 tion, particle intelligence and physical processes. These 1504 algorithms exhibit stochastic behavior and iteratively ex- 1505 plore design space using intelligent search strategies [177]. 1506 Meta-heuristic algorithms are typically adopted in opti- 1507 mization problems where there is a complex objective 1508 function landscape. They find global optima well and do 1509 not stagnate at local minima, as with certain other gradient- 1510 based approaches. These algorithms are also often applied 1511 to optimization problems with missing or imperfect infor- 1512 mation, which cannot be solved using other optimization 1513 techniques [178], for example, to optimize the geometri- 1514 cal arrangement of a metamaterial structure, to select the 1515 most suitable bulk materials from a database [179], or, 1516 to arrange a meta-atom assembly [180]. Meta-heuristic 1517 algorithms tend to find good or optimal solutions in a sig- 1518 nificantly shorter amount of time compared to brute force 1519 executive search methods. Hence, they are often used in 1520 design processes that have a considerable reliance on FEA 1521 or other computationally expensive metamaterial perfor- 1522 mance evaluation techniques. Nevertheless, meta-heuristic 1523 optimization algorithms are often considered computation- 1524 ally more expensive than other techniques. They have 1525 a tendency to find sub-optimal solutions, as finding pre- 1526 cise positions of global maxima is impractical due to the 1527 additional computational costs involved. Most of these al- 1528 gorithms do not scale well with the increasing complexity 1529 of a design problem, and their suitability should be consid- 1530 ered variable, depending on the extensiveness of available 1531 problem-specific information. 1532 3.3.1 Genetic Algorithms 1533 Evolutionary algorithms (EA), including but not limited 1534 to the genetic algorithms (GA) are a class of optimization 1535 algorithm that uses the individuals of a population as en- 1536 ablers to solutions in a given problem. These approaches 1537 work through genetic operators such as reproduction, muta- 1538 tion, recombination and selection. These operators develop 1539 solutions to problems by evolving the population. They are 1540highly exploratory approaches and since each individual 1541 within a generation acts as an independent agent search- 1542 ing for the optimum, the computations can be executed in 1543 parallel. As with all meta-heuristic approaches, EA cannot 1544 ensure that the solution is optimal due to the nature of the 1545 random search, but they do find a near-optimal solution if 1546 it exists [181]. The literature suggests that although evolu- 1547 tionary algorithms take significant time to converge and are 1548 not the most efficient approach, they are robust methods 1549 and perform consistently well in all optimization prob- 1550 lems [83]. GA algorithms can very efficiently elucidate 1551 minute geometrical alterations in mechanical metamateri- 1552 als to maximize mechanical performance [10], [182] and 1553 recently Cerniauskas and Alam [183] revealed >4000% 1554 improvements in specific shear properties were achievable 1555 from their base structures through GA regulated parametric 1556 optimization, Figure 12. 1557 The most widely applied meta-heuristic search algorithms 1558 in metamaterial design are variations of the genetic algo- 1559 rithm. A few GA variants have been applied to design 1560 and optimize metamaterials. Simple GA algorithms for ex- 1561 ample, have been used to design inhomogeneous acoustic 1562 metamaterials by varying the coordinates in the Helmholtz 1563 equation [184], specifically, the team combined results 1564 from the theory of homogenization with GA. Importantly, 1565 they validated their outputs by designing different types 1566 of cloaks (curved rectangles, split rings and crosses) and 1567 after comparison against existing technologies, which their 1568 metamaterial designs mimicked well. Wu and co-workers 1569 [185] also made use of simple GA, but in their case to 1570 direct the assembly of phononic meta-atoms based meta- 1571 materials. Their work evidenced the possibility of GA 1572 use in modularizing metamaterials with both maximum 1573 and tunable bandgaps, by taking a bottom-up approach 1574 to design, starting at the scale of meta-atoms. Wang et 1575 al.[84] used a multi-island GA to parametrically opti- 1576 mize re-entrant auxetic honeycomb structures. The team 1577 analyzed the structures using a Python script based on 1578 Castigliano’s theorem where mass was a constraint the 1579 Poisson’s ratio was used to form the objection function. 1580 A non-sorting genetic algorithm (NSGA-2) discussed by 1581 Garland et al. [40] is an example of a multi-objective GA, 1582 which when combined with a surrogate CNN model was 1583 used to optimize the structures of lattice metamaterials. 1584 In their work, they noted that they could train the CNN 1585 more rapidly and with fewer example designs via active 1586 learning, a method that essentially allows the algorithm 1587 to select its own new training data. Another example of 1588 multi-objective GA is in [87] where tunable stiffness meta- 1589 material structures in bone were optimized using NSGA-2. 1590 Here, the objectives were focused on tuning varied stiff- 1591 ness while minimizing von Mises stress. Design problems 1592 using GA often require lengthy computational times. To 1593 deal with this limitation, meta-heuristic searches can be 1594 combined with other techniques to exacerbate the process 1595 of metamaterial discovery. Luna et al. [86] for example, 1596 used GA with an ANN based surrogate model to reduce 1597 the computational time required for full wave numerical 1598 27 APREPRINT - SEPTEMBER 30, 2023 Figure 12: Specific shear stiffness comparison for 4-polytope projected mechanical metamaterial unit cells in shear at different levels of optimization with a 5-cell structure reaching >4000% improvements in specific shear properties over its original unit cell structure. Reproduced with the permission of [183] under a Creative Commons (CC-BY) licence. simulations in the design optimization of permittivity in 1599 electromagnetic metamaterials. It has further been shown 1600 that GA and ANN can be coupled for optimization and 1601 performance predictions in the development of modular 1602 mechanical metamaterials [185]. GA/ANN combinations 1603 have also been used in inverse design frameworks to gen- 1604 erate a compositional pattern-producing network (CPPN) 1605 [3], which was used to compound meta-atoms into metallic 1606 metamolecules for the controlled design of optical meta- 1607 surfaces. The team showed that even though metasurfaces 1608 typically have many intricate mechanisms as well as mul- 1609 tiple degrees of freedom in view of design optimization, 1610 the rate at which these surfaces can reach an apical de- 1611 sign is improved by CPPN. The design of dual polarizer 1612 metasurfaces was also optimized using an inverse design in 1613 combination with training through a deep learning network 1614 (DLN) serving as a forward model for GA, i.e. a deep 1615 learning forward genetic algorithm (DLF-GA) by Zhu et 1616 al.[186]. Here, the team focused on using the DLF-GA to 1617 predict the phase of meta-atoms for electromagnetic meta- 1618 surfaces where dual (linear) polarization was specifically 1619 orthogonal. Other variations of GA such as micro genetic 1620 algorithms [187] have been applied as optimization meth- 1621 ods for binary patterned nanostructures in tungsten based 1622 metasurface absorbers, and for developing inverse design 1623 frameworks with systematic arrangements such as mesh 1624 filters with grid patterns [188] of optical metamaterials. 1625 Hybrid genetic program (HGP) is a tree based genetic 1626 program that generates random initial population designs 1627 for the purpose of both design synthesis and subsequent 1628 optimization. HGP uses user-defined cost functions to eval- 1629 uate each individually generated design and the method 1630 has been shown to result in metamaterial based antennas 1631 with wider bandwidth absorbers than other nature inspired 1632 techniques [189]. A final extended GA technique, im- 1633 plemented for seismic metamaterials for the purpose of 1634 earthquake mitigation is Augmented Lagrangian Genetic 1635 Algorithm (ALGA) [190], which solves nonlinear opti- 1636 mization problems that have a combination of linear and 1637nonlinear constraints, and bounds and is an algorithm inte- 1638 grated within popular commercial modeling tools such as 1639 Matlab. 1640 3.3.2 Particle Swarm Intelligence Algorithms 1641 Ant Colony Optimization (ACO) is inspired by the forag- 1642 ing behavior of ants. It recreates a population of artificial 1643 ants that deposit pheromones on paths as they explore a 1644 solution space. Over time, paths with higher pheromone 1645 levels become more attractive, leading to the discovery 1646 of optimal or near-optimal solutions as evidenced from 1647 some of the earlier papers in this area [191], [192], which 1648 primarily consider electromagnetic metamaterial antennae 1649 and the prediction of bandwidth. ACO is often also used 1650 for combinatorial optimization problems [89]. Alharbi et 1651 al.for example recently proposed a Dipper-Throated (DT) 1652 optimization for ACO (DTACO) [193] as a viable method 1653 for predicting the bandwidth of electromagnetic metama- 1654 terials, which in turn is an enabler for the manipulation 1655 of bending, absorbing or amplification of electromagnetic 1656 waves. DT optimization is essentially one that assumes 1657 birds fly and swim to find food and the optimization takes 1658 into account bird numbers, velocities, and positions. The 1659 researchers note that though the method is reliable, its re- 1660 liability requires accurate models and simulations of the 1661 metamaterials under consideration, and that without accu- 1662 rate initial models, the method provides inaccurate predic- 1663 tions of bandwidth. The spatial filtering of electromagnetic 1664 waves is important when designing metamaterials that are 1665 frequency selective surfaces (FSS). While much of the 1666 work in this area is dealt with in 2D, Zhu et al. [194], 1667 [195] used a multi-objective lazy ACO (MOLACO) to 1668 consider both polarization problems and angle indepen- 1669 dent design in 3D FSS, generating unintuitive structures 1670 based on wire grid resolutions that superior to conventional 1671 planar, multilayer and 3D FSS in terms of angular perfor- 1672 mance and unlike traditional designs, were not polarization 1673 dependent. 1674 28 APREPRINT - SEPTEMBER 30, 2023 Particle Swarm Optimization (PSO) is another effective op- 1675 timization method of the geometrical parameters in meta- 1676 materials. It can be used to fine-tune the shape, size, and 1677 arrangement of metamaterial elements to achieve specific 1678 optical, acoustic, or thermal properties. PSO is also ap- 1679 plied in designing cloaking devices using metamaterials 1680 such as antennas [90] [196]. Choudhury et al. [197] devel- 1681 oped PSO to yield structural parameters in metamaterials 1682 relevant at specific resonant frequencies. The team op- 1683 timized square split ring resonator (SRR) metamaterials 1684 and noted that the increase in resonant frequency, was due 1685 to the square micro-sized split in the SRR, and that post- 1686 PSO optimization, the SRR showed improved directivity. 1687 Similar conceptual benefits are reported for circular patch 1688 split structures [198]. There has been some work focused 1689 on combining PSO with neural network (NN) methods 1690 to improve predictive capacity. Zhang et al. [199] for 1691 example, combined CNN and binary particle swarm op- 1692 timization (BPSO) to alter the reflection phases of meta- 1693 surfaces to match a desired target. BPSO algorithms have 1694 also been discussed [200] in terms of topology optimiza- 1695 tion for the 3D design of negative permeability dielectric 1696 metamaterials. Here, the researchers compared the PSO 1697 optimized topologies against those modeled using the com- 1698 mercial FEA tool ANSYS. After comparison, while FEA 1699 was found to yield accurate results, the CPU time was 1700 significant as compared to the PSO, the main benefit of 1701 which over FEA was computational time saved. Song et 1702 al.[201] provide another example, this time combining 1703 ANN and PSO. They used a radial basis function neural 1704 network RBF-NN as a surrogate between geometric param- 1705 eters and predicted frequency responses in electromagnetic 1706 metamaterial absorber structures. 1707 Bee Colony Optimization (BCO) algorithms have, albeit 1708 to a lesser degree than PSO and ACO, been used to op- 1709 timize metamaterials in various applications, including 1710 microwave absorbers, lenses, and beam steering devices. 1711 Artificial bees can be used to explore the design space and 1712 adapt the properties of the metamaterial to reach specific 1713 target requirements [91]. Gaynutdinov et al. [202] used a 1714 BCO technique to optimally parameterize metamaterials 1715 to generate an electromagnetic shield with operator speci- 1716 fied coefficients for reflection and transmission. Yu et al. 1717 [203] used BCO to accelerate the rate at which permittivity 1718 and thickness of wire metamaterials for use as resonant 1719 cavity antennas, could be determined. The optimization of 1720 magneto-mechanical metamaterials structures has received 1721 recent attention from Ma et al. [204], who used a discrete 1722 artificial bee colony to find optimal structure-distributions 1723 within these metamaterials for targeted strains. The au- 1724 thors trained a ResNet to replace FEA based outputs, thus 1725 significantly decreasing optimization output times. 1726 Firefly Algorithms (FA) can be applied to optimize the light 1727 manipulation properties of metamaterials, for example by 1728 creating metamaterials with unique refractive indices, or, 1729 light-focusing capabilities. Fireflies adjust the parameters 1730 of metamaterial elements to achieve desired optical effects 1731 [92]. While they are not as popular as ACO and BCO, there 1732are several instances where FA have found usefulness as a 1733 tool within the design process for, especially, electromag- 1734 netic metamaterials. Using FA, Saraswat [205] developed 1735 a hybrid fractal SRR metamaterial antenna for multiband 1736 operations such that the design contributed to resonance 1737 between 1.9-15.2GHz. In their earlier work Saraswat and 1738 Kumar [206] had also considered miniaturization methods 1739 for SRR metamaterials in quad based antennas. While FA 1740 is known to be a generally efficient and speedy algorithm 1741 that is concurrently easy to implement [207], it can expe- 1742 rience convergence problems through entrapment in local 1743 minima, with FA control parameters being a randomiza- 1744 tion parameter and an absorption coefficient. To counter 1745 FA convergence problems and thus improve the effective- 1746 ness of FA, Srivastava et al. [208] incorporated an inertial 1747 weighting within the FA to balance trade offs between local 1748 and global FA searches, and consequently also improving 1749 the precision and accuracy of fraction delay filter designs, 1750 which has relevance in antenna engineering. 1751 Cuckoo Search (CS) algorithms have been used to enable 1752 the optimization of structural parameters in metamaterials 1753 to improve specific electromagnetic or acoustic properties. 1754 Cuckoos can lay eggs (representing candidate designs) in 1755 nests (representing solutions) to explore and improve the 1756 design space [93]. Examples of CS in metamaterials de- 1757 sign include the work of Yang et al. [209] whose focus was 1758 on acoustic metamaterials with multiple parallel hexagonal 1759 Helmholtz resonators, more recently researched by [210], 1760 Yang et al. [211] whose work focused on the structural pa- 1761 rameterization of metamaterials for sound absorption, and 1762 the development of thin multi-layer metamaterial sound 1763 absorbers through CS based parameter optimization [212]. 1764 4 Discussion 1765 4.1 General Trends 1766 Our analyses reveal that the most popular techniques, by 1767 the number of publications, are evolutionary algorithms 1768 and artificial neural networks. A summary of methods 1769 is provided in four separate tables in the Appendix, fo- 1770 cusing on ANN (Appendix Table 1), ML excluding ANN 1771 (Appendix Table 2), nature inspired AI (Appendix Table 1772 3), and combined methods (Appendix Table 4). Approxi- 1773 mately half of all publications we reviewed use a variation 1774 of an evolutionary algorithm while a little under half relied 1775 on a neural network. Less than 10% of the papers used 1776 different approaches of AI and ML such as gradient-based 1777 methods and dimensionality reduction techniques. Figure 1778 13 shows the percentage share of different methodologies 1779 used over the last five years. This pie chart reflects the 1780 rapid increase in the preferred use of ANN for metamate- 1781 rial applications. The dominant application areas of the 1782 two techniques differ and the techniques are not mutu- 1783 ally exclusive. Whilst ANN is used for mainly 3 specific 1784 applications namely, inverse design, surrogate modeling 1785 and hyperparameter optimization, GA and its derivatives 1786 have a wider range of applications. They are primarily for 1787 29 APREPRINT - SEPTEMBER 30, 2023 optimization such as parameter optimization and inverse 1788 design. Additionally, there are several publications we dis- 1789 cussed that use hybrid approaches, combining the strengths 1790 of different AI techniques. We are now in a position to 1791 extend Figure 3 in a way that allows us to identify more 1792 specifically the AI areas applied to metamaterials design 1793 and engineering, Figure 14. While in this new figure we 1794 note that the exponentially increasing trend is still evident 1795 as was notably the case in Figure 3, we also note high 1796 surges in recent years in the NN, ML and EA histogram 1797 bars, with significant increases also noticeable with GA. 1798 Currently, ML algorithms are clearly very popular and ul- 1799 timately, each approach exacerbates the rate of discovery 1800 of new metamaterial architectures, which is indubitably 1801 the reason these AI methods have gained so much popular- 1802 ity. The main areas of AI application in the metamaterials 1803 design space are summarized in the pie chart of Figure 1804 15. Here, we note that mechanical metamaterials are the 1805 largest application area, followed by areas related to the 1806 manipulation of light-based waves i.e. biosensor-based, 1807 photonic, plasmonic, acoustic, electromagnetic, etc. 1808 4.2 Opportunities and Challenges 1809 Our review conveys the importance that artificial intelli- 1810 gence (AI) has in metamaterial design in modern times. AI 1811 enables the exploration of a broad range of possible new 1812 materials discoveries. The integration of AI has already led 1813 to several breakthrough discoveries in material science and 1814 engineering. Some of the key advantages and challenges of 1815 machine intelligence in metamaterials design as indicated 1816 by our review follow. 1817 4.2.1 Efficient Exploration of Design Space 1818 AI algorithms, such as genetic algorithms, neural networks, 1819 and reinforcement learning, can efficiently explore com- 1820 plex metamaterial design spaces to find novel and optimal 1821 structures, saving time and resources compared to brute 1822 force algorithms, analytical methods, or iterative trial-and- 1823 error methods. Time efficiency means informed decision- 1824 making is easier at the user-level [213], and the recognition 1825 of specific performance criteria can reduce the need for 1826 extensive testing, whether experimental or virtual [214]. 1827 In addition, complex patterns and relationships within the 1828 available data can be analyzed, as part of the fundamental 1829 design process. Existing material properties can also be 1830 analyzed and patterns suggested leading to desirable be- 1831 haviors and properties. Vast design spaces can be explored 1832 with a high degree of precision and detail that would be im- 1833 practical or impossible for human designers [215]–[217]. 1834 4.2.2 Multi-Objective Optimization 1835 Machine learning models like neural networks, can be 1836 trained to model the relationships between specified de- 1837 sign parameters and a range of different material prop- 1838 erties [218], [219]. These models approximate complex, 1839 nonlinear objective functions, enabling efficient optimiza- 1840 tion from baseline input structures. Genetic algorithms 1841(GA) and particle swarm optimization (PSO) explore this 1842 by iteratively generating and evaluating designs. Conver- 1843 gence towards Pareto-optimal solutions [220], [221] can 1844 further the efficient exploration of the vast metamaterials 1845 design space by generating candidate designs based on 1846 prior knowledge and simulations. Reinforcement learn- 1847 ing algorithms, for example, can learn optimal strategies 1848 for exploration, rather than solely focusing on perform- 1849 ing architectural optimizations [222]. AI can also create 1850 surrogate models of complex simulations, which are other- 1851 wise computationally expensive. Surrogate models, such 1852 as Gaussian processes [223], approximate the simulation 1853 results, allowing for faster iterations during the process of 1854 optimization. 1855 4.2.3 Materials Discovery 1856 AI as discussed in this review, is seen to accelerate the 1857 discovery of new materials by predicting their properties. 1858 This can be conducted at varying length scales, from the 1859 atomic-level to the macro-scale. Atomic-level design de- 1860 cisions leading to novel and unique atomic arrangements 1861 result in a knock-on effect at the macroscopic level [180], 1862 affecting macro-scale metamaterial properties and behav- 1863 iors. Machine learning algorithms and neural networks 1864 (CNN, GAN, CGAN), furthermore, can be used to ana- 1865 lyze broad datasets of materials properties and their atomic 1866 structures. By identifying patterns and correlations within 1867 the data, the properties of new metamaterials based on 1868 their composition, structure, and other factors [72] [73] 1869 [77] can be predicted. Genetic algorithms and evolution- 1870 ary optimization techniques generate and evolve candidate 1871 materials by iteratively modifying their compositions or 1872 structures to improve specified properties. As such, it is 1873 clear that AI enables the discovery of new materials that 1874 may not otherwise have been considered using traditional 1875 methods [184] [185]. By analyzing a broad range of can- 1876 didate compositions and structures, these techniques cre- 1877 ate unconventional metamaterials with potentially unique 1878 properties and behaviors. 1879 4.2.4 Data Availability and Quality 1880 Through our review, we noted that the availability and 1881 quality of data are two significant challenges when AI 1882 techniques are applied to metamaterial design. This is es- 1883 pecially true in data-driven models, which require large, 1884 high-quality datasets [224]. Metamaterial design often 1885 involves high-dimensional data, making it challenging to 1886 collect, store, and analyze the relevant information effec- 1887 tively. Obtaining comprehensive and reliable data on mate- 1888 rials properties and their corresponding structures can be 1889 challenging, particularly in the case of emerging materials. 1890 Some metamaterial properties or structures may also be 1891 unconventional, leading to sparse data points within the 1892 design space. Sparse data can make it difficult for AI mod- 1893 els to generalize effectively [225]. Moreover, gathering 1894 unique experimental data is expensive and time-consuming, 1895 30 APREPRINT - SEPTEMBER 30, 2023 25% 7% 13% 9%7%10%3%17%5%3%1%CNNs LSTMs RNNs GANs MLP Autoencoder GNNs Support Vector Machine(SVM) K-nearest neighbor (KNN) Naive Bayesin K-Mean Clustering Figure 13: Publications categorized by machine learning algorithms used in metamaterials design. CNN - convolutional neural network, LSTM - long short-term memory, RNN - recurrentneural network, GAN - generative adversarial network, MLP - multilayer perceptron, GNN - graph neural network. Figure 14: Increasing trend of different AI-based and nature-inspired techniques in metamaterials. 31 APREPRINT - SEPTEMBER 30, 2023 29% 20% 17%11%6%4%4%3%1%1%1% 1% 1% 1% Mechanical SensorBio Photonics Plasmonics Acoustic Electromagnetic Optics Thermal emission Functional coding Dialectric RETRACTED ARTICLE Antenna Nanophotonics Thermophotovoitaics Figure 15: Publications categorized by application area thus limiting the quantity and variety of data that can be 1896 available for training and validation. 1897 4.2.5 Computational Resources 1898 Training complex AI models, in particular deep learning 1899 models, are computationally intensive and powerful hard- 1900 ware may be needed to ensure that the full benefit of the 1901 AI is achieved. When it comes to metamaterials, a fur- 1902 ther challenge arises due to the complex nature of their 1903 design problems, which involve high-dimensional design 1904 spaces, intricate simulations, and computationally expen- 1905 sive optimization processes [226]. Metamaterial design 1906 tasks frequently require parallelization with numerical sim- 1907 ulations to evaluate the performance of candidate designs 1908 in real time. Such simulations are computationally and 1909 resource expensive and are also time-consuming. 1910 4.2.6 Interdisciplinary Expertise 1911 Effective metamaterial design requires collaboration be- 1912 tween materials scientists, physicists, engineers, and AI 1913 programmers [227]. Choosing the appropriate AI optimiza- 1914 tion algorithm is challenging, as different algorithms per- 1915 form differently depending on the problem. Researchers 1916 must carefully select the most suitable algorithm for a spe- 1917 cific metamaterial design task. AI approaches optimize 1918 designs based solely on data-driven patterns, without a 1919 deep understanding of the underlying metamaterial physics 1920 or materials science, leading often to unconventional or 1921 counter-intuitive designs that can be difficult to interpret. 1922Bridging the gap between these disciplines and integrating 1923 the knowledge is a challenge that has to be overcome for 1924 the design process to itself reach an optimum [228]. 1925 4.2.7 Robustness and Generalization 1926 AI models must be robust and capable of generalizing 1927 well beyond the training data. Since metamaterials are 1928 inherently complex, often exhibiting nonlinear behaviors 1929 [229]–[231], it can be challenging to create models that 1930 are sufficiently effective at generalization. Metamaterials 1931 behave differently under different conditions (temperature, 1932 pressure, humidity, etc.), and as such, the collection of 1933 data to circumvent specific variability is challenging and 1934 can lead to uncertainty in the predictions made. Moreover, 1935 models may memorize the training data [232], instead of 1936 learning from underlying principles, which itself can result 1937 in poorer levels of generalization. 1938 4.2.8 The Causal Relationship Problem 1939 Our review reveals that traditional ML does not consider 1940 causal relationships between variables and as such does 1941 not lead to direct solutions [233]. Many widely used neural 1942 network (NN) models, such as deep learning models, are 1943 often referred to as ‘black boxes’ because they lack trans- 1944 parency and interpretation [234]. They can make accurate 1945 predictions or classifications, but the challenge is then to 1946 understand how specific decisions have been made, and 1947 to subsequently explain the causal relationships between 1948 inputs and outputs. Deep neural networks, especially with 1949 32 APREPRINT - SEPTEMBER 30, 2023 complex architectures, can capture extremely complex pat- 1950 terns and relationships in data [235], [236]. While this 1951 is beneficial for prediction accuracy, it can make it very 1952 difficult to interpret the reasons underlying particular de- 1953 cisions. Due to high non-linearity in ANN models, small 1954 changes in input features can lead to large, unintuitive vari- 1955 ations in the model output, making it hard to understand 1956 cause-effect relationships. 1957 4.2.9 Hyperparameter Optimization and Hybrid 1958 ANN & GA 1959 Although the literature suggests that AI-based surrogate 1960 modeling is often used in developing metamaterials, the 1961 common issue remains the selection of hyperparameters 1962 [237] (e.g. batch sizes, activation and loss functions) and 1963 model architecture (e.g. shape of the network, number of 1964 nodes and layers). Hyperparameters are values that control 1965 the learning or optimization process. As is widely known, 1966 the selection of hyperparameters is a key factor that deter- 1967 mines both the quality of the outputs and the performance 1968 of the algorithms. This applies to both nature-inspired 1969 [204] and ML algorithms [238]. Experience and intuition- 1970 driven model setups typically rely on trial-and-error based 1971 approaches which can lead to sub-optimal performance. 1972 4.3 Future Perspectives 1973 We anticipate that the use of artificial neural networks 1974 in metamaterial discovery frameworks will continue to 1975 grow based on evidence on the current growth rate of 1976 the number of publications per year (cf. Figures 3 and 1977 14). AI approaches in metamaterial design hold significant 1978 promise in revolutionizing the field and unlocking new 1979 capacities for designing and engineering metamaterials 1980 [239] with unique properties and behaviors. Since ma- 1981 chine intelligence is both customizable and flexible [43], 1982 we anticipate high predictive power ANN-based models 1983 will be used alongside inverse design frameworks, map- 1984 ping thence, metamaterial performance to their parameters 1985 and further accelerating the discovery of novel metamate- 1986 rials. Generative models and reinforcement learning will 1987 expedite the discovery of new metamaterials [240], ac- 1988 counting for uncertainties in material properties and in 1989 fabrication processes. Moreover, as computational power 1990 continues to rise, the rate of machine intelligence based 1991 metamaterials discoveries will rise. AI will play a crucial 1992 role in solving complex inverse design problems, where 1993 the desired properties are specified, and machine intelli- 1994 gence will search for and find the relevant structures to 1995 meet the specifications. In terms of data-driven design, AI 1996 techniques, including machine learning and data analyt- 1997 ics, will leverage progressively larger materials properties 1998 datasets and over time, as the body of experimental ma- 1999 terial properties data expands, more explicably accurate 2000 designs will emerge [241], [242]. AI will be integrated 2001 into additive manufacturing and nanofabrication as design 2002 optimization will need to take into account limitations in 2003 manufacturing. The integration of machine intelligence 2004into architectural design concurrent with manufacturing 2005 parameterization will enable a rapid realization of complex 2006 structures that can also be made efficiently and at speed 2007 [243]. Greater interdisciplinarity and collaboration will 2008 be needed with experts from a variety of fields interacting 2009 to fine-tune discipline specific concepts into the design 2010 process. 2011 Evolutionary Algorithms (EA) are extremely powerful 2012 [244] and remain the most commonly used technique since 2013 they can optimize models when information is incomplete 2014 or only partially available. The hybridization of AI tech- 2015 niques is increasing, with primary combinations at present 2016 being machine learning and deep learning. Hybridization 2017 can enhance the effectiveness of EA by integrating data- 2018 driven insights, predictive models, and optimization. EA 2019 will become more proficient at handling constraints, in- 2020 cluding physical, manufacturing, or structural constraints, 2021 to ensure that the designed metamaterials meet specifi- 2022 cations. To reduce the computational cost of evaluating 2023 metamaterial designs, EA will increasingly rely on surro- 2024 gate models [97], [128], such as Gaussian processes or 2025 neural networks, as they approximate objective functions. 2026 This will in turn accelerate the optimization process [245], 2027 [246]. In summary, both Artificial Intelligence (AI) and 2028 Evolutionary algorithms (EA), including machine learn- 2029 ing, are powerful tools in metamaterials design. While 2030 AI with machine learning encompasses a broader range 2031 of techniques and methods, EA have their own unique 2032 strengths and applications [198], [247], [248]. The future 2033 of EA in metamaterial design is shaped by their distinct 2034 characteristics and specific challenges. AI offers powerful 2035 gradient-based optimization methods and data-driven in- 2036 sights for metamaterial design. EA remain valuable tools 2037 for global optimization, handling complex design spaces, 2038 and addressing challenges with noisy or expensive objec- 2039 tive functions. The future likely involves greater synergy 2040 between both approaches to unlock more potential within 2041 the metamaterials design space. 2042 Mechanical metamaterials as previously discussed in Fig- 2043 ure 15, are the largest specialized area where machine 2044 intelligence is used for design optimization. In this branch 2045 of metamaterials, the two main directions of research are 2046 performance prediction and inverse design. Since mechan- 2047 ical metamaterials exhibit by default, a hierarchy in de- 2048 sign for properties, machine intelligence will need to look 2049 beyond specifically performance prediction and inverse 2050 design, as other aspects such as manufacturing constraints 2051 [183], microstructural design for macroscopic properties, 2052 and the optimization of operation controls [249]. An emer- 2053 gent branch of mechanical metamaterials is, ‘unfolding 2054 structures’. These are often large deformation structures 2055 typically designed to fold from flat sheets into 3D struc- 2056 tures, taking inspiration from arts such as origami and 2057 kirigami [250]. While origami and kirigami metamaterials 2058 are receiving increasing attention [251]–[254], the utility 2059 of machine intelligence in their design and construction 2060 is still low compared to unit-cell based structures. Many 2061 papers in this area consider topology and unit cell arrange- 2062 33 APREPRINT - SEPTEMBER 30, 2023 ment for controlled deformation [255]–[258], and the total 2063 percentage of papers using machine intelligence in con- 2064 junction with origami or kirigami metamaterials is <2% 2065 overall, and in 2023, the percentage has dropped further 2066 to<1.5%. This is probably partly due to the popularity 2067 of unit-cell based metamaterials, but we presume that the 2068 rate of growth of machine intelligence in origami/kirigami 2069 metamaterials design will increase over time and that this 2070 will likely be a result of the development of more general- 2071 ized AI methods. 2072 The future of machine intelligence in metamaterial design 2073 is an exciting frontier. With time, it will be integrated 2074 into all levels of technology readiness, becoming common- 2075 place as an assistive tool in the computational design of 2076 metamaterials. 2077 Acknowledgements 2078 HS wishes to thank the HEC Pakistan for her PhD scholar- 2079 ship. 2080 Conflicts of Interest 2081 The authors declare no conflicts of interest. 2082 Author Contributions 2083 Conceptualization (GC, PA); Data curation (GC, HS, PA); 2084 Formal analysis (GC, HS, PA); Funding acquisition (HS, 2085 PA); Investigation (GC, HS, PA); Methodology (GC, HS, 2086 PA); Project administration (PA); Resources (HS, PA); 2087 Software (GC, HS, PA); Supervision (PA); Validation (GC, 2088 HS, PA); Visualisation (PA); Roles/Writing - original draft 2089 (GC, HS, PA); Writing - review and editing (GC, HS, PA). 2090 34 APREPRINT - SEPTEMBER 30, 2023 Appendix Table 1: Table summarizing papers that use ANN in metamaterials design. Publication Year Method(s) Application of AI Application field: Acoustic 1 Ma, Cheng, and Liu [259] 2018 Deep Learning Optimization framework 2 Xu, Guan, Bao, et al. [260] 2018 CNN Optimization framework 3 White, Arrighi, Kudo, et al. [246] 2019 ANN surrogate Model 4 Robeck, Cipolla, and Kelly [261] 2019 CNN Optimization framework 5 Hughes, Williamson, Minkov, et al. [262] 2019 ANN Inverse design 6 Sajedian, Kim, and Rho [263] 2019 CNN, RNN Optimization framework 7 Bostanabad, Chan, Wang, et al. [264] 2019 ANN Inverse Design 8 Wu, Liu, Wang, et al. [265] 2020 ANN Inverse Design 9 Weng, Ding, Hu, et al. [266] 2020 Deep Learning Classification 10 Melo Filho, Angeli, Ophem, et al. [267] 2020 ANN Inverse Design 11 Chen, Lu, Karniadakis, et al. [268] 2020 Deep Learning Inverse Design 12 Kollmann, Abueidda, Koric, et al. [269] 2020 Deep Learning Optimization Framework 13 Qu, Zhu, Shen, et al. [270] 2020 ANN Optimization Framework 14 Lai, Amirkulova, and Gerstoft [271] 2021 CNN, GAN Inverse Design 15 Gurbuz, Kronowetter, Dietz, et al. [272] 2021 GAN Inverse Design 16 Amirkulova, Tran, and Khatami [273] 2021 Deep Learning Inverse Design 17 Wu, Liu, Jahanshahi, et al. [274] 2021 ANN Inverse Design 18 Shah, Zhuo, Lai, et al. [275] 2021 RL Optimization Framework 19 Tran, Amirkulova, and Khatami [276] 2022 ANN Inverse Design 20 Wiest, Seepersad, and Haberman [277] 2022 GNN Inverse Design 21 Amirkulova, Zhou, Abbas, et al. [278] 2022 Deep Learning Inverse Design 22 Tran, Khatami, and Amirkulova [279] 2022 CNN Inverse Design 23 Li, Chen, Li, et al. [280] 2023 CNN Inverse Design 24 Li, Chen, Li, et al. [281] 2023 Deep Learning Inverse Design 25 Wang, Chen, Xu, et al. [282] 2023 ANN Inverse Design Application field: Electromagnetics 26 Jiang, Xiao, Liu, et al. [283] 2010 ANN and scaled conjugate Surrogate model gradient 27 Freitas, Rêgo, and Vasconcelos [140] 2011 ANN Surrogate model 28 Vasconcelos, Rêgo, and Cruz [139] 2012 ANN Surrogate model 29 Sarmah, Sarma, and Baruah [172] 2015 ANN optimization framework 30 Saha and Maity [138] 2016 ANN Surrogate model 31 Nanda, Sahu, and Mishra [108] 2019 ANN Inverse design 32 An, Fowler, Shalaginov, et al. [144] 2019 ANN Surrogate model 33 Yuze, Hai, and Qinglin [157] 2019 CNN Classification and clustering 34 Liu, Zhang, and Cui [156] 2019 CNN optimization framework 35 Hodge, Mishra, and Zaghloul [284] 2019 DC-GAN Inverse design 36 Hodge, Mishra, and Zaghloul [169] 2019 DC-GAN Inverse design 37 Hodge, Vijay Mishra, and Zaghloul [170] 2019 DC-GAN Inverse design 38 Kudyshev, Kildishev, Shalaev, et al. [166] 2019 DC-GAN Topology optimization 39 Fang and Zhan [285] 2020 ANN (Deep physical Surrogate model informed neural network) 40 Khatib, Ren, Malof, et al. [286] 2021 ANN(Deep Learning) Inverse design 42 Zhu, Qiu, Wang, et al. [287] 2021 Transfer Learning Inverse design 43 Xu, Luo, Luo, et al. [288] 2021 Transfer Learning Inverse design 44 Zhang, Qian, Fan, et al. [289] 2022 Transfer Learning Inverse design 45 Trinh, Guilleminot, Perrot, et al. [290] 2022 Transfer Learning Inverse design 46 Aybike and K ˙IL˙IMC ˙I [291] 2021 CNN Inverse design 47 Zhou, Qiu, Zhang, et al. [292] 2022 ANN Optimization framework 48 Zhou, Xiao, and Wang [293] 2022 Deep Learning Inverse design 49 Fantoni, Bacigalupo, Gnecco, et al. [294] 2023 Deep Learning Inverse design 50 Khatib, Ren, Malof, et al. [295] 2023 Deep Learning Optimization framework Application field: Mechanical 35 APREPRINT - SEPTEMBER 30, 2023 Continuation of Table 1 Publication Year Method(s) Application of AI 51 Gu, Chen, Richmond, et al. [134] 2018 CNN Surrogate model 52 Kumar, Tan, Zheng, et al. [146] 2020 ANN Inverse design 53 Ma, Zhang, Luce, et al. [105] 2020 ANN Inverse design 54 Xue, Beatson, Chiaramonte, et al. [136] 2020 ANN Surrogate model 55 Wilt, Yang, and Gu [107] 2020 CNN Inverse design 56 Kollmann, Abueidda, Koric, et al. [173] 2020 CNN Topology optimization 57 Fernández, Jamshidian, Böhlke, et al. [137] 2021 ANN Surrogate model 58 Whalen and Mueller [245] 2021 Transfer Learning Inverse design 59 Wang, Zhuang, Xian, et al. [296] 2021 Transfer Learning surrogate Modeling 60 Shah, Zhuo, Lai, et al. [297] 2021 Reinforcement Learning Inverse design 61 Kim, Kim, Yang, et al. [298] 2022 Transfer Learning Optimization framework 62 Wang, Zhuang, Xian, et al. [299] 2022 Transfer Learning surrogate Modeling 63 Wang, Zeng, Wang, et al. [300] 2022 Genetic Algorithms Inverse design 64 Mastrigt, Dijkstra, Hecke, et al. [301] 2022 CNN Classification 65 Mastrigt, Dijkstra, Hecke, et al. [301] 2022 ANN(Deep Learning) Rational design 66 Zeng, Zhao, Lei, et al. [302] 2023 ANN(Deep Learning) Inverse design 67 Zeng, Zhao, Lei, et al. [303] 2022 ANN(Deep Learning) Inverse design Application field: Optics 68 Ma, Cheng, and Liu [149] 2018 ANN Inverse design 69 Magnusson, Mueller, Juhl, et al. [304] 2018 ANN Surrogate model 70 Liu, Zhu, Rodrigues, et al. [305] 2018 DC-GAN Inverse design 71 He, He, Zheng, et al. [158] 2019 ANN Inverse design 72 An, Fowler, Zheng, et al. [151] 2019 ANN Inverse design 73 So, Mun, and Rho [150] 2019 ANN Inverse design 74 Chen, Zhu, Xie, et al. [109] 2019 ANN Inverse design 75 Akashi, Toma, and Kajikawa [306] 2019 ANN Surrogate model 76 Ma, Cheng, Xu, et al. [164] 2019 GAN Inverse design 77 Harper, Weber, and Mills [307] 2019 Pseudo autoencoder Inverse design 78 Ashalley, Acheampong, Besteir, et al. [148] 2020 ANN Inverse design 79 Hou, Tang, Shen, et al. [308] 2020 ANN Inverse design 80 Akashi, Toma, and Kajikawa [152] 2020 ANN Inverse design 81 Phan, Nguyen, Linh, et al. [153] 2020 ANN Inverse design 82 Tao, You, Zhang, et al. [145] 2020 ANN Surrogate model 83 Chen, Lu, Karniadakis, et al. [154] 2020 physics-informed neural Inverse design networks (PINNs) 84 Harper, Coyle, Vernon, et al. [110] 2020 Pseudo autoencoder Inverse design 85 Liu, Raju, Zhu, et al. [309] 2020 V AE Inverse design 86 Sa ˘gık, Karaaslan, Ünal, et al. [311] 2021 ANN Inverse design 87 Ma, Liu, Kudyshev, et al. [312] 2021 Deep Learning Inverse design 88 Zheng, Zhang, Chen, et al. [311] 2022 ANN Optimization framework 89 Trinh, Guilleminot, Perrot, et al. [290] 2022 Transfer Learning Inverse design 90 Zhang, Qian, Fan, et al. [313] 2022 Transfer Learning Inverse design 91 Fan, Qian, Jia, et al. [314] 2022 Transfer Learning Inverse design 92 Badloe, Lee, and Rho [315] 2022 ANN Inverse design 93 Zhou, Qiu, Zhang, et al. [316] 2022 ANN Optimization framework 94 Chen, Hu, Zhu, et al. [317] 2023 ANN Inverse design 95 Liu and Yu [318] 2023 Deep Learning Inverse design 96 Zhou, Qiu, Zhang, et al. [316] 2023 Deep Learning Inverse design End of Table 1 Table 2: Table summarizing papers that use ML (except ANN) in metamaterials design. Publication Year Method(s) Application of AI Application field: Acoustics 36 APREPRINT - SEPTEMBER 30, 2023 Continuation of Table 2 Publication Year Method(s) Application of AI 97 Bacigalupo, Gnecco, Lepidi, et al. [127] 2016 Nonlinear programming optimization framework 98 Amirkulova and Norris [103] 2018 Gradient-descent optimization framework 99 Ghattas, Chen, and Villa [319] 2019 Bayesian Optimization Optimization framework 100 Luo, Li, Li, et al. [320] 2020 Bayesian Optimization Fuzzy Design 101 Aybike and K ˙IL˙IMC ˙I [321] 2021 SVM,naive Bayes, decision trees Classification 102 Tran, Amirkulova, and Khatami [322] 2022 Bayesian Optimization, Gussian Process (GP) optimization framework 103 Jin, Zeng, Wen, et al. [323] 2022 Gradient-descent Inverse Design 104 Chen, Chen, Xu, et al. [324] 2022 Gradient-descent Inverse Design 105 Chang, Wang, and Dong [325] 2022 Bayesian Optimization Optimization framework 106 Hu, Wang, Du, et al. [326] 2023 Bayesian Optimization Optimization framework Application field: Electromagnetics 107 Sakurai, Yada, Simomura, et al. [327] 2019 Bayesian optimization optimization framework 108 Pita Ruiz, Amad, Gabrielli, et al. [101] 2019 Gradient-descent and Opological optimization framework -derivative-based optimization 109 Kurniawati, Putri, and Ningsih [123] 2020 Random forest regression optimization framework 110 Zhang, Wang, Xu, et al. [328] 2022 Bayesian Optimization Optimization framework 111 Chuma and Rasmussen [329] 2022 KNN,SVM,Bayesian Optimization Optimization framework 112 Jian, Alexandropoulos, Basar, et al. [330] 2022 KNN,SVM,Bayesian Optimization Optimization framework 113 Alharbi, Abdelhamid, Ibrahim, et al. [331] 2023 KNN,SVM,Gradient-based Optimization Optimization framework 114 Lin, Zheng, Hu, et al. [332] 2023 Bayesian Optimization Optimization framework Application field: Mechanical 115 Morris and Seepersad [333] 2018 Spectral clustering Classification and clustering 116 Bessa, Glowacki, and Houlder [130] 2019 Bayesian ML Classification and clustering 117 Singleton, Cheer, and Daley [100] 2019 Gradient-descent optimization framework 118 Dong, Chen, Zeng, et al. [129] 2019 Nelder-Mead optimization optimization framework 119 Stern, Arinze, Perez, et al. [334] 2020 Nonlinear programming optimization framework 120 Liu, Ye, Silva Izquierdo, et al. [335] 2022 SVM classification 121 Prasanna, Shantha, Pradeep, et al. [336] 2022 SVM ,KNN Optimization Framework 122 Zhai and Yeo [337] 2023 Bayesian Learning inverse design 123 Hu, Wang, Du, et al. [338] 2023 Bayesian Optimization inverse design 124 Hu, Zhan, Wang, et al. [339] 2023 LS-SVM Optimization Framework Application field: Optics 125 Brunton, Kutz, Fu, et al. [340] 2015 Sparse representation optimization framework for classification 126 Wormser, Wein, Stingl, et al. [102] 2017 Gradient-descent optimization framework 127 Mansouree and Arbabi [99] 2019 Gradient-descent optimization framework 128 Xu, Grinberg, Melati, et al. [341] 2019 PCA Classification and clustering 129 Chuma and Rasmussen [342] 2022 SVM,KNN,Naive bayes gaussian Classification 130 Hu, Li, Chen, et al. [343] 2022 LS-SVM Optimization Framework 131 Zhang, Wang, Xu, et al. [344] 2022 Bayesian Optimization Optimization Framework 132 Ahmadpour, Yetisen, and Tasoglu [345] 2023 Bayesian Optimization Optimization Framework 133 Khaleel and Mudhafer [346] 2023 Navie Bayes Classification End of Table 2 Table 3: Table summarzing papers that use nature-inspired AI in metamaterials design. Publication Year Method(s) Application of AI Application field: Acoustics 134 Hedayatrasa, Kersemans, Abhary, et al. [347] 2018 GA Topology optimization 135 Yeh and Harne [348] 2019 GA optimization framework 136 Meng, Chronopoulos, and Fabro [349] 2019 GA optimization framework 137 Yuan, Chen, Tan, et al. [350] 2019 GA optimization framework 37 APREPRINT - SEPTEMBER 30, 2023 Continuation of Table 3 Publication Year Method(s) Application of AI 138 Liu, Cai, and Wang [351] 2020 GA Inverse design 139 Pomot, Payan, Remillieux, et al. [184] 2020 GA optimization framework 140 Xie, Wang, and Mei [352] 2020 GA optimization framework 141 Wu, Hu, Wang, et al. [247] 2020 Evolutionary Algorithm Optimization Framework 142 Wang and Liu a82 2021 GA Optimization Framework 143 Wang and Liu [354] 2021 GA Optimization Framework 144 Qiu and Jin [355] 2021 GA Optimization Framework 145 Gao et al. [356] 2021 GA Inverse Design 146 Li, Luo, Zhang, et al. [357] 2021 GA Topology Optimization 147 Vazquez Torre, Brunskog, Cutanda Henriquez, et al. [358] 2021 GA Topology Optimization 148 Wu, Hu, Wang, et al. [353] 2022 GA Parametric Optimization 149 Dos Reis and Karathanasopoulos [359] 2022 GA Inverse Design 150 Sun, Zhang, Guo, et al. [360] 2022 GA Topology Optimization 151 Panahi, Hosseinkhani, Frangi, et al. [361] 2022 GA Optimization Framework 152 Long, Zhu, Gu, et al. [362] 2022 GA Inverse Design 153 Jian, Tang, Hu, et al. [363] 2022 Adaptive GA Optimization Framework 154 Morris, Wang, Plaisted, et al. [31] 2023 GA Optimization Framework 155 Zhang, Li, Wang, et al. [364] 2023 GA Optimization Framework 156 Xiong, Lee, and Qin [365] 2023 GA Optimization Framework 157 Steklain, Adames, and Ganacim [366] 2023 GA Optimization Framework 158 Li, Sun, Wu, et al. [367] 2023 GA Inverse Design Application field: Electromagnetics 159 Li, Chen, Zeng, et al. [368] 2017 Adaptive GA optimization framework 160 Zhang and Cuii [369] 2017 BPSO optimization framework 161 Ahmed, Chandra, and Al-Behadili [370] 2017 GA Inverse design 162 Han, Cao, Gao, et al. [371] 2017 GA optimization framework 163 Pfeiffer and Tomasic [372] 2017 GA optimization framework 164 Pelluri and Appasani [373] 2017 GA optimization framework 165 Feng, Chen, and Huang [374] 2017 GA optimization framework 166 Allen, Dykes, Reid, et al. [375] 2017 GA optimization framework 167 Ding, Zhang, Zhang, et al. [376] 2017 GA optimization framework 168 Mahdi and Taha [377] 2017 GA Topology optimization 169 Su, Lu, and Li [378] 2017 PSO optimization framework 170 Orlandi [379] 2018 Differential evolution optimization framework (DE) algorithm 171 Ba ˘gmancı, Karaaslan, Altınta¸ s, et al. [380] 2018 GA optimization framework 172 Lim, Song, Kim, et al. [381] 2018 GA optimization framework 173 Corrêa, Resende, Bicalho, et al. [382] 2018 GA optimization framework 174 Kumar, Behera, and Suraj [383] 2018 GA optimization framework 175 Clemens, Iskander, Yun, et al. [189] 2018 Hybrid genetic programming optimization framework 176 Soltani, Soltani, and Aguili [384] 2019 GA Inverse design 177 Ibili, Karaosmanoglu, and Ergul [385] 2019 GA optimization framework 178 Seshadri and Gupta [386] 2019 GA optimization framework 179 Nanda, De, Sahu, et al. [387] 2019 GA optimization framework 180 Assal, Benzerga, Sharaiha, et al. [388] 2019 GA optimization framework 181 Karatzidis, Kantartzis, Pyrialakos, et al. [389] 2019 GA optimization framework 182 Tian and Li [390] 2019 GA optimization framework 183 Yuan, Ma, Sui, et al. [391] 2019 GA Topology optimization 184 Yanzhang and Jinghao [392] 2019 GA Topology optimization 185 Sui, Ma, Chang, et al. [393] 2019 IAGA optimization framework 186 Steckiewicz and Choroszucho [394] 2019 PSO optimization framework 187 Hao, Du, and Zhang [395] 2019 PSO-FSP optimization framework 188 Gupta, Saxena, and Bhatia [396] 2020 BFPSO optimization framework 189 Xie, Li, and Dong [397] 2020 GA Inverse design 190 Abdulkarim, Deng, Luo, et al. [398] 2020 GA optimization framework 191 Hameed, Shawkat, and Al-Badri [399] 2020 GA optimization framework 192 Borzooei, Rezagholizadeh, and Biabanifard [400] 2020 GA optimization framework 193 Abdulkarim, Deng, Karaaslan, et al. [401] 2020 GA optimization framework 194 Duan, Zhang, Wang, et al. [179] 2020 GA optimization framework 195 Allen, Dykes, Reid, et al. [402] 2020 GA optimization framework 38 APREPRINT - SEPTEMBER 30, 2023 Continuation of Table 3 Publication Year Method(s) Application of AI 196 Zhu, Wang, Sun, et al. [403] 2020 GA optimization framework 197 Zhu, Wang, Sui, et al. [404] 2020 GA optimization framework 198 Tung, Ha-Van, and Seo [405] 2020 GA optimization framework 199 Suraj, Behera, and Badhai [406] 2020 GA optimization framework 200 Jiang, Li, Li, et al. [407] 2020 GA optimization framework 201 Whiting, Kang, Campbell, et al. [408] 2020 GA optimization framework 202 Cui, Xu, Yu, et al. [409] 2020 GA optimization framework 203 Thomes, Mosquera-Sánchez, and De Marqui [410] 2020 GA optimization framework 204 Nguyen, Bui Bach, Bui, et al. [411] 2020 genetic programming (GP) optimization framework 205 Li and Yang [412] 2020 NSCGA optimization framework 206 Li and Yang [413] 2020 NSCGA optimization framework 207 Yang, Huang, Song, et al. [414] 2020 PSO optimization framework 208 Papathanasopoulos and Rahmat-Samii [415] 2020 PSO optimization framework 209 Diaz, Burckel, Adomanis, et al. [416] 2021 GA optimization framework 210 Ahmadi, Vaezi, Harzand, et al. [417] 2021 GA optimization framework 211 Gonçalves, Mesquita, and Silva [418] 2021 GA optimization framework 212 Mayer, Bi, Griesse-Nascimento, et al. [419] 2022 GA Optimization framework 213 Zong, Zhu, Yu, et al. [420] 2022 GA Inverse Design 214 Liu, Wang, Olivier, et al. [421] 2022 GA Inverse Design 215 Luo, Lan, Nong, et al. [422] 2022 GA Inverse Design 216 Nguyen and Seo [423] 2022 GA Inverse Design 217 Lu, Gao, and Dai [424] 2022 GA Inverse Design 218 Sun, Jiang, and Wang [425] 2023 GA Optimization Framework 219 Feng, Zhang, Cheng, et al. [426] 2023 GA Optimization Framework 220 Zhang, Duan, Liu, et al. [427] 2023 GA Optimization Framework Application field: Mechanical 221 Kim, Yang, Hwang, et al. [428] 2017 GA optimization framework 222 Abdeljaber, Avci, Kiranyaz, et al. [429] 2017 GA optimization framework 223 Palermo, Vitali, and Marzani [190] 2018 Augmented Lagrangian optimization framework Genetic Algorithm (ALGA) 224 Callanan, Ogunbodede, Dhameliya, et al. [430] 2018 GA Inverse design 225 Bakır, Dalgaç, Ünal, et al. [431] 2019 GA optimization framework 226 Wang, Sun, Li, et al. [432] 2019 GA optimization framework 227 Meng, Chronopoulos, Fabro, et al. [433] 2020 GA optimization framework 228 Chen, Moughames, Ji, et al. [434] 2020 GA optimization framework 229 Qiu, Wang, Xie, et al. [435] 2020 GA optimization framework 230 Ghachi, Alnahhal, Abdeljaber, et al. [436] 2020 GA optimization framework 231 Królikowski, Blazejewski, and Knitter [437] 2020 GA optimization framework 232 Yang, Yang, and Lo [438] 2020 GA optimization framework 233 Bacigalupo, Gnecco, Lepidi, et al. [439] 2021 Adaptive optimization Surrogate model algorithm 234 Wang, Callanan, Ogunbodede, et al. [440] 2021 GA Inverse design 235 Hashemi, McCrary, Kraus, et al. [87] 2021 GA optimization framework 236 Li, Luo, Zhang, et al. [441] 2021 GA Topology optimization 237 Wang and Liu [84] 2021 multi island GA optimization framework 238 Chen, Yan, Feng, et al. [442] 2021 PSO optimization framework 239 Wang, Zeng, Wang, et al. [443] 2022 GA Inverse Design 240 Dos Reis and Karathanasopoulos [444] 2022 GA Inverse Design 241 Dong, Hu, Holmes, et al. [445] 2022 GA Optimization Framework 242 Dong and Wang [248] 2022 GA Optimization Framework 243 Panahi, Hosseinkhani, Frangi, et al. [446] 2022 GA Optimization Framework 244 Liu and Acar [447] 2023 GA Optimization Framework 245 Zeng, Duan, Zhao, et al. [448] 2023 GA inverse Design 246 Cerniauskas and Alam [10] 2023 GA Optimization Framework 247 Cerniauskas and Alam [182] 2023 GA Optimization Framework 248 Cerniauskas and Alam [183] 2023 GA Optimization Framework Application field: Optics 249 Sugino, Ishikawa, Hayashi, et al. [449] 2017 GA optimization framework 39 APREPRINT - SEPTEMBER 30, 2023 Continuation of Table 3 Publication Year Method(s) Application of AI 250 Sun, Cai, and Wang [450] 2017 PSO optimization framework 251 Huang, Pu, Zhao, et al. [451] 2018 GA optimization framework 252 Karaaslan, Ba ˘gmancı, Ünal, et al. [452] 2018 GA optimization framework 253 Bor, Babayigit, Kurt, et al. [453] 2018 GA optimization framework 254 Mayer and Lobet [454] 2018 GA optimization framework 255 Dubaji ´c, Dani ˇci´c, Vukovi ´c,et al. [455] 2018 GA optimization framework 256 Franco Rêgo, Gomes De Souza, 2018 GA optimization framework and Rodriguez-Esquerr [456] 257 Thompson and Pisano [188] 2018 micro Genetic Algorithm Inverse design 258 Ren, Liu, Dong, et al. [457] 2018 PSO optimization framework 259 Parvez, Rao, and Zanjani [458] 2019 GA Inverse design 260 Blechman, Almeida, Sain, et al. [459] 2019 GA optimization framework 261 Zanjani, Chaharmahali, Biabanifard, et al. [460] 2019 GA optimization framework 262 Mayer and Lobet [461] 2019 GA optimization framework 263 Ma and Zhang [462] 2019 GA optimization framework 264 Yeh and Harne [463] 2019 GA optimization framework 265 Li, Stan, Czaplewski, et al. [187] 2019 micro Genetic Algorithm optimization framework 266 Li, Bao, Jiang, et al. [464] 2019 PSO Inverse design 267 Chaharmahali, Soltani, Hoseini, et al. [465] 2020 GA Inverse design Shuai, Zhang, Xu, et al. [466] 2020 GA Inverse design 268 Jiang and Yi [467] 2020 GA Inverse design 269 Lari, Vafapour, and Ghahraloud [468] 2020 GA optimization framework 270 Cai, Sun, Wang, et al. [469] 2020 GA optimization framework 271 Liu, Maier, and Li [470] 2020 GA optimization framework 272 Melo, Ribeiro, Benevides, et al. [471] 2020 GA optimization framework 273 Li, Liu, Long, et al. [472] 2020 GA optimization framework 274 DIaz, Burckel, Adomanis, et al. [473] 2020 GA optimization framework 275 Mayer, Griesse-Nascimento, Bi, et al. [474] 2020 GA optimization framework 276 Briones, Carrillo, and Ruiz-Cruz [475] 2020 PSO optimization framework 277 Vafapour, Ghahraloud, Keshavarz, et al. [476] 2021 GA optimization framework 278 Jin, Yang, Xu, et al. [477] 2021 PSO optimization framework 279 Dadkhahfard [478] 2021 PSO optimization framework 280 Hong, Son, Kim, et al. [479] 2021 PSO optimization framework 281 Luo, Lan, Nong, et al. [480] 2022 GA Inverse Design 282 Dong, Hu, Holmes, et al. [481] 2022 GA optimization framework 283 Mayer, Bi, Griesse-Nascimento, et al. [482] 2022 GA optimization framework 284 Sahraeian and Emami [483] 2022 GA optimization framework 285 Zhang, Zhang, Zhang, et al. [484] 2022 GA Inverse Design 286 Zong, Zhu, Yu, et al. [485] 2023 GA optimization framework 287 Morris, Wang, Plaisted, et al. [486] 2023 GA optimization framework 288 Morris, Wang, Plaisted, et al. [487] 2023 GA optimization framework End of Table 3 Table 4: Table summarizing papers that use multiple AI techniques in metamateri- als design. Publication Year Method(s) Application of AI Application field: Acoustics 289 Bacigalupo, Gnecco, Lepidi, et al. [131] 2019 ANN and Quasi-Monte optimization framework Carlo sequences 290 Wu, Liu, Jahanshahi, et al. [5] 2021 ANN and Reinforced Surrogate model learning algorithm 291 Wu, Liu, Jahanshahi, et al. [488] 2021 ANN surrogate Model 292 Garland, White, Jensen, et al. [489] 2021 CNN, Genetic Algorithm Optimization framework 293 Jin, He, Wen, et al. [490] 2022 ANN Inverse Design 294 Kennedy, Lim, et al. [237] 2022 Deep Learning Optimization Framework 295 Li, Guo, Sun, et al. [491] 2022 Genetic Algorithm Topology Optimization 296 Liu, Xie, Huang, et al. [492] 2022 Deep Learning 3D-Printing 40 APREPRINT - SEPTEMBER 30, 2023 Continuation of Table 4 Publication Year Method(s) Application of AI 297 130 130 2023 Gradient-based Method Optimization Framework 298 Wang, Chen, Xu, et al. [493] 2023 ANN Inverse Design Application field: Electromagnetics 299 Pandeeswari, Raghavan, Krishnan, et al. [106] 2012 ANN and Scaled Surrogate model conjugate gradient 300 Luna, Vasconcelos, and Cruz [86] 2013 GA and ANN optimization framework 301 Orlandi [494] 2018 GA and ANN optimization framework 302 Zhang, Liu, Wan, et al. [199] 2019 CNN and binary PSO Surrogate model 303 Boddeti, Alabassi, Aggarwal, et al. [112] 2019 GA and Gradient-descent optimization framework 304 Mercy Kingsta and Seyatha [495] 2019 GA, pattern search optimization framework 305 Song, Zhao, and Tang [201] 2019 PSO and ANN optimization framework 306 Harper and Mills [175] 2020 ANN and Bayesian optimization framework optimization 307 Zhu, Qiu, Wang, et al. [186] 2020 GA and ANN Inverse design 308 Prabhakar, Babu, Adinarayana, et al. [496] 2020 GHGWO, PSO, BFO optimization framework 309 Bakale, Nandgaonkar, and Deosarkar [497] 2020 PSO, GA optimization framework 310 Calik, Belen, Mahouti, et al. [174] 2021 ANN and Bayesian optimization framework optimization 311 Yu and Liu [498] 2021 GA with load-resistors optimization framework matrix decomposition 312 Hamdan [499] 2023 K-Mean clustering Inverse Design 313 Hamdan [500] 2023 K-Mean clustering, ANN Design Space Reduction 314 Mahouti, Belen, Tari, et al. [501] 2023 Bayesian Optimization, ANN Inverse Design Application field: Mechanical 315 Chen, Skouras, Zhu, et al. [122] 2018 Gaussian mixture model Classification and clustering 316 White, Arrighi, Kudo, et al. [97] 2019 ANN and Gradient-based Topology optimization nonlinear programming 317 Bostanabad, Chan, Wang, et al. [117] 2019 Globally approximate Inverse design Gaussian process 318 Bostanabad, Chan, Wang, et al. [118] 2019 Globally approximate Inverse design Gaussian process 319 Singleton, Cheer, and Daley [502] 2019 PSO, GA, HGA optimization framework 320 Chen and Gu [95] 2020 ANN and Gradient-descent Inverse design 321 Wu, Liu, Wang, et al. [185] 2020 GA and CNN optimization framework 322 Glodež, Klemenc, Zupani ˇc,et al. [503] 2020 GA, DASA Inverse design 323 Wang, Chan, Liu, et al. [116] 2020 k-means and ShapeDNA Inverse design 324 Dong, Qin, and Xiao [128] 2020 Nelder-Mead, GA Surrogate model and ANN 325 Bonfanti, Guerra, Font-Clos, et al. [132] 2020 Reinforced annealing, optimization framework Monte Carlo and CNN 326 Wang, Chan, Ahmed, et al. [111] 2020 V AE, k-means and PCA Inverse design 327 Garland, White, Jensen, et al. [40] 2021 GA and ANN optimization framework 328 Ji, Chen, Liang, et al. [504] 2021 Optimal Latin hypercube optimization framework technique and GA 329 McMillan, Öztürk, and Acar [505] 2022 PCA Inverse Design 330 Indurkar, Karlapati, Shaikeea, et al. [506] 2022 GNN Optimization Framework 331 Zhao, Zhang, Zhang, et al. [30] 2022 Genetic Programming Optimization Framework 332 Mustapha, Alhiyafi, Shafi, et al. [507] 2023 SVM,ANN Inverse Design 333 Zhang, Qin, Shen, et al. [508] 2023 Conditional GAN Optimization Framework Application field: Optics 334 Comin and Hartschuh [29] 2018 GA and ANN optimization framework 335 Backer [98] 2019 ANN and Steepest-descent Inverse design 336 Liu, Zhang, Dan, et al. [124] 2019 ANN, DecisionTree, ExtraTree, Inverse design KNN and Random forest 337 Mukherjee, Bhattacharjee, Halder, et al. [126] 2019 ANN, Weighted KNN, Classification and clustering Complex tree, Linear SVM 41 APREPRINT - SEPTEMBER 30, 2023 Continuation of Table 4 Publication Year Method(s) Application of AI 338 Kiarashinejad, Abdollahramezani, 2019 Pseudo autoencoder, Classification and clustering 339 Zandehshahvar, et al. [120] autoencoder and PCA 340 Liu, Zhu, Lee, et al. [3] 2020 ANN and Cooperative Inverse design convolution algorithm 341 Moon, Choi, Lee, et al. [121] 2020 ANN, t-Stochastic Neighbor Classification and clustering Embedding AND k-means 342 Tao, Zhang, You, et al. [176] 2020 GA and ANN Surrogate model 343 Liu, Zhang, Dai, et al. [509] 2020 GA, PSO, multi-traversal optimization framework direct-binary search and simulated annealing 344 Du, You, Zhang, et al. [125] 2021 ANN and Forest regression Surrogate model and Support vector regression 345 Li, Chen, Xiong, et al. [28] 2022 MLP,CNN,RNN Optimization Framework 346 Li, Deng, Liu, et al. [510] 2022 Autoencoder, GAN Optimization Framework 347 Al-Ashwal, Al Soufy, Hamza, et al. [511] 2023 DNN,CNN,GAN Optimization Framework End of Table 4 42 APREPRINT - SEPTEMBER 30, 2023