![| Forward prediction ANNs for PnCs and EMs. a | Feature parameters and a testing result of the ANN, where the solid line is the experimental result, the dashed line is predicted by the ANN, and the dotted line is calculated by numerical simulations [83]. b | Some examples of samples, and the comparison of predicted (simulated) sound absorption coefficients (red dots) by the ANN and the ones measured (blue dots) [84]. c | The constructed CNN and MLP, as well as](https://www.researchgate.net/profile/Chen-Xu-Liu-2/publication/368331393/figure/fig1/AS:11431281118624004@1675817732357/Forward-prediction-ANNs-for-PnCs-and-EMs-a-Feature-parameters-and-a-testing-result_Q320.jpg)
![| Parameter design ANNs for PnCs and EMs. a | A MLP used to design 2D EMs according to the input band characteristics [96]. b | The forward prediction MLP, where the inputs are feature parameters and the outputs are lower and upper limits of bandgaps, and the design process by combining genetic algorithm and MLP [94]. c | 1D PnC simulated by multiple trained ANNs [101]. d | Probability-density-based ANN and its "one-to-may" designs [102]. e | A TNN composed of an inverse network and a pretrained forward network. f | An advanced TNN [93] where the encoder](https://www.researchgate.net/profile/Chen-Xu-Liu-2/publication/368331393/figure/fig2/AS:11431281118624005@1675817732436/Parameter-design-ANNs-for-PnCs-and-EMs-a-A-MLP-used-to-design-2D-EMs-according-to_Q320.jpg)
![| Topology design ANNs for PnCs and EMs. a | The AE-based model by Li et al. [106] where topological feature indicates compressed low-dimensional vector. b | The VAE-based model by Liu et al. [109] where the pretrained generative model is the trained VAE's decoder. c | The CGAN by Gurbuz et al. [113] where "TL" (conditions) is the transmission loss of the 2D EMs and "Z" is the input random variables. d | The CGAN by Jiang et al. [87] where the target refers to dispersion curves of 2D PnCs.](https://www.researchgate.net/profile/Chen-Xu-Liu-2/publication/368331393/figure/fig3/AS:11431281118607223@1675817732522/Topology-design-ANNs-for-PnCs-and-EMs-a-The-AE-based-model-by-Li-et-al-106-where_Q320.jpg)
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Journal of Computational Design and Engineering, 2023, 10, 602–614
DOI: 10.1093/jcde/qwad013
Advance access publication date: 6 February 2023
Review article
Deep learning for the design of phononic crystals and
elastic metamaterials
Chen-Xu Liu and Gui-Lan Yu *
School of Civil Engineering, Beijing Jiaotong University, 100044 Beijing, China
∗Corresponding author. E-mail: glyu@bjtu.edu.cn
Abstract
The computer revolution coming by way of data provides an innovative approach for the design of phononic crystals (PnCs) and elastic
metamaterials (EMs). By establishing an analytical surrogate model for PnCs/EMs, deep learning based on articial neural networks
possesses the superiorities of rapidity and accuracy in design, making up for the shortcomings of traditional design methods. Here,
the recent progresses on deep learning for forward prediction, parameter design, and topology design of PnCs and EMs are reviewed.
The challenges and perspectives in this emerging eld are also commented.
Keywords: deep learning, phononic crystal, elastic metamaterial, inverse design, articial neural network
1. Introduction
Phononic crystal (PnC) is a class of articial periodic structures for
elastic waves, which has important properties, such as bandgap
(Mohammadi et al.,2007) and band edge state (Qiu & Liu, 2006).
This concept was rst proposed by Kushwaha et al. (1993)by
analogy with photonic crystals (Yablonovitch, 1987). Many exper-
iments demonstrated the existence of PnCs’ bandgaps (Martínez-
Sala et al.,1995; Montero De Espinosa et al.,1998; Sánchez-Pérez et
al.,1998). In 2000, Liu et al. (2000) presented a novel periodic struc-
ture with elastic wave bandgaps in the deep subwavelength scale
on the mechanism of local resonance. And then, locally resonant
structures with negative refraction (Zhang & Liu, 2004), negative
elastic modulus (Li & Chan, 2004), and negative mass density (Mei
et al.,2006) were found one after the other in the eld of elas-
tic waves. These unconventional properties are not available in
natural materials and lead to the birth of the concept of elastic
metamaterials (EMs). With the deepening of related research, the
understanding of the connotation of PnCs and EMs is also devel-
oping. Some researchers believed that EMs are articial structures
showing some novel and counterintuitive effects with their fea-
ture sizes much smaller than wavelengths. Others extended the
concept of EMs to PnCs (Lu et al.,2009). In this review, the former
opinion is adopted.
As described above, PnCs and EMs possess some supernatural
physical properties and can effectively manipulate elastic waves.
They seek the attention of researchers in many elds, such as
acoustics (Cummer et al.,2016;Donget al.,2022;Liuet al.,2021),
mechanics (Hosseini & Zhang, 2021;Maet al.,2022;Wanget al.,
2021), and civil engineering (Yu & Miao, 2019; Zaccherini et al.,
2020;Zenget al.,2021,2022). Over the past three decades,we have
witnessed the rapid development of PnCs and EMs in theoretical
studies (Wang et al.,2020b;Zhouet al.,2009), experimental re-
searches (Brûlé et al.,2014;Liuet al.,2021; Ruan et al.,2021; Zhu
et al.,2012), and practical applications (Muhammad & Lim, 2022;
Wan g et al.,2020a; Zhu et al.,2015). Most of these researches ad-
dressed forward problems. However, they laid a solid foundation
for the inverse design of PnCs/EMs, which is a realistic challenge
and of great signicance in practical engineering.
There are three common approaches to designing structures or
materials, including Edisonian, theoretical analysis, and bioinspi-
ration. It is well known that trial-and-error in experiments and
simulations is central to the Edisonian approach and its nature
determines the inefciency of design (Zeng et al.,2020). The theo-
retical analysis is an important means to guide the design of PnCs
and EMs By building physics-based models, the inuence of dif-
ferent geometry and material parameters on properties can be
observed, and structural designs can be realized. However, this
method works in a limited space, and the design difculty will rise
with the increase of the quantity of design parameters. The bioin-
spiration approach benets from the real natural world,and a lot
of PnCs and EMs were designed in this way, such as hierarchical
PnCs (Zhang & To, 2013), spider web-inspired PnCs (Dal Poggetto
et al.,2021), and honeycomb EMs (Sui et al.,2015). They have good
performances, but the designs only meet limited demands.
With the rapid development of computer hardware, the com-
puting ability has been signicantly improved. Gradient-based al-
gorithms (such as level-set method or adjoint method) and evolu-
tionary approaches (e.g., genetic algorithm or particle swarm op-
timization) are widely used and gradually rened to solve the de-
sign problems of PnCs and EMs (Yi & Youn, 2016). The two meth-
ods can give satisfactory results and do not need too much human
intervention. Many researches proved the effectiveness and intel-
ligence of the two methods of designing PnCs and EMs (Dong et al.,
2014b; Liang & Du, 2020;Rong&Ye,2019; Vineyard & Gao, 2021).
However, for each design, both methods require many numerical
simulations, and a lot of time and computational efforts are nec-
essary (Kollmann et al.,2020).
Deep learning is a method solving problems rapidly and effec-
tively, where computational models consisting of multiple pro-
Received: October 17, 2022. Revised: January 30, 2023. Accepted: February 1, 2023
C
The Author(s) 2023. Published by Oxford University Press on behalf of the Society for Computational Design and Engineering. This is an Open Access article
distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse,
distribution, and reproduction in any medium, provided the original work is properly cited.
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Journal of Computational Design and Engineering, 2023, 10(2), 602–614 | 603
cessing layers are trained to learn representations of data with
various abstraction levels (Lecun et al.,2015). In the eld of com-
puter science, such as computer vision, natural language pro-
cessing, and decision-making, deep learning has obtained a large
number of excellent achievements (Nickel et al.,2015;Nodaet al.,
2015;Voet al.,2019; Voulodimos et al.,2018; Young et al.,2018).
Many researchers in other disciplines, such as physics, chemistry,
biology, medicine, and engineering, have been attracted succes-
sively and devoted to the interdisciplinary studies related to deep
learning (Angermueller et al.,2016;Gohet al.,2017; Jiang et al.,
2021;Khatibet al.,2021;Maet al.,2021;Nguyenet al.,2019;Wang
2021 年;Khatib等人。，2021 年;Ma等人。，2021 年;Nguyen等人。，2019 年;Wang
2021 年;Khatib等人。，2021 年;Ma等人。，2021 年;Nguyen等人。，2019 年;Wang
et al.,2019). Their researches demonstrated that deep learning
et al.，2019 年）。 他们的研究表明 ，深度学习
et al.，2019 年）。 他们的研究表明 ，深度学习
has the potential to overcome the shortcomings of traditional ap-
有可能 克服 传统 AP-
有可能 克服 传统 AP-
proaches, such as high labour costs, time consumption, and mas-
劳动力成本高、耗时 和 MAS 等问题
劳动力成本高、耗时 和 MAS 等问题
sive computing resources. As a data-driven method, a deep learn-
ing model can discover helpful information automatically and
simulate mapping relationships approximating the original laws
of materials/structures with restrictions. For PnCs and EMs, new
design methods based on deep learning have been explored and
studied over the past four years. However, except Jin et al. (2022a)
introduced the mechanisms of some main algorithms of machine
learning and the latest progress of structure design according to
the PnC and EM types, the relevant review literature has rarely
been reported. As an essential branch of machine learning, deep
learning plays a signicant role in the analysis and design of PnCs
and EMs. It is of practical signicance to review different deep
learning models in detail to illustrate further how deep learning
is combined with PnCs and EMs to solve problems.
In this article, we present a review of the recent studies on deep
learning for the design of PnCs and EMs which can manipulate
elastic waves in uid and/or solid media. First, we briey intro-
duce the progress in the design of PnCs and EMs as well as the his-
tory and development of deep learning. And then, relevant stud-
ies on deep learning for forward prediction, parameter design, and
topology design of PnCs and EMs are reviewed, where several ma-
jor deep learning models are discussed. Finally, we end up with
a set of comments on the challenges and perspectives in this in-
terdisciplinary eld, which possess the potential to provide more
possibilities for deep learning and PnCs/EMs.
2. The Design of PnCs and EMs
There are two main parts in the design of PnCs and EMs, which
are the forward design and the inverse design.The forward design
aims at discovering novel structural properties rather than spe-
cic designs. It relies on the experiences and theoretical knowl-
edge of designers, and a lot of outstanding achievements were ob-
tained. Liu et al. (2000) found a novel PnC/EM with subwavelength
bandgaps by theoretical and numerical analysis, of which the ba-
sic unit is a coated sphere embedded in the host material. Inspired
by the work of Liu et al. (2000), seismic metamaterials were pro-
posed to prevent buildings from waves induced by earthquakes
(Muhammad & Lim, 2022), which can isolate low-frequency vibra-
tions by small structure sizes. Based on the quantum hall effect in
condensed matter physics, Tian et al. (2020) designed a valley topo-
logical PnC to realize the lossless transmission of acoustic waves.
The forward design is mainly based on the forward analysis and
provides signicant references and foundations for the inverse
design. The inverse design focuses on specic designs and nd-
ing structures meeting explicit expectations. It is realized mainly
by the Edisonian approach, theoretical analysis, bioinspiration,
gradient-based algorithms, and evolutionary approaches, etc. The
Edisonian approach, namely the trial-and-error method, is based
on the results of forward analysis to conduct designs many times
对 正向分析 的结果 进行多次设计
对 正向分析 的结果 进行多次设计
articially until targets are accomplished, which is inefcient. If
人工直到目标完成，这是 无效的。 如果
人工直到目标完成，这是 无效的。 如果
the physical law between dependent and independent variables of
的因变量和自变量之间的 物理定律
的因变量和自变量之间的 物理定律
PnCs/EMs can be observed intuitively,the design can be guided by
PnCs/EMs 可以 直观地观察，设计 可以 通过以下方式进行指导
PnCs/EMs 可以 直观地观察，设计 可以 通过以下方式进行指导
analysis or numerical calculation. For example, Chen et al. (2022)
分析或数值计算。 例如，Chen 等人 （2022）
分析或数值计算。 例如，Chen 等人 （2022）
discussed the law of PnC bandgaps changing with elastic sup-
讨论了 PnC 带隙随弹性支撑变化 的定律
讨论了 PnC 带隙随弹性支撑变化 的定律
port constants, and designs meeting the expectations can be real-
端口常数和 满足 期望的设计可以是 真实的
端口常数和 满足 期望的设计可以是 真实的
ized. There are also some PnCs and EMs whose design inspiration
化。 还有 一些 PnC 和 EM 的设计 灵感
化。 还有 一些 PnC 和 EM 的设计 灵感
comes from creatures or structures in nature. Jin et al. (2022b)de-
来自 自然界中的生物或结构。 Jin et al. （2022b）de-
来自 自然界中的生物或结构。 Jin et al. （2022b）de-
signed a PnC inspired by honeycombs to block noise and vibration.
签署了一份受蜂窝启发的 PnC，以阻挡噪音和振动。
签署了一份受蜂窝启发的 PnC，以阻挡噪音和振动。
Referring to the characteristic that spider webs can compromise
指的是蜘蛛网可以妥协的 特性
指的是蜘蛛网可以妥协的 特性
between absorbing the impact of prey and effectively transmit-
在吸收猎物 的冲击和有效传递之间——
在吸收猎物 的冲击和有效传递之间——
ting information about the nature and location of the vibration
有关 振动性质和位置 的信息
有关 振动性质和位置 的信息
sources, Zhao et al. (2022) proposed a novel EM which increases
来源，Zhao 等人，2022 年）提出了一种新型 EM，它增加了
来源，Zhao 等人，2022 年）提出了一种新型 EM，它增加了
the relative density of periodic lattice and generates multiple lo-
cal resonant bandgaps. The above three design approaches need
a lot of human intervention and lack systematization. Gradient-
based algorithms and evolutionary approaches have been devel-
oped in recent years to improve the intelligence of the design of
PnCs and EMs. To maximize the bandgaps, Qiu et al. (2022)useda
joint topology optimization procedure to design two-dimensional
(2D) PnCs. Dong et al. (2022) combined genetic algorithm and nite
element method to customize an ultra-broadband EM which can
be used in noise control systems, medical ultrasonics, and con-
tactless particle control assembly. However, a lot of time and com-
puting resources are required since many numerical simulations
and iterations are needed. With the rapid development of articial
intelligence (AI) technology, new design methods based on deep
learning are expected to improve the intelligence and efciency
of PnC and EM design.
3. The History and Development of Deep
Learning
In history, deep learning has experienced three main develop-
ment waves (Goodfellow et al.,2016), and its name has also un-
dergone multiple changes. The rst was in the 1940–1960s when
deep learning was named cybernetics (Ashby, 1961). The second
was the time of connectionism in the 1980–1990s (Bechtel & Abra-
hamsen, 1991). The current resurgence is from 2006 to the present,
known as deep learning (Lecun et al.,2015). Generally, deep learn-
ing is a method based on articial neural network (ANN) which
has the ability to learn data representation. Its appearance can
be traced back to 1943, when an early model imitating the bi-
ological brain was presented (Mcculloch & Pitts, 1943). In 1958,
Rosenblatt created a perceptron model, which was a two-layer
computer learning network using simple addition and subtrac-
tion (Rosenblatt, 1958). In the 1980s, the back-propagation (BP) al-
gorithm was successfully used to train deep ANNs, which was a
milestone and effectively solved the exclusive-or problem (Rumel-
hart et al.,1986).Atthesametime,twocreativeANNs,recur-
rent neural network (Hopeld, 1982) and convolutional neural
network (CNN; LeCun, 1989), were invented and developed. They
have shown excellent performance in many elds so far (Dhillon
&Verma,2020;Yuet al.,2019). The popularity of deep learning
started around 2006, and the landmark event was that several
deeper ANNs were successfully trained (Bengio et al.,2006; Hinton
et al.,2006;Ranzatoet al.,2006). In the recent 10 years, more and
more outstanding algorithms and models have sprung up,such as
batch normalization (Ioffe & Szegedy, 2015), dropout (Srivastava
et al.,2014), Adam optimizer (Kingma & Ba, 2015), inception mod-
ule (Szegedy et al.,2015), AlexNet (Krizhevsky et al.,2012), ResNet
(He et al.,2016), DenseNet (Huang et al.,2017), variational autoen-
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604 | Deep learning for PnCs and EMs
Figure 1: MLP and its neurons.
coder (VAE; Kingma & Welling, 2014), and generative adversarial
network (GAN; Goodfellow et al.,2014).
For ease of understanding the deep learning framework, the
multilayer perceptron (MLP) as a simple and common deep learn-
ing model is introduced in this section. As shown in Fig. 1a, it
consists of multiple fully connected layers, including one input
layer, M-2 hidden layers, and one output layer. Each layer contains
many neurons, and a neuron is a mathematical expression, which
is usually written as
A(m)
n=g(m)w(m)
nTA(m−1)+b(m)
n=g(m)z(m)
n(1)
where A(m)
n,z(m)
n,w(m)
n,andb(m)
nare, respectively, the out-
put/activated value, pre-activated value, weight vector, and bias
of the nth neuron in the mth layer; A(m−1) is the output vector
from the (m−1)th layer, and it is the input data if mis equal to
1; g(m)(·) is the activation function in the mth layer, which is non-
linear and enables ANN to deal with complicated problems. Each
neuron is connected to all neurons in the next layer and sepa-
rated from other neurons in the same layer. One may notice the
MLP is essentially a complex function composed of many simple
functions. By adjusting weights and biases in all neurons, the MLP
has the ability to approximate an arbitrary function. For PnCs and
EMs, it is well known that their properties are unique if materials
and congurations are determinate, so there exists a relation from
materials and congurations to properties, which is an unknown
function. Hence, an MLP can be trained to simulate the mapping
relationship, and then predict the properties of PnCs and EMs ac-
cording to the input information. A brief description about the in-
puts and outputs of ANNs for PnCs/EMs is given in Table 1.
Before discussing the researches on deep learning for PnCs and
EMs, we briey introduce the main steps of using ANN to solve the
problems of PnCs and EMs. Firstly, it is necessary to build a dataset
by experiments or numerical simulations, which is supposed to
contain structural feature parameters (such as geometry param-
eters, material parameters, and topologies) and related properties
(such as bandgaps, dispersion curves, and frequency responses).
Secondly, a rational deep learning model or ANN must be con-
structed according to the characteristics of considered PnCs/EMs
and the problems to be solved. Finally, the training and testing of
the ANN with the dataset in a reasonable way is required.
4. Forward Prediction ANNs for PnCs and
EMs
Forward prediction by deep learning refers to using an ANN model
to predict the properties of PnCs/EMs according to the input fea-
ture parameters. As described before, ANN is a powerful analyti-
cal simulator which can approximate an arbitrary function. From
feature parameters to structural properties is a mapping, namely
a function. Unfortunately, for most complex problems, this func-
tion is unknown. Hence, we can train an ANN to simulate the un-
known function by data, and then the structural properties can be
predicted rapidly by the analytical ANN. The forward prediction of
PnCs/EMs is the foundation of the inverse design. Its realization
indirectly demonstrates the possibility of designing PnCs/EMs by
deep learning, and some design models are constructed on this
basis.
Back in 2006, Fuster-Garcia et al. (2006) built a simple ANN with
two layers to predict the frequency spectrums of 2D PnCs. In this
research, the dataset was obtained by experiments. As shown in
Fig. 2a, they considered ve feature parameters, including inci-
dence angle, cylinder radius, periodic constant, distance from the
measuring point to the PnCs’ centre, and row number. One hun-
dred fty-one samples were generated to form the dataset, where
145 of them were for training and the rest for testing. The results
showed that the predicted frequency spectrums were relatively
in good agreement with the testing samples. In 2021, Ciaburro
& Iannace (2021) constructed a dataset for EMs by experiments
and used it to train and test a three-layer ANN. They glued differ-
ent numbers of two types of buttons and polyvinyl chloride mem-
brane together to get the experimental data of the sound absorp-
tion coefcient (SAC), as shown in Fig. 2b. Fifteen types of samples
were made for experiments, and about 18000 sets of data were
obtained where each set included a frequency, a button weight,
a button number, a cavity thickness, and a SAC. They compared
the predicted (simulated) SACs by the ANN with the measured
SACs by experiments, and the general trends of the predicted and
measured values were relatively consistent. In these two studies,
ANNs were trained with experimental data to predict the struc-
tural properties. This treatment is closer to the reality, but the
workloads are enormous, and the problems solved by these types
of ANNs are very limited.
Relying on the development of computational theories and
methods, the properties of PnCs/EMs can be precisely simulated
by numerical approaches.Generating a dataset by numerical sim-
ulation is not only easy to realize but also can make the dataset
cover a wider range so that the ANN can handle more problems.
In 2019, Finol et al. (2019) built two datasets for 1D and 2D PnCs
with different material parameters in topology by the numerical
computing, and two types of ANNs, MLP and CNN, were compared
on the eigenvalue prediction, as shown in Fig. 2c. They concluded
that the CNN was superior to the MLP for the problems considered
and the predicted values were highly consistent with the testing
data. In the same year, Liu and Yu (2019) used an MLP to predict
the dispersion curves of 1D PnCs with consideration of different
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Journal of Computational Design and Engineering, 2023, 10(2), 602–614 | 605
Tab l e 1 : The inputs and outputs of ANNs for PnCs and EMs.
References ANNs Inputs Outputs
(Liu & Yu, 2019) MLP Geometry and material parameters Dispersion curves represented by a 3 ×101
eigenfrequency matrix
(Jiang et al.,2022) CNN Topology represented by a 50 ×50 matrix
consisting of 0 and 1
Dispersion curves represented by a 10 ×31
eigenfrequency matrix
(Miao et al.,2021) MLP +genetic algorithm Objective function of maximizing relative
bandgap
Nine geometry parameters
(Li et al.,2020)AE+MLP Pass bands and stop bands represented by 1
and 0, respectively
Topology represented by a 128 ×128 matrix
consisting of 0 and 1
(Liu & Yu, 2023)VAE+TNN The lower and upper limits of a bandgap Topology represented by a 20 ×20 ×3
matrix consisting of 0 and 1
Figure 2: Forward prediction ANNs for PnCs and EMs. (a) Feature parameters and a testing result of the ANN, where the solid line is the experimental
result, the dashed line is predicted by the ANN, and the dotted line is calculated by numerical simulations (Fuster-Garcia et al.,2006). (b) Some
examples of samples, and the comparison of predicted (simulated) SACs (red dots) by the ANN and the ones measured (blue dots) (Ciaburro & Iannace,
2021). (c) The constructed CNN and MLP, as well as the comparison of the eigenvalues predicted by the CNN and those analytically computed (Finol et
al.,2019). (d) Comparisons of predicted dispersion curves by the ANNs and analytically computed dispersion curves in Liu and Yu (2019), where the
black solid lines are calculated by transfer matrix method (TMM), and the red dotted and the cyan dashed lines are predicted by the MLP and the
radial basic function ANN, respectively. (e) The CNN used to predict dispersion curves of 2D PnCs/EMs with different topologies (Jiang et al.,2022).
lling fractions, mass density ratios, and shear modulus ratios. It
can be seen from Fig. 2d that the predicted results perfectly co-
incide with the dispersion curves obtained by TMM. Jiang et al.
(2022) predicted the dispersion curves of 2D PnCs/EMs with dif-
ferent topologies by CNN, as shown in Fig. 2e. Sadat and Wang
(2020) and Javadi et al. (2021) used MLPs to study the classication
of PnCs with or without bandgap and predict the width as well
as centre of bandgaps. There were also some studies (Liu & Yu,
2022b,c;Liuet al.,2019;Miaoet al.,2021;Wuet al.,2020) using MLPs
to directly predict bandgaps, where for the case without bandgap
between two adjacent bands, the maximum and minimum of the
corresponding bands were used to represent the lower and upper
limits of bandgap, respectively, or the set of data were deleted from
the dataset. Donda et al. (2021) used a CNN to predict the absorber
spectrums of EMs, and they found that the predicted spectrum by
the CNN was consistent with the experimental one.
From the researches on forward prediction, it can be seen that
ANNs, such as MLPs and CNNs, can accurately predict multiple
properties of PnCs/EMs, including dispersion curves, bandgaps,
and frequency spectrums. Deep learning can acquire the physi-
cal laws or experiences about PnCs and EMs. Therefore, the de-
sign problem on PnCs and EMs, containing parameter design
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606 | Deep learning for PnCs and EMs
Figure 3: Parameter design ANNs for PnCs and EMs. (a) An MLP used to design 2D EMs according to the input band characteristics (Zhang et al.,2022).
(b) The forward prediction MLP, where the inputs are feature parameters and the outputs are lower and upper limits of bandgaps, and the design
process by combining genetic algorithm and MLP (Miao et al.,2021). (c) 1D PnC simulated by multiple trained ANNs (Wu et al.,2021). (d)
Probability-density-based ANN and its ‘one-to-many’ designs (Luo et al.,2020). (e) A TNN composed of an inverse network and a pretrained forward
network. (f) An advanced TNN (Liu & Yu, 2022c) where the encoder and pretrained decoders, respectively, refer to inverse network and pretrained
forward networks, code-layer contains feature parameters, and ‘Case I’ and ‘Case II’ are two design situations.
and topology design, can be solved by a physics-based or well-
experienced ANN model.
5. Parameter Design ANNs for PnCs and
EMs
Parameter design generally refers to adjusting geometry and/or
material parameters to meet expectations. Common deep learn-
ing models for the parameter design of PnCs and EMs include MLP,
hybrid design model, probability-density-based ANN, and tandem
neural network (TNN).
An MLP is able to directly design the feature parameters of PnCs
or EMs, as shown in Fig. 3a (Zhang et al.,2022).Themechanismis
that an MLP is trained with band characteristics (targets) as in-
puts and feature parameters (design variables) as outputs, and a
relation from targets to design variables is built. However, the MLP
is only suitable for a simple issue. When such a situation occurs in
the dataset, i.e., one property value corresponds to different sets
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Journal of Computational Design and Engineering, 2023, 10(2), 602–614 | 607
of feature parameters or several very close property values corre-
spond to multiple sets of feature parameters which differ greatly,
the MLP is difcult to train for convergence, because the function
nature of the MLP is contradictory to the non-uniqueness of the
dataset. Liu et al. discussed this phenomenon in Liu et al. (2019),
where an MLP was used to directly output design variables of 1D
PnCs, and the results were satisfactory for one-parameter design
(the training set only contains 10 sets of data) but unsatisfactory
for three-parameter design (the training set contains 1000 sets of
data). To avoid the unstable training induced by this situation, He
et al. (2021) trained an MLP with the radius of scatterer as output
and bandgap width as well as 0/1 (if the radius is lower than one-
third of the periodic constant, 0 is input; otherwise, 1 is input) as
inputs. By setting the value of 0 or 1 to dene the radius less or
greater than one-third of the periodic constant, respectively, they
successfully made the relation from inputs to outputs mapping.
However, this approach is difcult to solve complicated problems
where such values cannot be dened easily.
To solve the above problems, including the unstable training
and the inability to deal with complicated issues, hybrid design
models were used by some researchers, which are composed of
ANNs and other algorithms. Miao et al. (2021) combined MLP and
genetic algorithm to design the geometry parameters of 2D PnCs,
as shown in Fig. 3b. The genetic algorithm and numerical simula-
tion method usually work in combination to optimize PnCs/EMs,
and there are many relevant studies for reference (Dong et al.,
2014a;Donget al.,2017; Han & Zhang, 2019). In the study by Miao
et al. (2021), the MLP was trained to predict bandgaps according to
the input geometry parameters, which solved the unstable train-
ing. The MLP was substituted for numerical simulations in the ge-
netic evolution process, making the hybrid design model able to
solve complex problems. Wu et al. (2021) proposed a new method
to design 1D PnCs, as shown in Fig. 3c. Similar to the idea of data-
driven nite element method, the simulations were carried out
by the elements which were all trained ANNs, and each element
transferred responses to the next according to the input geometry
parameters and responses. They used the interior-point algorithm
to adjust the geometry parameters of all elements to achieve an-
ticipated output responses.
Inspired by quantum mechanics, Luo et al. (2020)proposeda
probability-density-based ANN, as shown in Fig. 3d, which com-
bined deep learning with mixture Gaussian sampling. This ANN
rst mapped the target frequency spectrum into the parameters
of individual Gaussian distribution. Then these parameters were
used to construct a mixture Gaussian distribution to probabilisti-
cally sample output solutions for the design. This model changes
the nature of the traditional MLP. From input to output in the
probability-density-based ANN is no longer a mapping relation-
ship but a probability problem. It outputs a probability distribu-
tion according to an input target, and several peaks exist in the
probability distribution, which correspond to the designed multi-
ple sets of geometry parameters.
As described above,a single forward prediction MLP can output
the structural properties according to the input feature parame-
ters. However, if there is no restriction on the structural properties,
unrealistic solutions may be obtained. For example, using a single
forward prediction MLP to design an unrestricted geometry pa-
rameter may result in a negative one.This is because the negative
falls outside the dataset domain and the MLP is incapable of judg-
ing correctly. While the TNN can solve this problem. It consists of
an inverse network and a pretrained forward network, as shown
in Fig. 3e. The pretrained forward network is the forward predic-
tion MLP, and the inverse network limits the ranges of designed
feature parameters by activation functions. An anticipated prop-
erty is input into the inverse network, and then the inverse net-
work outputs designed feature parameters into the pretrained for-
ward network. Next, a predicted property will be output, and the
weights and biases of the inverse network will be updated by the
BP according to the difference between the anticipated and pre-
dicted properties until a satisfactory predicted value is obtained.
Finally, the designed results are output by the inverse network.
The TNN is also called autoencoder (AE) since the input is equal
to the output or unsupervised neural network since there is no
label during the training. Liu et al. (2019)andHeet al. (2022)used
the TNN to design 1D PnCs, and Liu et al. (2022) used this model to
design the geometry parameters of EMs. In addition, an advanced
TNN (Liu & Yu, 2022c) was proposed, as shown in Fig. 3f, where
the loss function and the activation function in the last layer of
the inverse network (encoder) were rened, and two pretrained
neural networks were used. This advanced TNN can solve more
complicated design problems involving three-component materi-
als, different site conditions, and full wave modes (shear and lon-
gitudinal wave modes).
For most inverse problems, one target generally corresponds to
multiple designs, which is called ‘one-to-many’ design. Apart from
the MLP learning the relation from properties to feature param-
eters, hybrid design model, probability-density-based ANN, and
TNN are all able to realize the ‘one-to-many’ design, which is con-
ducive to the customization of PnCs/EMs and makes more results
meeting expectations to be picked up. Liu and Yu (2022c)andLuo
et al. (2020) discussed this phenomenon in detail. In addition, Luo
et al. (2020) compared the probability-density-based ANN and the
TNN,and they found that the TNN was superior to the probability-
density-based ANN in terms of accuracy. Although they thought
the TNN had only one output, multiple different sets of outputs
may be obtained if several sets of the same targets are input into
the TNN simultaneously.
6. Topology Design ANNs for PnCs and EMs
It is the fact that the performance of a structure depends on the
geometric distribution of its constituent materials. Topology de-
sign refers to distributing constituent materials reasonably within
a given area (called design domain) to make the structure exhibit
expected performance under specic conditions. The design do-
main is usually divided into multiple same subareas,and one sub-
area is only allocated with one material. Therefore, a matrix com-
posed of integers or orthogonal unit vectors can represent mate-
rial distributions where one integer or orthogonal unit vector in-
dicates one material in a subarea. Compared with the parameter
design, the topology design is more difcult to realize due to its
high dimensionality and discreteness. There are many methods
to realize the topology design or topology optimization effectively,
such as density approach (Bendsoe & Sigmund,2003), level set ap-
proach (Vineyard & Gao, 2021), and discrete approach (Dong et al.,
2017). With the development of AI technology, the topology de-
sign based on deep learning has been explored. For PnCs and EMs,
there are mainly three ANNs to design topologies, including AE-,
VAE-, and GAN-based model.
Li et al. (2020) combined an AE and an MLP to design topologies
of 2D PnCs, as shown in Fig. 4a. The AE was responsible for com-
pressing 128 ×128 topology matrices into low-dimensional vec-
tors and inverse restoration. The MLP was responsible for build-
ing the relation from dispersion properties to the transformed
low-dimensional vectors. A targeted dispersion property was in-
put into the MLP to output a designed low-dimensional vector,
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608 | Deep learning for PnCs and EMs
Figure 4: Topology design ANNs for PnCs and EMs. (a) The AE-based model by Li et al. (2020) where topological feature indicates compressed
low-dimensional vector. (b) The VAE-based model by Liu and Yu (2022e) where the pretrained generative model is the trained VAE’s decoder. (c) The
conditional GAN (CGAN) by Gurbuz et al. (2021) where ‘TL’ (conditions) is the transmission loss of the 2D EMs and ‘Z’ is the input random variables. (d)
The CGAN by Jiang et al. (2022) where the target refers to dispersion curves of 2D PnCs.
and then the vector was input into the AE’s decoder to obtain
a designed topology. In the model shown in Fig. 4a, the AE is
for dimensionality reduction and data restoration, and the MLP
is for design, which is a clear and reasonable design idea. How-
ever, for more complicated design problems, this model may be
invalid due to the function nature of the MLP, which was dis-
cussed in the above section, and it cannot realize ‘one-to-many’
design.
Compared with the AE, the VAE has the ability to generate new
samples (Yao et al.,2019), which is conducive to improving the de-
sign space. The VAE is able to transform high-dimensional and
discrete topology information into low-dimensional and contin-
uous latent space consisting of latent vectors, and it can restore
inversely. Liu and Yu (2022e) proposed a VAE-based model to de-
sign topologies of 2D EMs. Two ANNs, a VAE,and a TNN, were used,
where the VAE’s decoder and the TNN were combined to design
topologies, as shown in Fig. 4b. Similar to the idea of Li et al. (2020),
the VAE is for dimensionality reduction and data restoration, and
the TNN is for the design of latent vectors. Compared with a sin-
gle MLP, the TNN has a more powerful design ability. In addition,
Liu and Yu (2022e) added two equations about frequencies and
sizes into the model to simultaneously design topologies and peri-
odic constants. The VAE-based model was also used to realize the
topology design of 2D PnCs for in-plane waves and out-of-plane
waves, simultaneously or respectively (Liu & Yu, 2022a,d,2023).
Wan g et al. (2022) discussed the performance of VAE, beta VAE,and
Gaussian mixture beta VAE, and they found the Gaussian mixture
beta VAE was better than others in terms of restoration accuracy
for their problem where the dataset includes 360 training topolo-
gies and 40 testing topologies. And then,they used a hybrid model
composed of genetic algorithm and Gaussian process regression
to design latent vectors.
The GAN is also able to generate new samples, although it is
more difcult to train in comparison with the VAE. A single GAN
can only deal with topology information, and structural proper-
ties must be considered additionally in design. Hence, Gurbuz et al.
(2021) used a CGAN to design 2D EMs, as shown in Fig.4c. The con-
ditions in the CGAN referred to the property (transmission loss) of
the structure. When the CGAN was trained, the conditions were
input into the generator and the discriminator, enabling the CGAN
to learn the property information besides topologies. Then, a tar-
geted property (condition) and a random vector were input into
the trained generator to obtain the designed PnCs. By adversarial
training, the generator built a relation from a probability distribu-
tion (random vectors were input into the generator) and structural
property to topology. One may notice that the CGAN can achieve
a ‘one-to-many’ design due to the variability of the random vec-
tors. Jiang et al. (2022) also adopted CGAN to realize the topology
design of 2D PnCs. Unlike Gurbuz et al. (2021), they combined the
generator and the forward prediction CNN to realize the inverse
design, as shown in Fig. 2d. The generator designed topologies, and
the CNN evaluated design accuracy, which was a helpful tool for
selecting effective designs.
There were also some other deep learning-based methods used
to design topologies of PnCs/EMs. Zhang et al. (2021) trained a
CNN to predict structural properties according to the input 5 ×5
topologies of 2D PnCs, and they realized designs by using the CNN
to generate a database containing all possibilities (225)whereaset
of data consisted of a topology and a property predicted by the
CNN.However, topology design usually needs a design domain di-
vided more nely to meet more expectations, and this method will
cause a computational disaster if a higher dimensional topology is
required. Zhao et al. (2021) combined CNN and genetic algorithm
to design 1 ×25 topologies of 1D EMs The CNN was used to pre-
dict the sound elds according to the input topologies, and the
genetic algorithm was used to design the topologies. The design
idea is similar to that of Miao et al. (2021). However, for a higher
dimensional topology, a larger dataset may be needed to achieve
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Journal of Computational Design and Engineering, 2023, 10(2), 602–614 | 609
a powerful CNN and the search time of the genetic algorithm will
increase.
Besides the above approaches,the conditional VAE may also en-
able the topology design of PnCs/EMs, which was applied to other
elds, such as optics (Tang et al.,2020), biology (Greener et al.,2018),
and chemistry (Lim et al.,2018).
7. Challenges and Perspectives
The research on performance prediction and structure design
of PnCs/EMs based on deep learning is just beginning. Although
some achievements have been made,there are still many compli-
cated problems to be solved.
7.1. Prediction of complex/higher order
dispersion curves
At present, the dispersion curves of PnCs/EMs in the deep learning
models are limited to the real parts, so only the frequency ranges
of attenuated waves can be determined, and the weakened am-
plitudes cannot be known. Complex dispersion curves possess in-
formation about the change of a wave amplitude over distance
according to Bloch theorem (Pu & Shi, 2019).If the relation of com-
plex dispersion curves and feature parameters is simulated by
deep learning, it is reasonable to believe that the specic arrange-
ment of basic units meeting the vibration mitigation requirement
can be designed. In addition, this approach is also conducive to
saving materials and controlling the sizes of PnCs/EMs, making
the inverse design more intelligent.
In addition, the eigenfrequency numbers of dispersion curves
predicted by deep learning are very limited. For example, only the
rst 10-order dispersion curves were predicted in Jiang et al. (2022).
There are more bandgaps between higher order dispersion curves,
which is conducive to utilizing more information to guide the de-
sign. However,a lot of computational efforts are needed to obtain
higher order dispersion curves. Hence, to provide more design in-
formation and improve design efciency, realizing the prediction
of higher order dispersion curves by deep learning is signicant
for PnCs and EMs.
7.2. Diversity of topology, lattice, and property
Different from the parameter design, the topology design is not
limited to a specic shape of scatterer and has more possibilities.
However, the high dimensionality of topology brings great compu-
tational difculties to discovering new structures.One may notice
that in many existing studies (Li et al.,2020; Liu & Yu, 2022a,d,
2023;Wanget al.,2022), the topologies in datasets are some spe-
cic shapes without randomness, limiting the discovery space of
novel congurations. To improve the design space and ensure the
sample quality, Liu and Yu (2022e) proposed a generation rule to
obtain ternary topologies with random shapes, which is a helpful
reference for PnCs/EMs with a single scatterer within a basic unit
and low-dimensional topology. For the same purpose, Jiang et al.
(2022) used the built-in imdilate function in Matlab, and Gurbuz
et al. (2021) proposed an algorithm to randomly generate topolo-
gies. In order to design more diverse structures and discover novel
PnCs/EMs with better performance by deep learning, generating
topologies need further exploration to enlarge the design space
reasonably.
It is also to be noticed that most of the studies about 2D PnCs
and EMs based on deep learning focused on the topology with
C4-symmetry and square lattice arrangement. However, there are
many other lattice layouts, such as hexagon, rectangle, cube, and
cuboid, as well as many other topologies, such as asymmetry or
C1-, C2-, C3-, and C6-symmetry, whose behaviour may differ from
that studied in the existing literature. Therefore, to enrich the di-
versity of design, ANNs for different types of PnCs and EMs are
worthy of further exploration. In addition, apart from the dis-
persion curves, bandgaps, and frequency spectrums which were
mainly considered in the existing studies on deep learning for
PnCs and EMs, other properties,such as negative refractive index,
Dirac cone, and valley state, are also signicant for the manipula-
tion of elastic waves. Thus, it is valuable to develop the ANNs for
more properties of PnCs and EMs.
7.3. Combination design
It is well known that the properties of a PnC/EM work in nite fre-
quency regions, and multiple PnCs/EMs with different ranges of
properties are usually combined to manipulate the waves within
an extensive frequency range.It is difcult for designers to nd an
appropriate combination considering saving materials and limit-
ing sizes. Some methods,such as the Edisonian approach and evo-
lutionary approach, will cost a lot of time and resources to realize
the combination design. However,deep learning has the potential
to handle this issue rapidly.New structures of ANNs and loss func-
tions may be necessary to simultaneously meet the expectations,
including broadbands, saving materials, and limiting sizes. ‘One-
to-many’ combination design is also possible to realize, providing
more choices.
7.4. Design for tunable PnCs and EMs
There are many types of tunable PnCs/EMs, which have the ad-
vantage of tuning feature parameters to meet changeable expec-
tations (Wang et al.,2020a). For example, by adjusting the electric
eld, the ranges of piezoelectric PnCs’ bandgaps will be altered to
satisfy a new anticipation. For a simple law between adjustable
parameters and properties, parameter analysis can be used to
build a function to approximate the mapping relationship. How-
ever, it is difcult for a complicated law to nd an appropriate
surrogate function. Fortunately, deep learning has the potential to
guide the adjustment of tunable PnCs/EMs, which is essentially
a design problem as described above. Due to the superiority in
speed, deep learning also has a broad application prospect on the
instant adjustment of PnCs/EMs.
7.5. Small data training in ANNs
The data with physical connotations for PnCs and EMs are ob-
tained mainly through experiments or numerical simulations. For
in-depth learning and exploration, massive data with physics-
based labels are required for ANNs. However, it is not easy to
collect large numbers of data through experiments, and numer-
ical simulations need a lot of time and computational resources.
Therefore,how to realize the function of ANNs through fewer data
is an important issue for the deep learning-based design of PnCs
and EMs.
Recently, ANNs have been used to solve partial differen-
tial equations (PDEs; Samaniego et al.,2020), which are named
physics-informed neural networks (PINNs; Cai et al.,2022). With-
out preparing labelled data, PINNs can nd a solution that satis-
es certain PDEs. Although PINNs are similar to numerical simu-
lation methods, which need many iterations to obtain an approxi-
mate solution satisfying constraint equations, they possess infor-
mation with physical connotation and may reduce the need for
data by transfer learning.
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610 | Deep learning for PnCs and EMs
7.6. Quantication of uncertainty in ANNs
Condence in the output of ANNs is essential for users, but ANNs
can neither deliver certainty estimates nor output the credibil-
ity of predictions. Hence, many scholars have been working on
quantifying the uncertainty in ANNs (Abdar et al.,2021;Eaton-
Rosen et al.,2018; Gawlikowski et al.,2021), but most of the stud-
ies focused on that about the forward prediction problem. What
method is suitable for the uncertainty quantication in ANNs for
the inverse design of PnCs and EMs needs to be studied, where
the difference between parameter design and topology design is
suggested to consider.
7.7. Interpretability of ANNs
ANNs have achieved high accuracy in the prediction and design
of PnCs and EMs, but they are still regarded mostly as black boxes.
However, it is important for users to suitably weigh the decisions
aided by ANNs. Hence, the understanding and reasoning about
the output of ANNs, namely the interpretability of ANNs, is nec-
essary for users. The interpretability of ANNs is similar to the hu-
man thought process, which is helpful for users to justify the out-
put by a series of logically consistent and understandable choices
in the ANN and improve the level of trust. Unfortunately, despite
several years of research effort in computer science, progress on
the interpretability of ANNs remains limited (Lipton, 2018). How
to interpret ANNs for PnCs and EMs is a very challenging issue.
8. Conclusions
Deep learning developed from imitating biological brains,but now
its revolutions are emerging to solve more practical problems, of
which the idea is no longer limited to biological neurons. By con-
structing multilevel abstraction of data, deep learning is a pow-
erful tool to handle highly complicated problems and simulate
unknown intricate relations.
In this review, we presented the recent progress in deep learn-
ing for performance prediction and structural design of PnCs and
EMs, including forward prediction ANNs, parameter design ANNs,
and topology design ANNs. Relatively, the forward prediction for
PnCs/EMs is easy to realize if enough data is provided. From fea-
ture parameters of PnCs and EMs to their properties is a mapping
law in real, and this law can be simulated by deep learning no mat-
ter how complicated it is. The parameter design methods based on
deep learning in PnCs and EMs are developed very well, and there
are several excellent models, such as the hybrid design model,
probability-density-based ANN, and TNN. Lots of studies demon-
strated the high efciency of deep learning models designing pa-
rameters. Due to the high dimensionality and discreteness, the
topology design of PnCs/EMs is an extremely challenging problem.
Some deep learning models, containing AE-based model, VAE-
based model, and GAN-based model, were used to design topolo-
gies of PnCs and EMs, respectively. Still, there are few researches
using a large number of verications based on numerical simula-
tions to prove the topology design performance by deep learning.
Several challenges and perspectives of further problems in
deep learning of PnCs and EMs are proposed. The prediction of
complex/higher order dispersion curves by deep learning is sig-
nicant, which can provide more information for the design. The
approach of generating topologies is worthy of further exploration
to extend the design space and discover novel structures. More
topologies, lattices, and properties are worthy of being studied for
the design diversity and more exible wave manipulation. Com-
bination design is conducive to broadband, material saving, and
size limitation. For tunable PnCs/EMs, deep learning possesses the
advantages of intelligence and rapidity in complicated problems
and instant adjustment. To improve condence in deep learning,
studies of the uncertainty quantication and interpretation for
deep learning models are necessary, which are very challenging
issues.
This interdisciplinary study merges the cutting-edge knowl-
edge of computer science, material science, physics and mechan-
ics, etc. Although some remarkable achievements have been ob-
tained over the past few years, further breakthroughs need efforts
from researchers with different disciplinary backgrounds. Deep
learning scholars are supposed to team up with scientists in PnCs
and EMs to exploit new algorithms to more effectively solve re-
lated problems, such as less data, higher dimensional topology,
faster speed, and more precise manipulation. Both two can also
be mutually reinforcing. For this purpose, the PnCs and EMs com-
munity should share resources, like datasets and automatic data
acquisition methods, to speed up the development of relevant re-
searches. A gene bank including diverse PnCs and EMs should be
built, and hierarchical AI algorithms should be developed to more
efciently achieve the ‘global’ optimal design for a given target.AI
era is coming, and intelligent design in PnCs and EMs is emerging.
Acknowledgments
This work is supported by the National Natural Science Founda-
tion of China (No. 12172037) and Beijing Natural Science Founda-
tion (No. 8222024).
Conict of interest statement
None declared.
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