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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

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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