Deep learning for Dirac dispersion engineering in sonic crystals

Abstract

Band structure and Dirac degeneracy are essential features of sonic crystals/acoustic metamaterials to achieve advanced control of exciting wave effects. In this work, we explore a deep learning approach for the design of phononic crystals with desired dispersion. A plane wave expansion method is utilized to establish the dataset relation between the structural parameters and the energy band features. Subsequently, a multilayer perceptron model trained using the dataset can yield accurate predictions of wave behavior. Based on the trained model, we further impose a re-learning process around a targeted frequency, by which Dirac degeneracy and double Dirac degeneracy can be embedded into the band structures. Our study enables the deep learning approach as a reliable design strategy for Dirac structures/metamaterials, opening up the possibilities for intriguing wave physics associated with Dirac cone.

Full text

Deep learning for Dirac dispersion engineering
in sonic crystals
Cite as: J. Appl. Phys. 135, 244303 (2024); doi: 10.1063/5.0206258
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k
Submitted: 29 February 2024 · Accepted: 5 June 2024 ·
Published Online: 26 June 2024
Xiao-Huan Wan, Jin Zhang, Yongsheng Huang,
a)
and Li-Yang Zheng
a)
AFFILIATIONS
School of Science, Shenzhen Campus of Sun Yat-sen University, Shenzhen 518107, People’s Republic of China
a)
Authors to whom correspondence should be addressed: huangysh59@mail.sysu.edu.cn and zhengly27@mail.sysu.edu.cn
ABSTRACT
Band structure and Dirac degeneracy are essential features of sonic crystals/acoustic metamaterials to achieve advanced control of exciting
wave effects. In this work, we explore a deep learning approach for the design of phononic crystals with desired dispersion. A plane wave
expansion method is utilized to establish the dataset relation between the structural parameters and the energy band features. Subsequently,
a multilayer perceptron model trained using the dataset can yield accurate predictions of wave behavior. Based on the trained model, we
further impose a re-learning process around a targeted frequency, by which Dirac degeneracy and double Dirac degeneracy can be embed-
ded into the band structures. Our study enables the deep learning approach as a reliable design strategy for Dirac structures/metamaterials,
opening up the possibilities for intriguing wave physics associated with Dirac cone.
© 2024 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution-NonCommercial 4.0
International (CC BY-NC) license (https://creativecommons.org/licenses/by-nc/4.0/). https://doi.org/10.1063/5.0206258
I. INTRODUCTION
Phononic crystals (PCs) and acoustic metamaterials (AMs) are
novel artificially designed materials that exhibit extraordinary physi-
cal properties not found in natural materials,
1–3
including negative
elastic modulus and negative density,
1,4,5
zero refractive index,
6,7
neg-
ative refraction,
8,9
and acoustic invisibility.
10,11
The unique dispersion
characteristics and dynamic properties of these materials make them
potential candidates for the manipulation of sound.
12–15
There are
mainly two mechanisms for the formation of phononic bandgaps,
namely, the Bragg scattering mechanism
16
and the local resonance
mechanism.
17,18
Parameters that influence the characteristics of
bandgaps include material properties,
19
structural parameters,
20
topo-
logical optimization,
21
etc. In addition, recent research activities have
been focused on the unique band degeneracy properties, such as the
Dirac cone and the double Dirac cone.
22–24
These specific band
structures exhibit a range of interesting physical phenomena, includ-
ing pseudo-diffusive transport,
25,26
topological edge states,
27
Zitterbewegung tremor,
28
and acoustic phase-reconstruction.
29
The
realization of Dirac dispersion brings new applications for the design
of acoustic metamaterials and devices.
In recent years, deep learning (DL) has attracted much atten-
tion owing to its fast response and efficiency to solving various
complex problems. One of the deep learning models is based on
the artificial neural network consisting of multiple processing layers
that are trained to intelligently extract statistical rules and learn rep-
resentations of big data.
30
It has been shown that deep learning is
capable of predicting excellent results in natural language process-
ing and decision making.
31–34
These findings suggest that deep
learning has the potential to overcome the shortcomings of tradi-
tional methods, such as the ability to reduce labor costs, decrease
time loss, and save significant computational resources.
35–40
New
design methods for PCs and AMs based on deep learning have also
become a hot topic in recent years.
41–47
As a data-driven approach,
datasets for deep learning models are mainly constructed through
experiments or numerical simulations, etc., which can automati-
cally reveal useful information. Examples include the correspon-
dence between structural parameters (e.g., geometrical parameters,
material parameters, and topology features) and related properties
(e.g., bandgap, dispersion curves, and frequency response) of PCs
and AMs.
48
In this work, we propose a multilayer perceptron-based deep
learning model to design 2D PCs with targeted properties. We
use the plane wave expansion (PWE) method to establish the
dataset relation of the band properties of PCs with its geometrical
parameters. This allows us to obtain tons of data for the use of
training, testing, and validation for the deep learning model.
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Based on the trained DL model, we are able to design 2D pho-
nonic crystals with desired bandgap properties. We also propose a
re-learning process for the DL model, which ensures the ability to
tune the local properties of energy bands, allowing for the engi-
neeringoftheDiracconeontospecificlocationsoftheBrillouin
zone. The study of this work can facilitate the design of phononic
crystals and Dirac dispersion that can lead to applications with
novel phenomena, including quantum spin Hall effect, unidirec-
tional propagation, robust transmission, etc.
II. METHODS
A workflow is shown in Fig. 1. We consider a rectangular PC
with scatterers embedded in air, a unit cell of which is depicted
in Fig. 1 (left panel). The lattice constants are ax¼10:5 mm and
ay¼ffiffiffi
3
pax=3. We consider the scatterers made of aluminum cylin-
ders (radius R¼1 mm) with ρ¼2700 kg/m3and νA¼6260 m/s.
The center distance of two cylinders in the xaxis is d1and in the y
axis is d2. Given the values of the structural parameters (d1,d2),
there is a corresponding band structure (BS) [i.e., Fig. 1 (right
panel)] describing the propagation property of sound in the PC.
The correspondence between the structural parameters and the BS
can be revealed by using the plane wave expansion method. In the
forward calculation process, we are able to establish the relation of
input (d1,d2) and its output (the BS) based on the PWE
calculation. After numerous calculations of the BSs for many PC
samples, the relation dataset between the structural parameters and
the BS is established. In the inversion design process, a multilayer
perceptron network constructed by seven fully connected neural
layers is used and trained to be the DL model. During the training
process, the BS spectrum is the input and the output is the struc-
tural parameters (d1,d2). Based on the relation dataset obtained
from the forward calculation process, the DL model can be well
trained to predict structural parameters (d1,d2) with great precision
when a targeted BS is aimed. It is worth noticing that the unit cell
contains four cylinders while only two structural parameters are
considered. This leads to the existence of certain symmetries in the
unit cell. However, we emphasize that symmetry is not necessary
for the DL model and symmetry constraint can be avoided if
enough structural parameters are included.
A. Forward calculation
The PWE method has been shown to be an elegant tool for
the calculation of dispersion curves in various setting contexts.
49,50
One of the most essential advantages of PWE is that it can quickly
return the BS of a given PC with high accuracy. This fast response
becomes more and more appealing, especially when we have a great
amount of calculation for thousands of PC samples. Considering a
PC unit cell in Fig. 1 (left panel), sound wave dynamics is described
FIG. 1. Schematic diagram of PC design based on a deep learning model. Forward calculation allows us to obtain the band structures of PCs for given parameters d1
and d2using the PWE method. For the inverse design process, a machine learning model is trained to predict the geometric parameters d1and d2when a PC with a spe-
cific band structure is aimed.
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J. Appl. Phys. 135, 244303 (2024); doi: 10.1063/5.0206258 135, 244303-2
©Author(s)2024

by the wave equation
1
λ(r)
@2Φ
@t2¼∇1
ρ(r)∇Φ

, (1)
where λ(r) is the bulk modulus and ρ(r) is the mass density. Φis the
pressure of sound that can be expressed in a Bloch wave form due to
the periodicity,
Φ(
~
r)¼ei~
k~
rΦ
~
k(
~
r), (2)
where ~
kis the Bloch wave vector. Due to periodicity, 1
λ(
~
r),1
ρ(
~
r),and
Φ
~
k(
~
r) can be expanded in the forms of Fourier series,
ξ(
~
r)¼X
~
G
ξ~
Gei~
G~
r, (3)
where ξ(
~
r) stands for 1
λ(
~
r),1
ρ(
~
r),andΦ
~
k(
~
r). ~
Gis the reciprocal lattice
vector.
For a single cylinder, the Fourier coefficient ξGcan be
expressed as
ξG¼S1ðd2rξ(
~
r)ei~
G~
r
¼ξAlFþξB(1 lF), ~
G¼0,
(ξAξB)F(~
G), ~
G=0,
((4)
where Sis the area of the cell and lFdenotes the filling ratio of cyl-
inders in the unit cell. ξAand ξBcorrespond to material parameters
of media Aand B, respectively. F(~
G) is the structure function,
depending on the shape of the scatterer. For two-dimensional PCs
with a rectangular dot matrix, the metric area S¼ax*ayin the
above equation. Since the cross-sectional shape is a circle, its occu-
pancy in the unit cell is lF¼πR2=S, and the structure function
F(~
G)¼S1ðR
0ð2π
0
eiGrcos
w
rdrd
w
¼2lF
J1(GR)
GR :(5)
Here, J1is the first-order Bessel function of the first kind and Gis
the module of ~
G. According to the translation property of the
Fourier transform, if ξ(
~
r) shifts by an increment ~
r0, then its Fourier
transform is multiplied by a factor ei~
G~
r0, as follows:
ξ(
~
rþ~
r0) !ei~
G~
r0ξ~
G:(6)
Thus, for complex lattices whose primitives contain multiple cyl-
inders, the Fourier transform can be obtained by simple summation,
X
ri
ξ(
~
rþ~
ri),X
ri
ei~
G~
riξG, (7)
where ~
riis the position vector of each cylinder in the primitive.
Substituting Eqs. (4) and (3) into Eq. (1), the eigen-valued
equation can be obtained as follows:
50,51
X
~
G0
ω2λ1
~
G~
G0ρ1
~
G~
G0(~
kþ~
G)(~
kþ~
G0)
hi
Φ
~
k~
G0¼0:(8)
Let the wave vector ~
krun over the irreducible Brillouin zone bound-
aries: Γ!X!M!Γ; the energy band structure ω(~
k)isobtained
by solving the eigenvalue problem of Eq. (8), leading to dispersion
curves (one of the results) shown in the right panel of Fig. 1.
B. Inverse design
The DL model is shown in Fig. 2(a), where a multilayer per-
ceptron consisting of one input layer, one output layer, and five
hidden layers is applied. Each layer contains a certain number of
neurons (nodes), and the neurons in each layer are interconnected
with all the neurons in the neighboring layers. The mathematical
relation for the neurons is usually expressed as
48
A(m)
n¼g(m)(z(m)
n), z(m)
n¼ω(m)
n

TA(m1) þb(m)
n, (9)
where A(m)
n,z(m)
n,ω(m)
n, and b(m)
ndenote the output/activated value,
the pre-activated value, the weight vector, and the bias of the nth
neuron in the mth layer. A(m1) is the output vector from the
(m1)th layer. For m¼1, A(0) is the data from the input layer.
The activation function of the mth layer g(m)is introduced to incor-
porate nonlinear properties that help us to increase the expressive
power of the neural network, allowing the network to handle very
complex data representations. Some of the popular activation func-
tions include sigmoid, rectified linear units (ReLU), hyperbolic
tangent, etc. In this work, we use ReLU as an activation function.
The numbers of nodes in each hidden layer are set to be 502,
251, 126, 63, and 31 (from left to right). The output layer has two
nodes corresponding to the structural parameters d1and d2. For the
input layer, it contains 501 nodes that can be regarded as a data
vector with 501 components. Thus, we have to incorporate the eigen-
frequency spectrum information onto the vector as an input. Taking
the ΓM direction as an example, we focus on the eigenfrequency
spectrum ranging from 0 to 70 kHz as shown in Fig. 2(b).Firstofall,
the frequency span is uniformly divided into 501 intervals with a
correspondence to the input of 501 components. Each interval is
rendered with different integers given whether its corresponding
eigenfrequency is located in the passband (valued as 1) or in the
stopband (valued as 0). Since there exist bandgaps around
27:85–33:34 kHz and 57:04–60:27 kHz [cyan in Fig. 2(b)], the inter-
val span corresponding to the two bandgaps is marked as 0 and the
rest corresponding to the passband as 1. Thus, the energy band
property in the ΓM direction is interpreted onto the data vector.
Following this methodology, the BS information for the ΓX and the
XM directions can also be coded onto the data vector; consequently,
the global band property of PCs can be interpreted as an input for
the DL model.
To train the DL model, a dataset with 11 336 PC samples in
total is constructed using the PWE calculation with the structural
parameters ranging in d1[[2, 5:3] mm, d2[[2, 4:1] mm. 80%
portions of total data are used for the model training (the remaining
10%for validation and 10%for testing). A loss function estimating
the accuracy of model prediction is defined by the mean-square error
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©Author(s)2024

(MSE) based on the structural parameters,
MSE ¼1
nX
n
l¼1
(^
dldl)2, (10)
where nis the number of structural parameters (i.e., n¼2) and ^
dl
corresponds to the predicted value. During the training, a small
batch gradient descent algorithm is applied with 1000 iteration
times. To speed up the calculation convergence, the Adam opti-
mizer of the learning rate α¼0:0001 is applied. The changing of
the cost function during 100 iterations is shown in Fig. 2(c),
where the curve of the cost function (blue line in symbol a)
decreases rapidly as the increasing of iterations and reaches
quickly to convergency after 40 iterations. To verify that the DL
model is well trained, we now use the remaining 10%portions of
data for validation. The convergency of the loss function for the
validation dataset is also shown in Fig. 2(c) by red dots. It can
be seen that the result of the validation set has an excellent
agreement with the training set. In comparison, we also show in
Fig. 2(c) the model performance of two smaller datasets with
1974 samples (symbol b) and 5037 samples (symbol c), respec-
tively. It can be seen that the convergence speed and the change
in error during iterations are relatively slower for the two cases of
smaller samples. Thus, we conclude that with a proper number of
learning samples, the model performance can be well trained to
have fast convergency. Additionally, it should be noted that the
constructed multilayer perceptron model is coded in Python,
referring to the Keras machine learning (ML) library within
TensorFlow during its training.
52
FIG. 2. (a) The DL model constructed by a fully connected multilayer neural network with an input layer, an output layer, and five hidden layers. The number of neurons
for each layer is labeled. The output layer corresponds to the structure parameters d1and d2. (b) The band spectra in the ΓM direction, in which the whole frequency
span is divided into 501 parts, and the passband and stopband are remarked as 1 and 0, respectively. (c) Loss function vs the number of iterations trained by 11 336
samples labeled as a, by 1974 samples labeled as b, and by 5037 samples labeled as c.
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III. DISCUSSIONS
Based on the trained DL model, we can verify its accuracy and
stability using the testing dataset. To demonstrate, we first calculate
the global BS of a PC with d1¼3:35 mm and d2¼3:34 mm, the
result of which is shown in Fig. 3(a) by blue curves. Then, we take
the BS as an input for the DL model; it returns the predicted struc-
tural parameters, i.e., ^
d1¼3:64 mm and ^
d2¼2:98 mm. The corre-
sponding BS from the DL model is also shown in Fig. 3(a) by red
dots. It can be seen that the results of the two cases have a good
agreement, demonstrating the accuracy and stability of the trained
DL model. We also evaluate the performance of the trained DL
model for the partial band structure. In Fig. 3(c),thedispersion
curves of sound propagating along the XM direction are displayed by
blue curves with d1¼3:11 mm and d2¼2:94 mm. The predicted
values from the DL model of the same BS are ^
d1¼3:07 mm and
^
d2¼3:05 mm. The predicted BS marked by red dots in Fig. 3(c) is
seen to have a very good agreement with the original one.
The well-trained DL model can be used to design PCs with
desired stopbands at aimed frequency and bandwidth. Suppose a
PC with a global bandgap at 28.55–33.55 kHz is targeted, we only
need to set the values of components in the input vector corre-
sponding to this frequency span to be 0, and the rest is set to be 1.
Imposing this input vector to the DL model, a pair of structural
parameters ^
d1¼4:01 mm and ^
d2¼3:01 mm are predicted. Both
predicted (magenta dots) and targeted (black line) bandgaps are
displayed in the bottom panel of Fig. 3(b), where the horizontal
axis corresponds to the frequency and the vertical axis with labels
“1”and “0”indicating the propagating and forbidden bands.
FIG. 3. (a) Band structures of a PC. Red dots are the results predicted by the DL model. Blue lines correspond to the original results. (b) Targeted bandgap labeled in
lines and a predicted bandgap marked by dots from the model. (c) Similar to (a) but band structures only in the XM direction are considered. (d) When partial bandgaps
along the XM marked by blue are aimed, the DL model can predict the structure parameter of a PC with partial bandgaps marked by red dots.
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It can be seen that the predicted bandgap is in the range of
28:5533:47 kHz, indicating a good agreement with our targeted
bandgap. However, when a larger bandgap at 27:133 kHz is
aimed [blue line in the top panel of Fig. 3(b)], the model predicts
aPC(
^
d1¼4:19 mm and ^
d2¼3:01 mm) with a bandgap at
28.3–33.2 kHz (red dots). This discrepancy is due to the width of
the targeted bandgap exceeding the maximum of the allowed
bandgap in the current PC configuration.
The DL model can also be used to design PCs with an aimed
partial bandgap. For instance, along the XM direction as shown in
Fig. 3(d), we expect a PC with two bandgaps (blue lines) at
f¼28:136 kHz and f¼45:547:5 kHz, respectively. The DL
model predicts gaps (red dots) at f¼27:535 kHz and
f¼46:347:5 kHz with the structural parameters d1¼3:18 mm
and d2¼3:05 mm.
In addition, the DL model enables the possibility of fine
tuning the local behavior of sound at specific locations in the BS.
This ability is important, especially for the design of Dirac disper-
sion, which currently is at the hotspot in the field of topological
photonics/phononics.
22–24,53
Such a linear crossing of energy
bands, typically happening to occur at a specific location of the BS,
brings rich wave physics associated with the topology property. We
emphasize that using the DL model, it is possible to embed such
topological band objects to PCs by focusing on the local properties
of energy bands. As seen in the BS of Fig. 1 (right panel), there
exists a bandgap between the second and third energy bands at the
FIG. 4. (a) The global machine learning (ML) model predicts the possibility of the existence of the Dirac cone highlighted by a box around Γfor d1¼2:54 mm and
d2¼2:96 mm. (b) Zoomed view of the boxed area in (a). By a re-learning process, the local property of bands around this region can be precisely predicted. The
re-learning frequency window is chosen to be [f3,f4] = [31.625, 30.604] kHz. After a re-training process, a local ML model is established. This leads to a perfect Dirac
cone shown in blue lines in (c) with d1¼2:45 mm and d2¼2:97 mm. The red dots are the results from simulation. (d) The two eigenmodes of the Dirac point. The color
level indicates the amplitude of sound pressure.
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Γpoint, and the Dirac cone might potentially appear if we can
close the bandgap. We implement this idea using the DL model,
which retunes the dispersion curves in Fig. 4(a) with the predicted
structural parameters d1¼2:54 mm and d2¼2:96 mm. However,
one can observe that the two bands do not really degenerate at the
Γ(highlighted by a box). This is due to the fact that the DL model
is carried out in a frequency span from 0 to 50 kHz, with insuffi-
cient information for the local properties around the Γpoint,
leading to deviations between the aimed Dirac cone and prediction.
From hereon, we refer to this DL model as a global machine learn-
ing (ML) model.
To shed more light on the ability of the DL model for tuning
the local properties of energy bands, we impose a re-learning
process upon the trained model and focus on the local property
of bands at high symmetry points. We refer to this new model as
a local ML model. The main idea is the following. Based on the
predicted results of the global ML model, we assure that it is pos-
sibletoobtainaDiracconeattheΓin the frequency range
[30:854, 31:375] kHz. We, thus, implement a second round of the
DL process by setting the input vector (501 components) only
carrying the band information around [30:854, 31:375] kHz.
The zoomed view is shown in Fig. 4(b), where we mark
f1¼31:375 kHz, f2¼30:854 kHz, f3¼(1 þβ)f0¼31:625 kHz,
and f4¼(1 β)f0¼30:604 kHz with the central frequency
f0¼(f1þf2)=2¼31:115 kHz and β¼0:016 labeling a small fre-
quency variation region for the local ML model. Thus, the
relearning process is carried out in the frequency window [f4,f3],
and this allows us to thoroughly dictate and control the band
property at the high symmetry points. To construct the dataset
for the local ML model, we calculate the BS of 10 201 PCs with d1
varying in [2.44, 2.64] mm and d2in [2.86, 3.06] mm. Once the
local ML model is well trained, the power of the model is exhib-
ited in Fig. 4(c), where Dirac degeneracy is predicted accurately at
the Γpoint with d1¼2:45 mm and d2¼2:97 mm obtained by
the local ML model. The blue lines represent the energy band
spectra from the local ML model, and the red dots are from the
simulation. The inset shows a perfectly linear crossing of two
bands, verifying the appearance of the Dirac cone. The eigen-
modes of the Dirac point are depicted in Fig. 4(d).Bytuningthe
structural parameters, the two modes from the second and third
bands can accidentally degenerate at the Γpoint, leading to the
Dirac cone.
Using the local ML model, one can also obtain a double Dirac
cone at the high symmetry points. By performing the same meth-
odology mentioned above the M point, we are able to fine-tune the
degeneracy of four modes with d1¼5:26 mm and d2¼3:04 mm,
forming a double Dirac cone as shown in Fig. 5(a). The blue lines
represent the BS from the model, and the red dots correspond to
the simulation result. The origin of this Dirac cone can be regarded
as a consequence of band folding in the case when the unit cell
transforms into a rectangular lattice. The four corresponding
degenerate modes are depicted in Fig. 5(b), in which the modes
can be engineered to exhibit Z2topological transport and other
interesting related wave phenomena.
54,55
IV. CONCLUSIONS
To summarize, we present a deep learning model that can
be used to design 2D phononic crystals with desired bandgap
properties. The forward calculation and the inverse design pro-
cesses have been detailed. It has been shown that the well-
trained DL model can be a powerful tool for the prediction of
PC structures when desired band properties are aimed. We also
propose a locally deep learning model based on a re-learning
process, which ensures the ability to tune the local properties of
energy bands, allowing for the engineering of a Dirac cone onto
specific locations of the Brillouin zone. The study of this work
can facilitate the design of phononic crystals and Dirac disper-
sion that can lead to applications related to novel phenomena,
including quantum spin Hall effect, unidirectional propagation,
robust transmission, etc.
ACKNOWLEDGMENTS
This work was supported by the National Natural Science
Foundation of China (NNSFC) under Grant No. 12204553.
FIG. 5. (a) Double Dirac cone predicted by the local ML model with
d1¼5:26 mm and d2¼3:04 mm. Blue lines are the predicted results, and red
dots are the simulation. (b) The four eigenmodes at the Dirac point. The color
level indicates the amplitude of sound pressure.
Journal of
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J. Appl. Phys. 135, 244303 (2024); doi: 10.1063/5.0206258 135, 244303-7
©Author(s)2024

AUTHOR DECLARATIONS
Conflict of Interest
The authors have no conflicts to disclose.
Author Contributions
Xiao-Huan Wan: Conceptualization (equal); Data curation (equal);
Formal analysis (equal); Methodology (equal); Validation (equal);
Writing –original draft (lead); Writing –review & editing (equal).
Jin Zhang: Formal analysis (equal); Validation (equal). Yongsheng
Huang: Investigation (equal); Validation (equal); Visualization
(equal). Li-Yang Zheng: Conceptualization (equal); Funding acqui-
sition (equal); Supervision (equal); Validation (equal); Writing –
review & editing (equal).
DATA AVAILABILITY
The data that support the findings of this study are available
from the corresponding author upon reasonable request.
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