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Chin. Phys. B 33, 104301 (2024)
Inverse design of nonlinear phononic crystal configurations based on
multi-label classification learning neural networks
Kunqi Huang(黄坤琦)1,2, Yiran Lin(林懿然)1,2, Yun Lai(赖耘)1,2,†, and Xiaozhou Liu(刘晓宙)1,2,3,‡
1Key Laboratory of Modern Acoustics, Institute of Acoustics, Nanjing University, Nanjing 210093, China
2School of Physics, Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China
3State Key Laboratory of Acoustics, Institute of Acoustics, Chinese Academy of Sciences, Beijing 100190, China
(Received 22 June 2024; revised manuscript received 31 July 2024; accepted manuscript online 6 August 2024)
Phononic crystals, as artificial composite materials, have sparked significant interest due to their novel characteristics
that emerge upon the introduction of nonlinearity. Among these properties, second-harmonic features exhibit potential
applications in acoustic frequency conversion, non-reciprocal wave propagation, and non-destructive testing. Precisely
manipulating the harmonic band structure presents a major challenge in the design of nonlinear phononic crystals. Tradi-
tional design approaches based on parameter adjustments to meet specific application requirements are inefficient and often
yield suboptimal performance. Therefore, this paper develops a design methodology using Softmax logistic regression and
multi-label classification learning to inversely design the material distribution of nonlinear phononic crystals by exploiting
information from harmonic transmission spectra. The results demonstrate that the neural network-based inverse design
method can effectively tailor nonlinear phononic crystals with desired functionalities. This work establishes a mapping
relationship between the band structure and the material distribution within phononic crystals, providing valuable insights
into the inverse design of metamaterials.
Keywords: multi-label classification learning, nonlinear phononic crystals, inverse design
PACS: 43.25.+yDOI: 10.1088/1674-1056/ad6b85
1. Introduction
When mechanical waves propagate in phononic crystals,
they tend to exhibit unique bandgap properties due to wave
scattering or local resonance in the structure.[1]In recent years,
the incorporation of nonlinearity into phononic crystal sys-
tems has significantly expanded the realm of wave propaga-
tion not achievable in linear systems. For instance, nonlin-
ear systems can display novel wave phenomena such as soli-
ton propagation, higher-order harmonic generation, and nonre-
ciprocal wave transmission, which hold considerable practical
significance.[2–5]In nonlinear phononic crystals, as mechani-
cal waves travel through the material, energy can be converted
into waves at higher frequencies. This is a process known as
second-harmonic generation. Second harmonic generation is
a common occurrence in nonlinear crystals, but designing an
ideal nonlinear phononic crystal remains a challenging task
due to the inherent non-uniformity of the material and the
complexity in calculations.[5–7]
Over the past few decades, researchers have designed
phononic crystals by tuning the physical parameters of the
materials to match specific dispersion characteristics.[8,9]This
traditional method requires empirical and experimental sup-
port, and the design process is complex, consuming a lot of
time and computer memory. In recent years, artificial intel-
ligence technology has developed rapidly. Neural networks,
as a crucial component of artificial intelligence technology,
have found widespread application across various fields in-
cluding computer vision, speech recognition, and natural lan-
guage processing. Inspired by advances in artificial intelli-
gence technology, the use of neural networks in the design
of metamaterials has increasingly garnered attention.[10–14]
For example, Peurifoy et al. trained a neural network to
approximate the light scattering properties of multilayered
nanospheres.[15]Finol et al. used a deep convolutional neu-
ral network and a conventional densely connected neural net-
work to predict the eigenvalues of phononic crystals and con-
cluded that the convolutional neural network outperformed
the conventional densely connected neural network.[16]Koll-
mann et al. applied the ResUNet convolutional neural net-
work model for the optimization design of metamaterials.[17]
Li et al. employed an autoencoder and multilayer perceptron
models to establish an intrinsic connection between the two-
dimensional phononic crystal image features, structural topol-
ogy features, and bandgap characteristics.[18]Gurbuz et al.
proposed a method based on generative adversarial neural net-
works to design acoustic metamaterials, successfully imple-
menting inverse design of unit cells.[19]Wei et al. proposed
a general method based on the gradient descent method to
study the inverse design of one-dimensional (1D) phononic
crystals.[20]The variational autoencoder model have been used
†Corresponding author. E-mail: laiyun@nju.edu.cn
‡Corresponding author. E-mail: xzliu@nju.edu.cn
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