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He et al. Microstructures 2023;3:2023037 https://dx.doi.org/10.20517/microstructures.2023.29 Page 9 of 24

Figure 2. ML for the design of band structure in infinite meta-structures. (A) Combining GAN and CNN to realize inverse design of
digitally coded metamaterials with anticipated band structures [94] . Reproduced with the permission of Ref. [94] Copyright 2022,
Elsevier. (B) Design phononic crystals with anticipated bandgaps by combining AE and MLP [99] . Reproduced with the permission of
Ref. [99] Copyright 2020, Elsevier. (C) Design phononic crystals with anticipated bandgaps by combining GA and MLP [100] . Reproduced
with the permission of Ref. [100] Copyright 2023, Taylor & Francis. (D) Employ MLP to design lightweight meta-structures with low-
[101]
frequency broadband vibration isolation functions .

optimal wave attenuation . It is worth noting that this work characterizes the wave attenuation ability of
[95]
the structure through complex band structures. Although the design process focuses on the real part of the
band structures, the final screening process from alternative structures is achieved by comparing the
imaginary part of band structures.

Tailoring bandgaps
From an application perspective, it is usually not necessary to determine the complete band structures but
only focuses on the wave information provided by the bandgaps in the band structures. In contrast, the
design based on bandgaps simplifies the difficulty of model training and has a stronger design purpose, so a
lot of work has been carried out in this area.

Liu et al. employed MLP and TNN to achieve inverse design from bandgaps to structures for the layered
[96]
phononic crystals . The basic conclusion is that for inverse design with single (filling fraction) or dual
(shear modulus ratio and mass density ratio) parameters, MLP and TNN perform equally. However, for
inverse design with three parameters, TNN has obvious advantages, while MLP has difficulty in
convergence. This is due to the increase in the number of design parameters deepens the nonlinearity of the
mapping, leading to the gradual exposure of data inconsistency issues. As mentioned in the introduction,
TNN can effectively solve this training bottleneck. Dong et al. proposed using GA to optimize MLP
architecture for fast prediction of bandgap width . The starting point of this study is to serve as an efficient
[97]
means to avoid the significant computational costs required for repeated finite element analysis of elastic
meta-structures. Wu et al. explored a design and optimization scheme of modular metamaterial using
[98]
ML . In their work, modular metamaterials are composed of a certain number of four candidate materials
through different configurations to form phononic crystals. By using GA and MLP, they realize the optimal
configuration design of one-dimensional and two-dimensional (2D) modular metamaterial according to the
bandgap target. Li et al. combined an AE with MLP to achieve 2D phononic crystal design with anticipated
bandgaps, as shown in Figure 2B . The RVE of phononic crystals is generated through random functions,
[99]
and the band structure data are obtained through a finite element method. The implementation of this
design consists of three steps. Firstly, the AE is trained to extract the topological features of the RVE