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Page 10 of 24 He et al. Microstructures 2023;3:2023037 https://dx.doi.org/10.20517/microstructures.2023.29

configuration. Secondly, an MLP model is trained to describe the relationship between the anticipated
bandgap and topological features. Finally, the encoder in the AE is replaced with MLP. Miao et al.
conducted another study on the design of 2D phononic crystals described by random functions, as shown in
[100]
Figure 2C . In this work, they first employed MLP to predict the bandgap and then used MLP combined
with GA to achieve the inverse design of the structure. In the inverse design scheme, GA is taken as the
main body, and the fitness function is constructed with the predicted bandgap of MLP and the target
bandgap in the iterative process, and the optimal individual that can adapt to the target bandgap is obtained
through iteration.

In terms of meta-structure design with high-quality vibration reduction function, Jin et al. employed MLP
to inverse design the Archimedes spiral meta-structure with deep subwavelength vibration isolation
[101]
function, as shown in Figure 2D . The double-layer corrugated core sandwiched structure between two
spiral plates can provide low-frequency bandgaps through a local resonance mechanism. However, it is
difficult to analyze the relationship between the bandgap and the parameters of the spiral plate. The trained
MLP model avoided the analytical process of inverse design and obtained a structure with low-frequency
broadband vibration isolation performance, which showed good consistency with the experiment. On et al.
modified the TNN architecture and realized the design of arch meta-structure with anticipated bandgap
vibration reduction function . Specifically, they inverted the pre-trained forward network and inverse
[102]
design network in traditional TNN, where the input is a structural parameter, while the intermediate layer
outputs the bandgap frequency. After training, preserving the inverse network of the backend can achieve
the design of bandgap frequencies to structures.

The application of RL in band structures is mainly to maximize the bandgap width or optimize the specific
range and focuses on one-dimensional structures with analytical dispersion relation. According to the
analytical dispersion relation of layered phononic crystals, Luo et al. used RL to optimize the component
[103]
widths and realized two functions: maximizing the bandgap width and customizing the bandgap range .
Wu et al. employed RL to optimize the masses of one-dimensional atomic chains to achieve custom
bandgaps . He et al. analyzed the longitudinal wave dispersion of periodically variable cross-section
[104]
[105]
beams and optimized three length parameters using RL to achieve maximum bandgap width .
Design of wave propagation characteristics in finite meta-structures
Different from the ideal infinite period meta-structures, the meta-structures in practical engineering can
only be composed of periodic or aperiodic finite distributions. Analyzing the propagation characteristics of
acoustic/elastic waves in finite meta-structures is an important step toward achieving practical engineering
applications for meta-structures. In this section, we review the finite meta-structure design works around
propagation characteristics. Table 2 provides a brief overview of ML for the design of wave propagation
characteristics in finite meta-structures.

Enhancing noise reduction
Arranging sound absorption structures is one of the main methods for controlling environmental noise,
which can be divided into porous sound absorption structures, resonant sound absorption structures, and
special sound absorption structures, and has been widely used. The ability of a structure to absorb sound
energy is usually characterized by calculating its sound absorption coefficient. Due to the complexity and
diversity of the structure, design is an important step to meet practical needs. Researchers have conducted
extensive explorations in this area using ML. For example, Donda et al. employed CNN to characterize the
[106]
acoustic absorption performance of acoustic absorbing metasurfaces . Subsequently, they implemented
the inverse design of the metasurface using CGAN . Zhang et al. realized the accelerated topological
[107]