Fig 4 - uploaded by Chen-Xu Liu

Content may be subject to copyright.

# | Topology design ANNs for PnCs and EMs. a | The AE-based model by Li et al. [106] where topological feature indicates compressed low-dimensional vector. b | The VAE-based model by Liu et al. [109] where the pretrained generative model is the trained VAE's decoder. c | The CGAN by Gurbuz et al. [113] where "TL" (conditions) is the transmission loss of the 2D EMs and "Z" is the input random variables. d | The CGAN by Jiang et al. [87] where the target refers to dispersion curves of 2D PnCs.

Source publication

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 artificial neural networks (ANNs) possesses the superiorities of rapidity and accuracy in design, making up for t...

## Contexts in source publication

**Context 1**

... et al. [106] combined an AE and a MLP to design topologies of 2D PnCs, as shown in Fig. 4a. The AE was responsible for compressing 128×128 topology matrices into low-dimensional vectors and inverse restoration. The MLP was responsible for building the mapping relationship from dispersion properties to the transformed low-dimensional vectors. A targeted dispersion property was input into the MLP to output a designed ...

**Context 2**

... The MLP was responsible for building the mapping relationship from dispersion properties to the transformed low-dimensional vectors. A targeted dispersion property was input into the MLP to output a designed low-dimensional vector, and then the vector was input into the AE's decoder to obtain a designed topology. In the model shown in Fig. 4a, the AE is for dimensionality reduction and data restoration, and ...

**Context 3**

... and discrete topology information into low-dimensional and continuous latent space consisting of latent vectors, and it can restore inversely. Liu et al. [108] proposed a VAE-based model to design topologies of 2D EMs. Two ANNs, a VAE and a TNN, were used, where the VAE's decoder and the TNN were combined to design topologies, as shown in Fig. 4b. Similar to the idea of Li et al. [106], the VAE is for dimensionality reduction and data restoration, and the TNN is for the design of latent vectors. Compared with a single MLP, the TNN has a more powerful design ability. In addition, Liu et al. added two equations about frequencies and sizes into the model to simultaneously design ...

**Context 4**

... to design latent vectors. The GAN is also able to generate new samples, although it is more difficult to train in comparison with the VAE. A single GAN can only deal with topology information, and structural properties must be considered additionally in design. Hence, Gurbuz et al. [113] used a conditional GAN (CGAN) to design 2D EMs, as shown in Fig. 4c. The conditions in the CGAN referred to the property (transmission loss) of the ...

**Context 5**

... 2021), and discrete approach (Dong et al., 2017). With the development of AI technology, the topology design based on deep learning has been explored. For PnCs and EMs, there are mainly three ANNs to design topologies, including AE-, VAE-, and GAN-based model. Li et al. (2020) combined an AE and an MLP to design topologies of 2D PnCs, as shown in Fig. 4a. The AE was responsible for compressing 128 × 128 topology matrices into low-dimensional vectors and inverse restoration. The MLP was responsible for building the relation from dispersion properties to the transformed low-dimensional vectors. A targeted dispersion property was input into the MLP to output a designed low-dimensional ...

**Context 6**

... inverse restoration. The MLP was responsible for building the relation from dispersion properties to the transformed low-dimensional vectors. A targeted dispersion property was input into the MLP to output a designed low-dimensional vector, and then the vector was input into the AE's decoder to obtain a designed topology. In the model shown in Fig. 4a, the AE is for dimensionality reduction and data restoration, and the MLP is for design, which is a clear and reasonable design idea. However, for more complicated design problems, this model may be invalid due to the function nature of the MLP, which was discussed in the above section, and it cannot realize 'one-to-many' ...

**Context 7**

... and discrete topology information into low-dimensional and continuous latent space consisting of latent vectors, and it can restore inversely. Liu and Yu (2022e) proposed a VAE-based model to design topologies of 2D EMs. Two ANNs, a VAE, and a TNN, were used, where the VAE's decoder and the TNN were combined to design topologies, as shown in Fig. 4b. Similar to the idea of Li et al. (2020), the VAE is for dimensionality reduction and data restoration, and the TNN is for the design of latent vectors. Compared with a single MLP, the TNN has a more powerful design ability. In addition, Liu and Yu (2022e) added two equations about frequencies and sizes into the model to simultaneously ...

**Context 8**

... GAN is also able to generate new samples, although it is more difficult to train in comparison with the VAE. A single GAN can only deal with topology information, and structural properties must be considered additionally in design. Hence, Gurbuz et al. (2021) used a CGAN to design 2D EMs, as shown in Fig. 4c. The conditions in the CGAN referred to the property (transmission loss) of the structure. When the CGAN was trained, the conditions were input into the generator and the discriminator, enabling the CGAN to learn the property information besides topologies. Then, a targeted property (condition) and a random vector were input into the ...

## Similar publications

We describe use of an optimization algorithm to produce three-dimensional, quasi-conformal transformation acoustics. The results indicate that the anisotropy of the transformed material can be made arbitrarily small by increasing the auxiliary function degrees of freedom. A boundary function is defined to prevent the algorithm from affecting the or...

## Citations

... [ Data-driven machine learning In current machine learning models for PMSMs, data-driven approach is the mainstream since it only requires data rather than complex physical equations. The data are normally generated by numerical calculation methods [156], such as the finite element method and plane wave expansion method. Machine learning models can be trained with these data. ...

Flexible manipulation of elastic and acoustic waves through phononic meta-structured materials (PMSMs) has attracted a lot of attention in the last three decades and shows a bright future for potential applications in many fields. Conventional engineering design methods for PMSMs rely on changing the material composition and empirical structural configurations, which often result in limited performance due to the limited design space. Recent advances in the fields of additive manufacturing, optimization, and artificial intelligence have given rise to a plethora of creative meta-structured materials that offer superior functionality on demand. In this Review, we provide an overview of inverse design of phononic crystals, phononic-crystal devices, phononic metamaterials, phononic metamaterial devices, phononic metasurfaces, and phononic topological insulators. We first introduce fundamental wave quantities including dispersion relations, scattering characterizations, and dynamic effective parameters, and then discuss how these wave quantities can be leveraged for systematic inverse design of PMSMs to achieve a variety of customized phononic functionalities with highly customizable full-wave responses, intrinsic physical parameters, and hybrid local–global responses. Furthermore, we show representative applications of some inverse-designed PMSMs and look at future directions. We outline the concept of phononic structures genome engineering (PSGE) through key developments in PMSM inverse design. Finally, we discuss the new possibilities that PSGE brings to wave engineering.

... Inspired by advances in artificial intelligence technology, the use of neural networks in the design of metamaterials has increasingly garnered attention. [10][11][12][13][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 neural network and a conventional densely connected neural network to predict the eigenvalues of phononic crystals and concluded that the convolutional neural network outperformed the conventional densely connected neural network. ...

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. Traditional 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.

... Examples include the correspondence 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. ...

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.

... Machine learning has opened up a promising avenue with efficiency and intelligence in science and engineering [66]. It has been instrumental in addressing a wide array of challenges across various domains, including medicine, chemistry, and mechanics [67][68][69]. In the field of origami structures, Zhang et al. [70] harnessed data-driven machine learning models for predicting structural mechanical behaviors. ...

Origami structures have the advantages of foldability and adjustability, with applications spanning numerous engineering fields. However, there remains a dearth of intelligent and convenient methods that can effectively tackle both potential energy prediction and design problems on origami structures. This study proposes a novel physics-informed neural network (PINN) to predict and design potential energy curves of Kresling origami structures without labelled data. A sorting operation is coupled into the PINN, ensuring the prediction correctness. The accuracy of the potential energy curves predicted by the PINN is demonstrated through comparison with a reference and the exhaustive method. A prediction only takes less than one second and the precision of the PINN significantly surpasses that of the exhaustive method, proving the extremely high efficiency and credibility of the PINN. Furthermore, two design cases for Kresling origami structures, matching a target potential energy curve and a set of target potential energy points, are performed. The designed structures meet the expectations and each design takes a few seconds, showing the efficiency and applicability of the PINN in inverse design. The presented physics-driven approach without labelled data offers an innovative tool with learning ability to predict and design. It also provides a valuable reference for the force and stiffness design of Kresling origami structures. In addition, the code of the PINN is shared online.

... Phononic crystals (PnCs) involve the arrangement of artificial unit cells in a periodic manner [1][2][3]. The acceleration of research in PnCs has greatly expanded the horizons for tailoring the transfer of elastic wave energy, including its quantity and orientation. ...

This research aims to advance the existing analytical model, which was based on the Euler–Bernoulli beam theory, for predicting the velocity amplitude of bending waves excited while amplified by defective phononic crystals (PnCs) with piezoelectric elements. The previous analytical model falls short for thick defective PnCs and is limited to low frequencies. Hence, the Timoshenko beam theory is newly incorporated to address these issues, considering shear deformations and rotational inertia. Its performance is validated against a finite element model. It confirms that the proposed analytical model outperforms its predecessor, which was suitable for thin defective PnCs with a slenderness ratio above 10 and frequencies around 2 kHz. Remarkably, the improved analytical model demonstrates excellent predictive capabilities up to 40 kHz for thick defective PnCs with a slenderness ratio of around 2.5. With a relative error of less than 1% compared to the finite element model, this advanced approach also demonstrates time-efficient and stable computational performance. This work introduces three contributions. First, it pioneers the proposal of an electromechanically coupled analytical model for predicting the performance of defect-mode-enabled bending wave excitation. Second, this enhanced analytical model surpasses existing counterparts in terms of both accuracy and computational stability, particularly for thick defective PnCs, even at high frequencies. Last, the analytical model dramatically reduces the computation time.

... It can be seen the number of publications has significantly increased in the past five years, and pertinent research has peaked in the last two years. Some review papers, including Jin et al. [7], Muhammad et al. [8], He et al. [9], as well as Liu and Yu [10], discussed the application of machine learning to the property prediction and inverse design of phononic metamaterials. Nonetheless, there is currently a dearth of in-depth reviews that elucidate the specific roles different machine learning models can fulfill in the field of phononic metamaterials. ...

Machine learning opens up a new avenue for advancing the development of phononic crystals and elastic metamaterials. Numerous learning models have been employed and developed to address various challenges in the field of phononic metamaterials. Here, we provide an overview of mainstream machine learning models applied to phononic metamaterials, discuss their capabilities as well as limitations, and explore potential directions for future research.

... The concept broadly refers to a design practice that builds on the common thread: discovering patterns from a finite collection of observational data, rather than domain knowledge, and then harnesses the patterns to expedite the otherwise arduous multiscale design procedure [22]. Evidenced by a growing volume of reviews from diverse perspectives [22][23][24][25][26][27][28][29][30], the new paradigm has drawn immense attention in a broad realm of engineering sciences, perhaps with the vision that it may possibly unlock the potential of metamaterials. ...

Multifunctional metamaterials (MMM) bear promise as next-generation material platforms supporting miniaturization and customization. Despite many proof-of-concept demonstrations and the proliferation of deep learning assisted design, grand challenges of inverse design for MMM, especially those involving heterogeneous fields possibly subject to either mutual meta-atom coupling or long-range interactions, remain largely under-explored. To this end, we present a data-driven design framework, which streamlines the inverse design of MMMs involving heterogeneous fields. A core enabler is implicit Fourier neural operator (IFNO), which predicts heterogeneous fields distributed across a metamaterial array, thus in general at odds with homogenization assumptions, in a parameter-/sample-efficient fashion. Additionally, we propose a standard formulation of inverse problem covering a broad class of MMMs, and gradient-based multitask concurrent optimization identifying a set of Pareto-optimal architecture-stimulus (A-S) pairs. Fourier multiclass blending is proposed to synthesize inter-class meta-atoms anchored on a set of geometric motifs, while enjoying training-free dimension reduction and built-it reconstruction. Interlocking the three pillars, the framework is validated for light-by-light programmable plasmonic nanoantenna, whose design involves vast space jointly spanned by quasi-freeform supercells, maneuverable incident phase distributions, and conflicting figure-of-merits involving on-demand localization patterns. Accommodating all the challenges without a-priori simplifications, our framework could propel future advancements of MMM.

... Jiang et al. [53] constructed a convolutional neural network (forward analysis model) that predicted dispersion curves and a generative adversarial network (inverse design model) that generated PnC designs for target dispersion curves. In 2023, Liu and Yu reported a literature review of deep-learning-enabled PnC designs [54]. Unfortunately, research on developing deep-learning-enabled PnCdesign methods is still in its infancy. ...

This paper proposes a deep-learning-based inverse design framework for a onedimensional, defect-introduced phononic crystal (PnC) as a narrow bandpass filter under longitudinal elastic waves. The purpose of the design optimization problem is to maximize the transmittance at the defect-band frequency, which is equal to the target frequency. The framework comprises three steps: (i) inverse design generation and filtering, (ii) forward analysis of frequencies and filtering, and (iii) forward analysis of transmittance and optimal design selection. Four deep-learning models are considered in the inverse model: a deep neural network, a tandem neural network, a conditional variation autoencoder, and a conditional generative adversarial network. The frameworks developed with each deep-learning model are evaluated using a test dataset and an arbitrarily defined defect-band frequency and phononic band-gap range. The results show that the frameworks proposed using the conditional variation autoencoder and the conditional generative adversarial network effectively present the best performance by solving the nonunique response-to-design mapping problem through probabilistic approaches. The deep-learning-based framework reduces the need for manual intervention and simplifies the inverse design process, making it a promising approach for finding the optimal design solution for the use of defect-introduced PnCs as narrow bandpass filters.

... Ma et al., 2021), mechanics (C. X. Liu & Yu, 2023), and engineering (Giglioni et al., 2023;Saneii et al., 2023;Song et al., 2023). These methods can generate topologies within seconds using neural networks instead of numerical simulations. ...

A novel deep learning‐based optimization (DLBO) methodology is proposed for rapidly optimizing phononic crystal‐based metastructure topologies. DLBO eliminates the need for pre‐optimized data by leveraging the learned relation from metastructure features to bandgaps. It enables optimization based on qualitative/quantitative descriptions and forms a regular generalization domain to avoid misjudgments. DLBO achieves similar or better results to genetic algorithm (GA) and only requires 0.01% of the time GA costs. Metastructures with different periodic constants and filling fractions are also optimized, offering insights for balancing space, material, and vibration isolation. Based on a newly defined objective function, an economical metastructure is customized for subway‐induced vibrations; and its performance on vibration isolation is verified through a 3D finite element model. Additionally, the datasets and codes in this study are shared.

... The recent advances in deep learning and similar techniques from an area of artificial intelligence enable prediction of the bandgap structures and even inverse design of the metamaterials with desired behavior. 27,28 Every predictive network requires an adequate selection of the input and output variables. While here an obvious input is just a four-element vector y ¼ ðy 1 ; y 2 ; y 3 Þ defining the geometry of the Bezier spline, we considered two possible options for output variables. ...

... used the concept of bandgap vector. [27][28][29] The considered frequency range 0 < x x max was split into n ¼ 500 narrow frequency bands x max i=n x i < x max ði þ 1Þ=n, where 0 i < n for each band, if no wavevectors at the perimeter of IBZ had corresponding normalized frequency inside the band, and then the value of 1 representing stopband was assigned. Otherwise, the band was represented by the value of 0 and classified as a passband, resulting in a 500-element bandgap vector representing a whole passband-stopband structure. ...

This Letter discusses elastic metamaterials incorporating curved beams in their architecture. Through employing Bezier splines, we reveal a wide versatility of geometrical designs of the unit cells and the consequent programmability of bandgap structures. By analyzing more than ten thousand possible specimens altogether, we highlight the similarity between dynamic properties of metamaterials formed by curves with different geometries defined via three variables only that correspond to the coordinates of control points of the Bezier spline. In particular, we establish the importance of such parameter as effective curve length in defining the probable positions of bandgaps. This study shows, in particular, that the bandgap ratio can reach 71% for metamaterials with proposed curved beams—a noticeable contrast with no bandgaps in their counterpart with straight elements. The employment of the deep learning model enables us to effectively predict passband–stopband structure in such metamaterials with satisfactory accuracy, potentially accelerating the design of metamaterials assembled from versatile unit cells.