Deterministic and probabilistic deep learning models for inverse design of broadband acoustic cloak

Concealing an object from incoming waves (light and/or sound) remained science fiction for a long time due to the absence of wave-shielding materials in nature. Yet, the invention of artificial materials and new physical principles for optical and sound wave manipulation translated this abstract concept into reality by making an object acoustically invisible. Here, we present the notion of a machine learning-driven acoustic cloak and demonstrate an example of such a cloak with a multilayered core-shell configuration. Importantly, we develop deterministic and probabilistic deep learning models based on autoencoder-like neural network structure to retrieve the structural and material properties of the cloaking shell surrounding the object that suppresses scattering of sound in a broad spectral range, as if it was not there. The probabilistic model enhances the generalization ability of design procedure and uncovers the sensitivity of the cloak parameters on the spectral response for practical implementation. This proposal opens up new avenues to expedite the design of intelligent cloaking devices for tailored spectral response and offers a feasible solution for inverse scattering problems.

In the last decade, acoustic cloaks via scattering cancellation 19,20 has become a topic of interest due to their robust designs, operating spectral range and ease of fabrication. In such schemes, isotropic layers of specific thickness, mass density and bulk modulus can be carefully tailored to cancel the first few scattering orders, which significantly reduce the scattering cross-section of the system, to make the object nearly-undetectable at a particular frequency. Consequently, the scattering cancellation approach generally employs acoustic metamaterials to realize ondemand cloaking devices. However, practical applications often desire more flexibility in the operating frequency band and require designing materials with positive physical properties (density and bulk modulus). Yet, the design of cloaking shells operating over broad frequency ranges with realistic material parameters remains a challenging endeavor. For instance, the broadband cloaking operation requires some additional layers in the core-shell configuration to cancel the higher scattering orders and, as a consequence, the design complexity grows and thus makes it extremely challenging to tune the geometry and material properties with conventional optimization techniques 21 . To mitigate such issues, data-driven approaches based on machine learning have provided a promising platform where artificial neural networks are trained to intelligently learn the intrinsic relation between various structural parameters and their spectral responses, and significantly reduce the overall computational time by predicting the solution immediately after the training phase 22,23 .
Despite such significant advancement in this area, the reported studies to date mostly emphasize on solving inverse electromagnetic problems in a deterministic fashion, while robust deep learning models for inverse acoustic scattering problems are yet to be developed. Here, we propose deep learning models as a practical tool to design broadband acoustic cloaks using a core-shell configuration. The proposed model utilizes fully connected DNNs to capture and generalize the nonlinear intricate relation between the design parameters and the spectral response for the forward and the inverse problem [see Fig.1(a)]. The implementation of the forward problem is straightforward, and consists in training the neural network that maps the design parameters directly to the spectral response. Yet, the inverse design is intrinsically challenging due to non-uniqueness of the solution and inherent convergence problems 30,31 . To address these issues, we design an autoencoder-like structure consisting of two DNNs where the pretrained forward network is cascaded behind the inverse neural network that maps the spectral response to the design parameters in either a deterministic [See Fig.1(b)] or a probabilistic manner [See Fig.1(c)]. The deterministic inverse design network provides only one set of design parameters for a given spectral response. However, a practical implementation generally demands more flexibility and diversity in the design due to external perturbation.
Hence, we introduce a novel model to provide probabilistic distributions of the design parameters, which flexibly generates the desired spectral response [see Fig.1(c)]. Fig. 1 Framework of the deep learning network for inverse design of the acoustic cloak. a Schematic illustration of the core-shell acoustic cloak and its spectral response (ratio between total scattering cross-section spectra of the cloak cloak and that of the object, i.e., object ) where the neural network learns the relation from (design parameters) to ℛ ( spectral response ) and from ℛ to for forward and inverse design, respectively. b-c Proposed deep learning models for inverse design of the cloak. b Deterministic model where the pretrained forward network acts as decoder to predict the spectral response. c Probabilistic model where the design space is transformed into the latent space z, with a standard Gaussian distribution. The physical design parameters are sampled from that distribution in the form of latent variables to generate the desired spectral response.
The probabilistic design with parameter distributions is more advanced than the deterministic design with fixed parameters on the refection of twofold benefits (i) the capability to generating a variety of design parameters for one desired spectral response and (ii) the ability to uncovering the sensitivity of the design parameters on the cloaking effect. We find that the bulk modulus is less sensitive to external perturbation than the thickness of the layers in designing the acoustic cloaks. We also perform numerical simulations with the finite element method (FEM) to confirm the theoretical cloaking predictions.

Results:
Deep learning model. To study acoustic invisibility, we consider a four-layered shell configuration where each layer is parametrized with different material and outer radius as shown in Fig. 1a. The properties of the cylindrical scatterer are described by its radius , volume mass density , and bulk modulus , while the cloak's layers are sequentially numbered as =1,2,3,4 to represent the outer radius and material properties ( , ) of the mth layer.
In order to analyze the scattering response of this system, we make use of the transfer matrix method (TMM) and compute the total scattering cross-section (SCS) spectra [See Methods section for details]. To quantify the cloak's performance, we define the ratio of the SCS spectra of the cloaked object cloak and the bare object object , i.e., cloak object ⁄ (or normalized SCS). This ratio reveals how well the object becomes acoustically invisible with the presence of the designed cloak. The ideal cloaking behavior is achieved by optimizing the design parameters to yield a SCS as close to zero as possible at the operation frequency. In our study, we consider, without loss of generality, a cylindrical scatterer with parameters: = 1 m, = and the inverse model to map the spectrum ℛ to the design . Both networks are trained by optimizing the neural network weights. For our analysis, we generate 68000 data samples for random design parameters, which are split into three distinct groups: 60000 data samples for training, 4000 data samples for validation, and 4000 data samples for final testing. The training data is used to train the network by optimizing the neural network weights, while the validation data set serves for checking and avoiding the overfitting issue, and the testing data set examines the prediction accuracy of the network.
Forward-modeling network. We first design the forward-modeling network to accurately predict the frequency-dependent SCS for given design parameters. The forward model builds a fully connected network between the design space as the input layer and SCS spectra ℛ as the output layer, as shown in Fig. 1(a). We normalize the data before training, to expedite the convergence of the network. In the training process, the training data is fed into the network and the weights are continuously optimized to minimize the loss function defined as ℒ = We predict the spectral response of the trained network for testing data and compare the results with those obtained by the TMM. We report the relative absolute spectral error on the testing data sets as: = ∑ | −̂|⁄ where is the discretized value for the target spectral response and ̂ is the corresponding predicted spectral response. The results for relative spectral error are plotted in Fig. 2(b) with a mean error of 3.47% for testing data, which proves the high prediction accuracy (over 96%) of the network. Figures (c)-(e) depict the spectral response of three representative cases from the testing data that clearly indicate the predicted results perfectly match with those of the TMM. Deterministic inverse-modeling network. Typically, practical applications demand cloaking effect at a particular frequency or broad frequency band, which necessitates the experimentally realizable design parameters to produce the desired spectral response. Yet, there is no common tool available for the accurate inverse design of clocking devices. The development of such tools significantly reduces the computational time for design optimization and accelerates the generation of the desired SCS spectra. To achieve this goal, we attempt to train the network inversely, which takes the spectral response as the input and the design parameters as the output.
However, we are unable to train the network successfully in the inverse direction due to nonunique solutions in response-to-design mapping that leads to non-convergence issues [See the SI for more details]. To resolve this issue, we implement the autoencoder-like network where we cascade the inverse network to the independently trained (or pretrained) forward-modeling network, as shown in Fig. 1(b). The forward network is trained separately to substitute the TMM simulation and acts as a data generator in training. During the training process, the pretrained forward network has fixed weights and biases while the weights of the inverse network are updated iteratively to minimize the loss function to the output response of designed structure predicted by neural network. The designed structure refers to the intermediate layer in the autoencoder-like network that predicts the eight-design parameter for the desired spectral response.   Probabilistic inverse-modeling network. In the deterministic inverse design, we choose one precise set of design parameters to generate the desired spectral response, yet, the practical implementation demands diversity in design parameters due to the possible unavailability of actual materials and unexpected deviations in the original design parameters. In this scenario, it is essential to enhance the generalization and robustness of our network by introducing the probabilistic prediction. To achieve this goal, we propose the stochastic inverse design that uses the latent space concept 34 for the probabilistic representation of the physical design parameters.
Our probabilistic inverse design network is basically a generative model which transforms the input spectral response into a mean vector and a variance vector 2 to approximate the distribution of the latent variables corresponding to the design space, and then the pretrained forward network, acting as a decoder, generates the same spectral response as the input by sampling latent variable vector from the Gaussian prior distribution illustrated in Fig.1(c).
The loss function for the probabilistic inverse network consists of mean absolute error for reconstruction ℒ , a Kullback-Leibler (KL) divergence error ℒ , and a precision parameter, i.e., the inverse of the variance = 1 2 ⁄ . The KL divergence ensures the generated latent space distributions follow the assumed Gaussian distribution, (0,1), and also 9 provides the discrepancy between the predicted distributions and the standard normal distribution. The precision term is incorporated to avoid the zero-variance problem for the generated distributions of the latent space. The model is trained to minimize the following loss function: where N is the total number of training samples, is the weight of probabilistic learning, and is the regularization parameter. These hyper parameters can be tuned by cross-validation during the training process. The detail of the derived cost function is provided in the Methods section.  properties, which can be scaled depending on the choice of the host medium and scatterer. For example, for a scatterer of size of 10 cm immersed in water, the designed cloak perfectly works over a broad spectrum ranging from a few Hz to 5 kHz. To demonstrate this idea, we consider a specific core material to achieve broadband cloaking by tuning the material and geometry of the four layered core-shell system. However, the approach can be applied to design invisible cloaks for any given core in such systems.
To summarize, we demonstrate the machine learning driven broadband acoustic cloak with multilayer core-shell configuration. In particular, we develop deterministic and probabilistic deep learning models for inverse design of acoustic cloak that efficiently solves the inverse design problem. The proposed models utilize the encoder-decoder like structure to solve the onmany mapping problem and retrieve the design parameters for the given spectral response. The where 0 is the wavenumber in the host medium, sca represents the complex scattering coefficient, is the Bessel function of order n and 1 is the Hankel function of the first kind and of order n. For convenience, we introduce the parameter being 1 for = 0 and 2 for ≥ 1 , that stretch the summation index from one to infinity, but needs to be truncated for convergence.
The pressure field in each cloaking layer is expressed using both Bessel functions, as there is no singularity (i.e., ≠ 0) computed from the components of T-matrix is used to determine the SCS that estimates the total power scattered by the cloaked cylinder, and is hence a measure of its far-field 'visibility', given as, To cloak the object, we need to minimize the SCS by searching the suitable bulk modulus and thickness of the cloaking layers. In the data generation process, the spectral response computed from cloak / object is the anticipated output. We apply this approach for data generation due to higher computational efficiency and accuracy as compared to finite element and finite difference time domain numerical methods.
Probabilistic design. The probabilistic inverse model provides the distribution of the design parameters, given the spectral response and the latent variable . This model uses the variational inference approach to approximate the distribution from which the latent variable is drawn.
Therefore, the loss function is modified in comparison to the deterministic case. The loss function contains the KL divergence term that guarantees our learned distribution ( | ) to be analogous to the predefined distribution ( ), which is assumed to be Gaussian, for each dimension of the latent space. In addition, we include the precision parameter to avoid the zero variance issue arose in biased data. The modified loss function for the stochastic network is defined as where the superscript donates the ℎ training sample, is the dimensionality of the latent space z, is the weight of loss on KL-divergence, and is the regularization parameter. For the Gaussian prior distribution (0,1), the precision parameter is the inverse of the variance ( ) = 1/ 2 and the KL divergence between two Gaussian distributions can be expressed as

Supplemental Material
In this supplementary material, we discuss non-convergence and overfitting issue in the inverse problem, details about optimized hyper parameters and presents more examples for the designed deterministic and stochastic inverse networks.

Overfitting problem in designing direct inverse network
As discussed in the main manuscript, the inverse design involves training of the network to retrieve the design parameters from the spectral response. Nevertheless, the direct training of inverse network suffers from overfitting problem that is caused due to one-to-many possible mapping and, as a result, the inverse function does not converge. Figure   composed of 100-500-500-500-400-8 nodes.

Optimal setting of hyper parameters for the designed forward and inverse network
The training results of the neural network is largely determined by the structure of the network and every hyper parameter. After various tests with the same training data samples, the hyper parameters can be fixed for the forward and inverse networks, which are shown in Table S1.

Design parameters for the cases considered in the main text
The design parameters of the results discussed in the main text are summarized in Table S2. In all cases, the densities of the designed layers are = [0.308, 1.961, 1.557, 0.579]. Note that we provide the normalized parameters which can be easily transformed to realistic parameters depending the on the choice of the host medium.

Additional examples for forward and inverse design network
We present some additional results from the testing data to show the accuracy of our designed forward and inverse networks.

III. Probabilistic inverse design:
In probabilistic design, the Gaussian distributions for the design parameters are determined for a given spectral response. As we showed in the main text, the spectral response are categorized on the basis of the standard deviation in generated distribution of designs parameters. Here, we show some examples where different sets of bulk modulus are used to generate the same spectral response.