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 optically and 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. We develop deterministic and probabilistic deep learning models based on autoencoderlike 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's parameters on the spectral response for practical implementation. This proposal opens up avenues to expedite the design of intelligent cloaking devices for tailored spectral response and offers a feasible solution for inverse scattering problems.

Figure 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 ${\sigma }_{\mathrm{cloak}}$ and that of the object, i.e., ${\sigma }_{\mathrm{object}}$) where the neural network learns the relation from $\mathcal{D}$ (design parameters) to $\mathcal{R}$ (spectral response) and from $\mathcal{R}$ to $\mathcal{D}$ 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 a 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.

Figure 2

Forward network learning of acoustic cloak. (a) Learning curves for training and validation data sets as a function of training epochs. (b) Histogram of relative spectral error for testing samples. The red vertical dashed line shows the mean spectral error. (c)–(e) Comparison of the spectral response for three representative examples obtained by machine learning model and TMM. The shaded area shows the absolute error between the predicted and target response, which indicates that the machine learning results are in perfect agreement with those of TMM. The design parameters are provided in the SM [45].

Figure 3

Deterministic inverse design of acoustic cloak. (a) Learning curves for training and validation data sets as functions of the training epochs. (b) Histogram of the relative spectral error for the testing data samples. The red vertical dashed line shows the mean spectral error. (c)–(e) Comparison of the spectral response for three representative cases obtained with machine learning, TMM, and comsol. These results clearly show that machine learning accurately predicts the target response. The designed responses require positive mass density and bulk modulus values that are provided in the SM [45].

Figure 4

Numerical simulations for designed broadband acoustic cloak. (a) Pressure field distribution produced by an incident plane wave for the bare object and (b) the cloaked object. The three different normalized frequencies $k_{0}a=1,1.5,2$ are picked from the spectral response shown in Fig. 2.

Figure 5

Probabilistic inverse design of acoustic cloak. (a) Learning curves for the training and validation data sets over training epoch. (b) Classification of testing samples based on maximum standard deviation (±2% and ±4% deviation from the mean value) in the generated distributions of design parameters. Bulk modulus parameters in design space exhibit large standard deviation. (c) Classification of testing samples based on ±2% and ±4% deviation from the mean value in the generated distributions of different combinations of bulk modulus parameters. (d)–(g) Gaussian distributions of bulk moduli showing maximum standard deviation for the representative cases. Comparison of the spectral responses generated from randomly sampled bulk moduli in the corresponding distribution (dashed red and green curves) and reference response (solid blue curve).