Deep Convolutional Mixture Density Network for Inverse Design of Layered Photonic Structures

Abstract

Machine learning (ML) techniques, such as neural networks, have emerged as powerful tools for the inverse design of nanophotonic structures. However, this innovative approach suffers some limitations. A primary one is the nonuniqueness problem, which can prevent ML algorithms from properly converging because vastly different designs produce nearly identical spectra. Here, we introduce a mixture density network (MDN) approach, which models the design parameters as multimodal probability distributions instead of discrete values, allowing the algorithms to converge in cases of nonuniqueness without sacrificing degenerate solutions. We apply our MDN technique to inversely design two types of multilayer photonic structures consisting of thin films of oxides, which present a significant challenge for conventional ML algorithms due to a high degree of nonuniqueness in their optical properties. In the 10-layer case, the MDN can handle transmission spectra with high complexity and under varying illumination conditions. The 4-layer case tends to show a stronger multimodal character, with secondary modes indicating alternative solutions for a target spectrum. The shape of the distributions gives valuable information for postprocessing and about the uncertainty in the predictions, which is not available with deterministic networks. Our approach provides an effective solution to the inverse design of photonic structures and yields more optimal searches for the structures with high degeneracy and spectral complexity.

Global Inverse Design across Multiple Photonic Structure Classes Using Generative Deep Learning

“…Other types of networks that fall in this category include variational autoencoders (VAEs) [ 52 ] and mixture density networks (MDNs). [ 53 ] Recent developments in GAN technology have led to numerous GAN‐variants, including but not limited to: the Self‐Attention GAN (SAGAN), [ 36 ] Deep Regret Analytic GAN (DRAGAN), [ 37 ] StyleGAN, [ 38 ] Wasserstein GAN (WGAN), [ 39 ] and the Least Squares GAN (LSGAN). [ 40 ] Here, as an initial proof of concept, we tested our framework using a modified cDCGAN architecture, as shown in Figure a.…”

“…We note that our definition of the L G differs from the original GAN implementation, where log[1– D ( G ( z,y ))] is minimized instead, since this was shown to not provide sufficient gradients. [ 53,54 ] To improve the performance of the cDCGAN, we applied one‐sided label smoothing and mini‐batch discrimination. [ 44,45 ] Unlike previous cDCGAN implementations, our approach relies on adversarial training without explicitly guiding the generator towards known images, [ 21 ] thereby achieving a greater degree of generalization that is unconstrained by pre‐existing images.…”

Probabilistic parameter estimation using a Gaussian mixture density network: application to X-ray reflectivity data curve fitting

“…An MDN enables probabilistic prediction by obtaining the conditional probability density pðx j j R j l ; wÞ of the output for a given input (Bishop, 1994;Unni et al, 2020;McLachlan & Basford, 1988). Unlike a general FCP, which gives only one best-fit parameter set for a given input as an output, an MDN produces values expressing the probability density distribution of parameters as an output.…”

“…However, when using a general FCP or a single-Gaussian density network, only one best-fit parameter is given for each parameter. Therefore, this network is not a problem if it satisfies one-to-one correspondence between input and output training data, whereas the relationship between the input and output is difficult to predict when there are two or more outputs for one input (Carmona-Loaiza & Raza, 2021;Unni et al, 2020). However, in the case of an MDN that uses several Gaussian mixtures, when there are several outputs for a given input, it can be expressed in the form of a probability distribution of several peaks.…”

A deep learning model for the topological design of 2D periodic wave barriers

“…Deep learning (DL) is a method based on artificial neural networks (ANNs) with representation learning, and it has been extensively researched and applied in civil engineering, such as damage detection (Cha et al., 2017; Zhang et al., 2017), health monitoring (Azimi & Pekcan, 2020; Ni et al., 2020; Rafiei & Adeli, 2018), and vibration control (Gutierrez Soto & Adeli, 2019). For periodic structures, DL also shows its capabilities both in solving forward and inverse problems in the field of electromagnetic waves, and great achievements have been made in the past 4 years, such as forward prediction (Christensen et al., 2020; da Silva Ferreier et al., 2018; Qu, et al., 2019), parameter design (Liu et al., 2018; Ma et al., 2018; Unni et al., 2020), and topological configuration design (Ma et al., 2021; So & Rho, 2019; Wang et al., 2020). While little work has been done on the intelligent prediction and design of periodic structures in the field of elastic waves, the present authors have realized the forward prediction and inverse design of 1D periodic structures using multilayer perceptrons (MLPs) and auto‐encoders (AEs; Liu & Yu, 2019; Liu et al., 2019).…”