Inverse Design Problem Research Articles

Compact 200 line MATLAB code for inverse design in photonics by topology optimization: tutorial

We provide a compact 200 line MATLAB code demonstrating how topology optimization (TopOpt) as an inverse design tool may be used in photonics, targeting the design of two-dimensional dielectric metalenses and a metallic reflector as examples. The physics model is solved using the finite element method, and the code utilizes MATLAB’s fmincon algorithm to solve the optimization problem. In addition to presenting the code itself, we briefly discuss a number of extensions and provide the code required to implement some of these. Finally, we demonstrate the superiority of using a gradient-based method compared to a genetic-algorithm-based method (using MATLAB’s ga algorithm) for solving inverse design problems in photonics. The MATLAB software is freely available in the paper and may be downloaded from https://www.topopt.mek.dtu.dk.

Tackling Photonic Inverse Design with Machine Learning.

Machine learning, as a study of algorithms that automate prediction and decision‐making based on complex data, has become one of the most effective tools in the study of artificial intelligence. In recent years, scientific communities have been gradually merging data‐driven approaches with research, enabling dramatic progress in revealing underlying mechanisms, predicting essential properties, and discovering unconventional phenomena. It is becoming an indispensable tool in the fields of, for instance, quantum physics, organic chemistry, and medical imaging. Very recently, machine learning has been adopted in the research of photonics and optics as an alternative approach to address the inverse design problem. In this report, the fast advances of machine‐learning‐enabled photonic design strategies in the past few years are summarized. In particular, deep learning methods, a subset of machine learning algorithms, dealing with intractable high degrees‐of‐freedom structure design are focused upon.

Computational design of innovative mechanical metafilters via adaptive surrogate-based optimization

Architected materials and metamaterials are a challenging frontier for the development of optimal design strategies targeted at the active and passive control of elastic wave propagation. Within this research field, the microstructural optimization of mechanical metamaterials for achieving desired spectral functionalities may require considerable computational resources. Based on this motivating framework, the present paper illustrates a machine learning methodology to attack the inverse design problem concerning the optimization of the dispersion properties characterizing a novel layered mechanical metamaterial, conceived starting from the bi-tetrachiral periodic topology. Specifically, an adaptive technique is adopted to surrogate and maximize the objective function purposely defined to determine the optimal beam lattice microstructure characterized by the largest stop bandwidth at the lowest centerfrequency (low-cutting mechanical metafilter). The technique is computationally efficient in identifying the existing optimal solution in the physically admissible parameter space. The designed bi-tetrachiral metamaterial provides satisfying broadband low-frequency filtering performances, not achievable by the component tetrachiral layers.

A Nonlinear Inverse Design Problem for a Pipe Type Heat Exchanger Equipped with Internal Z-Shape Lateral Fins and Ribs

A non-linear three-dimensional inverse shape design problem was investigated for a pipe type heat exchanger to estimate the design variables of continuous lateral ribs on internal Z-shape lateral fins for maximum thermal performance factor η. The design variables were considered as the positions, heights, and number of ribs while the physical properties of air were considered as a polynomial function of temperature; this makes the problem non-linear. The direct problem was solved using software package CFD-ACE+, and the Levenberg–Marquardt method (LMM) was utilized as the optimization tool because it has been proven to be a powerful algorithm for solving inverse problems. Z-shape lateral fins were found to be the best thermal performance among Z-shape, S-shape, and V-shape lateral fins. The objective of this study was to include continuous lateral ribs to Z-shape lateral fins to further improve η. Firstly, the numerical solutions of direct problem were solved using both polynomial and constant air properties and then compared with the corrected solutions to verify the necessity for using polynomial air properties. Then, four design cases, A, B, C and D, based on various design variables were conducted numerically, and the resultant η values were computed and compared. The results revealed that considering continuous lateral ribs on the surface of Z-shape lateral fins can indeed improve η value at the design working condition Re = 5000. η values of designs A, B and C were approximately 13% higher than that for Z-shape lateral fins, however, when the rib numbers were increased, i.e., design D, the value of η became only 11.5 % higher. This implies that more ribs will not guarantee higher η value.

Interpretable Forward and Inverse Design of Particle Spectral Emissivity Using Common Machine-Learning Models

Author(s): Elzouka, M; Yang, C; Albert, A; Prasher, RS; Lubner, SD | Abstract: Radiative particles are ubiquitous in nature and in various technologies. Calculating radiative properties from known geometry and designs can be computationally expensive, and trying to invert the problem to come up with designs specific to desired radiative properties is even more challenging. Here, we report a machine-learning (ML)-based method for both the forward and inverse problem for dielectric and metallic particles. Our decision-tree-based model is able to provide explicit design rules for inverse problems. Furthermore, we can use the same trained model for both the forward and the inverse problem, which greatly simplifies the computation. Our methodology shows the promise of augmenting optical design optimizations by providing interpretable and actionable design rules for rapidly finding approximate solutions for the inverse design problem. Inverse design is usually done by expensive, slow, iterative optimization. Elzouka et al. show how a simple machine-learning model (decision trees) can efficiently perform inverse design in one shot, while recovering design rules understandable by humans. Inverse design of the spectral emissivity of particles is used as an example.

Hierarchical combinatorial design and optimization of non-periodic metamaterial structures

Abstract As advanced manufacturing techniques like additive manufacturing have enabled the fabrication of complex geometries, it is worthwhile to explore potential techniques that can be used to design metamaterial structures to achieve desired behaviors or physical properties. Designing the fine-scale geometry of a metamaterial structure to perform a specific behavior defines an inverse problem that is often intractable. To make this inverse problem tractable, most researchers represent metamaterial structures relying on a single-type periodic unit cell that is repeated in regular grids. Such a representation simplifies modeling and simulation tasks but at the cost of conceivably limiting the spectrum of physical properties that can be achieved by using more than one type of unit cells. This paper defines a non-periodic representation via implicit functions that generate two types of unit cells in pairs, which cope with the geometric frustration of interface connection in non-periodic metamaterial design. To solve the inverse problem of non-periodic metamaterial design, we propose a genetic algorithm-based combinatorial optimization scheme to design light-weight and high-strength functional parts by optimizing relevant parameters in the generating functions of non-periodic unit cells.

Applying Machine Learning to the Optics of Dielectric Nanoblobs

Dielectric nanostructures are the basic building blocks for photonic metasurfaces exhibiting designer optical responses. As their optical responses are nonintuitive, design procedures often consider only primitive geometries such as circles, ellipses, and rectangles. Despite these simplified geometries, achieving a target response still requires the forward design problem of solving Maxwell's equations to build a database of geometric parameters and their spectral responses. Herein, this work aims to leverage on the strength of deep neural networks (DNN) in image recognition to tackle the intractable inverse design problem of complex geometries, in which geometric parameters cannot be extracted. The work focuses on nanoblob geometries, i.e., irregular structures with rounded corners. When given a desired spectral response, the work investigates the ability of a well‐trained DNN in generating suitable geometries of the dielectric nanostructure and a corresponding source polarization.

A Novel Approach of Unsteady Adjoint Lattice Boltzmann Method Based on Circular Function Scheme

A new approach of combination of the LBM and adjoint method based on the circular function idea is developed for optimization of unsteady incompressible/compressible inviscid flow in 1D and 2D problems. The circular function distribution is used for capturing the compressibility effect in the flowfield and the continuous adjoint method allows designers to implement large number of design variables in actual optimization problems. The adjoint approach is successfully derived for the first time based on the compressible LBM with terminal conditions for estimation of the cost function gradient vector in 1D/2D flow inverse design problem. To validate the new derived flow solver with new lattice, firstly, accuracy of the flow simulation is shown for the inviscid compressible discontinuous flows with shock wave. Secondly, for validation of novel derived adjoint optimization algorithm, three optimization problems in form of the inverse design, including the smooth and shock wave inviscid compressible flowfields, are presented. Also, trend of optimization algorithm is studied in all cases. The results indicate that the presented numerical optimization approach gives desirable accuracy in 1D/2D inverse design of inviscid unsteady compressible flowfields.

Application of the Homotopy Method for Fractional Inverse Stefan Problem

The paper presents an application of the homotopy analysis method for solving the one-phase fractional inverse Stefan design problem. The problem was to determine the temperature distribution in the domain and functions describing the temperature and the heat flux on one of the considered area boundaries. It was demonstrated that if the series constructed for the method is convergent then its sum is a solution of the considered equation. The sufficient condition of this convergence was also presented as well as the error of the approximate solution estimation. The paper also includes the example presenting the application of the described method. The obtained results show the usefulness of the proposed method. The method is stable for the input data disturbances and converges quickly. The big advantage of this method is the fact that it does not require discretization of the area and the solution is a continuous function.

Fast design of plasmonic metasurfaces enabled by deep learning

Metasurfaces is an emerging field that enables the manipulation of light by an ultra-thin structure composed of sub-wavelength antennae and fulfills an important requirement for miniaturized optical elements. Finding a new design for a metasurface or optimizing an existing design for a desired functionality is a computationally expensive and time consuming process as it is based on an iterative process of trial and error. We propose a deep learning (DL) architecture dubbed bidirectional autoencoder for nanophotonic metasurface design via a template search methodology. In contrast with the earlier approaches based on DL, our methodology addresses optimization in the space of multiple metasurface topologies instead of just one, in order to tackle the one to many mapping problem of inverse design. We demonstrate the creation of a Geometry and Parameter Space Library (GPSL) of metasurface designs with their corresponding optical response using our DL model. This GPSL acts as a universal design and response space for the optimization. As an example application, we use our methodology to design a multi-band gap-plasmon based half-wave plate metasurface. Through this example, we demonstrate the power of our technique in addressing the non-uniqueness problem of common inverse design. Our network converges aptly to multiple metasurface topologies for the desired optical response with a low mean absolute error between desired optical response and the optical response of topologies searched. Our proposed technique would enable fast and accurate design and optimization of various kinds of metasurfaces with different functionalities.

An inverse shape design problem in determining the optimal snowflake-shaped fins

A three-dimensional shape design problem is considered to determine the optimal snowflake-shaped fins (SSF), based on the minimization of maximum domain temperature of fin. The Levenberg-Marquardt method (LMM) and software package CFD-ACE+ are used as the design tools. The SSF can be obtained by modifying the helm-shaped fins (HSF), the fine surfaces of HSF can be increased by splitting central part of the extended bodies from the center line outward and perforations can therefore be formed. The validity of CFD-ACE+ is first performed to verify the accuracy of the numerical solutions, and two categories of test cases are examined in the present study. The estimated optimal SSF are then compared with the HSF. Results indicate that the optimal SSF can achieve lower maximum fin temperature than HSF since (i) for single IHS problems, the decrease percentages of dimensionless maximum thermal resistance (DMTR) of SSF vary from 0.182% to 11.553% for various fin heights when compared with the values of HSF, and (ii) For multiple IHS problems, the decrease percentages of DMTR can be as large as 14.358% and 17.631% for two and four IHSs, respectively, as convective heat transfer coefficient a = 0.1.

A survey of the state-of-the-art swarm intelligence techniques and their application to an inverse design problem

This paper encompasses a detailed review of state-of-the-art swarm-based algorithms, with a focus on their applications along with a discussion on the merits and limitations of each algorithm. Further, a recently developed advanced particle swarm optimization (APSO) algorithm is compared with the different state-of-the-art algorithms through solving an electromagnetic inverse problem. Results show that the APSO algorithm outperforms the other algorithms. This research provides a scientific guideline for the comparison of different swarm-based algorithms and their utilization regarding specific applications.

Optimization of depth-graded multilayer structure for x-ray optics using machine learning

We present a general machine-learning-based approach to solve the inverse design problem of depth-graded multilayer structures (so-called supermirrors) for x-ray optics. Our model uses Monte Carlo tree search (MCTS) with policy gradient in combination with a reflectivity simulation. MCTS is an iterative design method that showed competitive efficiency in materials design and discovery problems. A policy gradient algorithm with a neural network was added to optimize the tree expansion. The policy gradient is a reinforcement learning method that optimizes parametrized policies toward an expected return using gradient descent. This approach is applied to design a depth-graded multilayer structure that maximizes mean reflectivity in an angular range for Cu Kα radiation by selecting the optimal thickness and material for each layer in the structure. Mean reflectivity of 0.80 was achieved in an angular range of 0.45–0.55°. Alternating materials are selected from a predetermined set of materials. We confirmed that the policy gradient enhances the efficiency of MCTS. This approach can be applied autonomously on several x-ray applications without any parameter tuning or pre-available data.

A reduced-order model for gradient-based aerodynamic shape optimisation

This work presents a reduced order model for gradient based aerodynamic shape optimization. The solution of the fluid Euler equations is converted to reduced Newton iterations by using the Least Squares Petrov-Galerkin projection. The reduced order basis is extracted by Proper Orthogonal Decomposition from snapshots based on the fluid state. The formulation distinguishes itself by obtaining the snapshots for all design parameters by solving a linear system of equations. Similarly, the reduced gradient formulation is derived by projecting the full-order model state onto the subspace spanned by the reduced basis. Auto-differentiation is used to evaluate the reduced Jacobian without forming the full fluid Jacobian explicitly during the reduced Newton iterations. Throughout the optimisation trajectory, the residual of the reduced Newton iterations is used as an indicator to update the snapshots and enrich the reduced order basis. The resulting multi-fidelity optimisation problem is managed by a trust-region algorithm. The ROM is demonstrated for a subsonic inverse design problem and for an aerofoil drag minimization problem in the transonic regime. The results suggest that the proposed algorithm is capable of aerodynamic shape optimization while reducing the number of full-order model queries and time to solution with respect to an adjoint gradient based optimisation framework.

Machine learning and evolutionary algorithm studies of graphene metamaterials for optimized plasmon-induced transparency

Machine learning and optimization algorithms have been widely applied in the design and optimization for photonics devices. We briefly review recent progress of this field of research and show data-driven applications, including spectrum prediction, inverse design and performance optimization, for novel graphene metamaterials (GMs). The structure of the GMs is well-designed to achieve the wideband plasmon induced transparency (PIT) effect, which can be theoretically demonstrated by using the transfer matrix method. Some traditional machine learning algorithms, including k nearest neighbour, decision tree, random forest and artificial neural networks, are utilized to equivalently substitute the numerical simulation in the forward spectrum prediction and complete the inverse design for the GMs. The calculated results demonstrate that all algorithms are effective and the random forest has advantages in terms of accuracy and training speed. Moreover, evolutionary algorithms, including single-objective (genetic algorithm) and multi-objective optimization (NSGA-II), are used to achieve the steep transmission characteristics of PIT effect by synthetically taking many different performance metrics into consideration. The maximum difference between the transmission peaks and dips in the optimized transmission spectrum reaches 0.97. In comparison to previous works, we provide a guidance for intelligent design of photonics devices based on machine learning and evolutionary algorithms and a reference for the selection of machine learning algorithms for simple inverse design problems.

Definition and Exploration of the Integrated CO2 Mineralization Technological Cycle

This paper is part of a multi-disciplinary research program on development and application of an integrated CO2 mineralization (ICM) framework for development of carbon mineralization as a CO2 mitigation solution. ICM is viewed as a three concentric layer system: technological, industrial integration, and decision-making. The search for viable ICM solutions in a given societal and economic context, which could be posed as an inverse design problem, begins with the identification and characterization of every system component. As an early writing on the development and applicability of the proposed ICM framework, this contribution focuses on ICM's inner technological layer. Several technological pathways, each one defined as a set of processing and transformation steps that connect a feedstock to a specific marketable product, can coexist within this layer. The paper addresses the characterization of one such technological pathway, whose cycle is divided into three successive blocks: feedstock, carbonation and valorization. The proposed concepts are illustrated through the valorization of ferronickel slag from New Caledonia as supplementary cementitious material or cement constituent, a case study that targets the production of “greener” construction materials. The data presented in the paper confirm the feasibility of characterizing the chosen ICM technological pathway, giving credit to the proposition that ICM can be approached as an inverse design problem. While exemplifying the significance of the characterization work necessary for one particular ICM technological pathway, the paper argues that development of ICM requires working on a scale considerably larger than that of standard mineral carbonation process research. Indeed, where grams of carbonated products are sufficient to investigate mineral carbonation processes, kilograms are mandatory to test and validate the use performance of final marketable products. Without precluding the merits of seeking innovative solutions, the authors argue that unit operations and transformation processes whose validity is proven at an industrial scale should be favored for timely development of viable ICM solutions.

Aerodynamic-driven topology optimization of compliant airfoils

A strategy for density-based topology optimization of fluid-structure interaction problems is proposed that deals with some shortcomings associated to non stiffness-based design. The goal is to improve the passive aerodynamic shape adaptation of highly compliant airfoils at multiple operating points. A two-step solution process is proposed that decouples global aeroelastic performance goals from the search of a solid-void topology on the structure. In the first step, a reference fully coupled fluid-structure problem is solved without explicitly penalizing non-discreteness in the resulting topology. A regularization step is then performed that solves an inverse design problem, akin to those in compliant mechanism design, which produces a discrete-topology structure with the same response to the fluid loads. Simulations are carried out with the multi-physics suite SU2, which includes Reynolds-averaged Navier-Stokes modeling of the fluid and hyper-elastic material behavior of the geometrically nonlinear structure. Gradient-based optimization is used with the exterior penalty method and a large-scale quasi-Newton unconstrained optimizer. Coupled aerostructural sensitivities are obtained via an algorithmic differentiation based coupled discrete adjoint solver. Numerical examples on a compliant aerofoil with performance objectives at two Mach numbers are presented.

Inverse design of broadband highly reflective metasurfaces using neural networks

Metamaterials exhibit optical properties not observed in traditional materials. Such behavior emerges from the interaction of light with precisely engineered subwavelength features built from different constituent materials. Recent research into the design and fabrication of metamaterial-based devices has established a foundation for the next generation of functional materials. Of particular interest is the all-dielectric metasurface, a two-dimensional metamaterial exploiting shape-dependent resonant features while avoiding losses through the use of dielectric building blocks. However, even this simple metamaterial class has a nearly infinite number of possible configurations; researchers now require new methods to efficiently explore these types of design spaces. In this work, we employ rigorous coupled wave analysis to calculate reflection and transmission spectra associated with a class of open-cylinder all-dielectric metasurface. By altering the geometric parameters of open-cylinder metasurfaces, we generate a sparse training data set and construct artificial neural networks capable of relating metasurface geometries to reflection and transmission spectra. Here, we successfully demonstrate that pseudo autodecoder neural networks can suggest device geometries based on a requested optical performance---inverting the design process for this metasurface class. As an example, we query for and discover a particular open-cylinder metasurface displaying a reflection band $R\ensuremath{\ge}99%$ centered at ${\ensuremath{\lambda}}_{0}=1550\phantom{\rule{0.16em}{0ex}}\mathrm{nm}$ that is much broader $\mathrm{\ensuremath{\Delta}}\ensuremath{\lambda}=450\phantom{\rule{0.16em}{0ex}}\mathrm{nm}$ than anything reported for optical metasurfaces. We then analyze the modal interplay in the open-cylinder metasurface to better understand the underlying physics driving the broadband behavior. Ultimately, we conclude that neural networks are ideally suited for generally approaching these types of complex inverse design problems.

Direct steering of de novo molecular generation with descriptor conditional recurrent neural networks

Deep learning has acquired considerable momentum over the past couple of years in the domain of de novo drug design. Here, we propose a simple approach to the task of focused molecular generation for drug design purposes by constructing a conditional recurrent neural network (cRNN). We aggregate selected molecular descriptors and transform them into the initial memory state of the network before starting the generation of alphanumeric strings that describe molecules. We thus tackle the inverse design problem directly, as the cRNNs may generate molecules near the specified conditions. Moreover, we exemplify a novel way of assessing the focus of the conditional output of such a model using negative log-likelihood plots. The output is more focused than traditional unbiased RNNs, yet less focused than autoencoders, thus representing a novel method with intermediate output specificity between well-established methods. Conceptually, our architecture shows promise for the generalized problem of steering of sequential data generation with recurrent neural networks. The rise of deep neural networks allows for new ways to design molecules that interact with biological structures. An approach that uses conditional recurrent neural networks generates molecules with properties near specified conditions.

Azimuthal multiplexing 3D diffractive optics

Diffractive optics have increasingly caught the attention of the scientific community. Classical diffractive optics are 2D diffractive optical elements (DOEs) and computer-generated holograms (CGHs), which modulate optical waves on a solitary transverse plane. However, potential capabilities are missed by the inherent two-dimensional nature of these devices. Previous work has demonstrated that extending the modulation from planar (2D) to volumetric (3D) enables new functionalities, such as generating space-variant functions, multiplexing in the spatial or spectral domain, or enhancing information capacity. Unfortunately, despite significant progress fueled by recent interest in metasurface diffraction, 3D diffractive optics still remains relatively unexplored. Here, we introduce the concept of azimuthal multiplexing. We propose, design, and demonstrate 3D diffractive optics showing this multiplexing effect. According to this new phenomenon, multiple pages of information are encoded and can be read out across independent channels by rotating one or more diffractive layers with respect to the others. We implement the concept with multilayer diffractive optical elements. An iterative projection optimization algorithm helps solve the inverse design problem. The experimental realization using photolithographically fabricated multilevel phase layers demonstrates the predicted performance. We discuss the limitations and potential of azimuthal multiplexing 3D diffractive optics.