HyperHyperNetworks for the Design of Antenna Arrays Shahar Lutati 1 Lior Wolf 1 2 Abstract Designing antennas is a challenging inverse problem: while mapping the structure of the antenna to its radiation prop- We present deep learning methods for the design erties is possible (but inefficient) by numerically solving of arrays and single instances of small antennas. Maxwell’s equations, the problem of obtaining a struc- Each design instance is conditioned on a target ture that produces the desired radiation pattern, subject radiation pattern and is required to conform to to structural constraints, can only be defined as an opti- specific spatial dimensions and to include, as part mization problem, with a large search space and various of its metallic structure, a set of predetermined trade-offs (Wheeler, 1975). Our novel approach for design- locations. The solution, in the case of a single ing a Printed Circuit Board (PCB) antenna that produces antenna, is based on a composite neural network the desired radiation pattern, resides in a 3D bounding box, that combines a simulation network, a hypernet- and includes a predefined set of metallic locations. We then work, and a refinement network. In the design present a method for the design of an antenna array that of the antenna array, we add an additional de- combines several such antennas. sign level and employ a hypernetwork within a hypernetwork. The learning objective is based The single antenna method first trains a simulation network on measuring the similarity of the obtained radia- h that replaces the numerical solver, based on an initial tion pattern to the desired one. Our experiments training set obtained using the solver. This network is used demonstrate that our approach is able to design to rapidly create a larger training set for solving the inverse novel antennas and antenna arrays that are com- problem, and, more importantly, to define a loss term that pliant with the design requirements, considerably measures the fitting of the obtained radiation pattern to the better than the baseline methods. We compare desired one. The design networks that solve the inverse the solutions obtained by our method to existing problem include a hypernetwork (Ha et al., 2016) f that is designs and demonstrate a high level of overlap. trained to obtain an initial structure, which is defined by the When designing the antenna array of a cellular functional g. This structure is then refined by a network phone, the obtained solution displays improved t that incorporates the metallic locations and obtains the properties over the existing one. final design. For the design of an antenna array, on top of the parameters of each antenna, it is also necessary to determine the number of antennas and their position. For this task, we introduce the hyper-hypernetwork framework, 1. Introduction in which an outer hypernetwork q determines the weights Since electronic devices are getting smaller, the task of of an inner hypernetwork f, which determines the weights designing suitable antennas is becoming increasingly impor- of the primary network g. arXiv:2105.03838v1 [cs.LG] 9 May 2021 tant (Anguera et al., 2013). However, the design of small Our experiments demonstrate the success of the trained mod- antennas, given a set of structural constraints and the de- els in producing solutions that comply with the geometric sired radiation pattern, is still an iterative and tedious task constraints and achieve the desired radiation pattern. We (Miron & Miron, 2014). Moreover, to cope with an increas- demonstrate that both the hypernetwork f and the refine- ing demand for higher data rates in dynamic communication ment network t are required for the design of a single an- channels, almost all of the current consumer devices include tenna and that the method outperforms the baseline methods. antenna arrays, which adds a dimension of complexity to In the case of multiple antennas, the hyperhypernetwork, the design problem (Bogale & Le, 2016). which consists of networks q, f, g, outperforms the baseline 1Tel Aviv University 2Facebook AI Research. Correspon- methods on a realistic synthetic dataset. Furthermore, it is dence to: Shahar Lutati <[email protected]>, Lior Wolf able to predict the structure of real-world antenna designs <[email protected]>. (Chen & Lin, 2018; Singh, 2016) and to suggest an alterna- th tive design that has improved array directivity for the iPhone Proceedings of the 38 International Conference on Machine 11 Pro Max. Learning, PMLR 139, 2021. Copyright 2021 by the author(s). HyperHyperNetworks for the Design of Antenna Arrays 2. Related Work ple, making it too slow to support an efficient search for a geometry given the desired radiation pattern, i.e., solve the Misilmani & Naous(2019) survey design methods for large inverse problem. Additionally, since it is non-differentiable, antennas, i.e., antennas the size of λ/2−λ/4, where λ is the its usage for optimizing the geometry is limited. corresponding wavelength of their center frequency. Most of the works surveyed are either genetic algorithms (Liu Therefore, although our goal is to solve the inverse problem, et al., 2014) or SVM based classifiers (Prado et al., 2018). we first build a simulation network h. This network is used None of the surveyed methods incorporates geometrical to support a loss term that validates the obtained geometry, constraints, which are crucial for the design of the small and to propagate gradients through this loss. The simulation antennas we study, due to physical constraints. network h is given two inputs (i) the scale in terms of wave- length S and (ii) a 3D voxel-based description of the spatial A limited number of attempts were made in the automatic structure of the metals V . h returns a 2D map U describing design of small antennas, usually defined by a scale that the far-field radiation pattern, i.e., U = h(S; V ). is smaller than λ/10 (Bulus, 2014). Hornby et al.(2006); Liu et al.(2014) employ genetic algorithms to obtain the Specifically, S 2 R3 specifies the arena limits. This is target gain. Santarelli et al.(2006) employ hierarchical given as the size of the bounding box of the metal structure, Bayesian optimization with genetic algorithms to design in units of the wavelength λ corresponding to the center an electrically small antenna and show that the design ob- frequency. V is a voxel grid of size 64 × 64 × 16, which is tained outperforms classical man-made antennas. None of sampled within the 3D bounding box dimensions provided these methods employ geometric constraints, making them by S. In other words, it represents a uniform sampling on unsuitable for real-world applications. They also require a grid with cells of size [S1=64;S2=64;S3=16]. The lower running the antenna simulation over and over again during resolution along the z axis stems from the reduced depth of the optimization process. Our method requires a one-time many mobile devices. Each voxel contains a binary value: investment in creating a training dataset, after which the 0 for nonmetallic materials, 1 for conducting metal. The design process itself is very efficient. output tensor is a 2D “image” U(θ; φ), sampled on a grid of size 64 × 64, each covering a region of π=64 × 2π=64 A hypernetwork scheme (Ha et al., 2016) is often used squared arc lengths. The value in each grid point denotes to learn dynamic networks that can adjust to the in- the radiation power in this direction. put (Bertinetto et al., 2016; von Oswald et al., 2020) through multiplicative interactions (Jayakumar et al., 2020). It con- The directivity gain D = N (U) is a normalized version of tains two networks, the hypernetwork f, and the primary U: U(θ; φ) network g. The weights θg of g are generated by f based on D(θ; φ) = (1) R 2π R π f’s input. We use a hypernetwork to recover the structure φ=0 θ=0 U(θ; φ) sin(θ) dθ dφ of the antenna in 3D. Hypernetworks were recently used to obtain state of the art results in 3D reconstruction from a The design network solves the inverse problem, i.e., map single image (Littwin & Wolf, 2019). from the required antenna’s directivity gain D to a repre- sentation of the 3D volume V . We employ a hypernetwork We present multiple innovations when applying hypernet- scheme, in which an hypernetwork f receives the design pa- works. First, we are the first, as far as we can ascertain, rameters D and S and returns the weights of an occupancy to apply hypernetworks to complex manufacturing design network g. g : [0; 1]3 ! [0; 1] is a multi-layered perceptron problems. Second, we present the concept of a hyperhyper- (MLP) that maps a point p in 3D, given in a coordinate network, in which a hypernetwork provides the weights of system in which each dimension of the bounding box S is another hypernetwork. Third, we present a proper way to between zero and one, into the probability o of a metallic initialize hyperhypernetworks, as well as a heuristic for se- material at point p. lecting which network weights should be learned as conven- tional parameters and which as part of the dynamic scheme θg = f(D; S) ; o = g(p; θg) (2) offered by hypernetworks. The weights of the hypernetwork f are learned, while the weights of the primary network g are obtained as the output 3. Single Antenna Design of f. Therefore g, which encodes a 3D shape, is dynamic Given the geometry of an antenna, i.e. the metal structure, and changes based on the input to f.