TY - CONF

T1 - BLASCHKE PRODUCT NEURAL NETWORK (BPNN)

T2 - 10th International Conference on Learning Representations, ICLR 2022

AU - Dong, Juncheng

AU - Ren, Simiao

AU - Deng, Yang

AU - Khatib, Omar

AU - Malof, Jordan

AU - Soltani, Mohammadreza

AU - Padilla, Willie

AU - Tarokh, Vahid

N1 - Funding Information:
This work was supported in part by the Office of Naval Research (ONR) under grant number N00014-21-1-2590. YD, OK, SR, JM, and WJP acknowledge funding from the Department of Energy under U.S. Department of Energy (DOE) (DESC0014372).
Publisher Copyright:
© 2022 ICLR 2022 - 10th International Conference on Learning Representationss. All rights reserved.

PY - 2022

Y1 - 2022

N2 - Numerous physical systems are described by ordinary or partial differential equations whose solutions are given by holomorphic or meromorphic functions in the complex domain. In many cases, only the magnitude of these functions are observed on various points on the purely imaginary jω-axis since coherent measurement of their phases is often expensive. However, it is desirable to retrieve the lost phases from the magnitudes when possible. To this end, we propose a physics-infused deep neural network based on the Blaschke products for phase retrieval. Inspired by the Helson and Sarason Theorem, we recover coefficients of a rational function of Blaschke products using a Blaschke Product Neural Network (BPNN), based upon the magnitude observations as input. The resulting rational function is then used for phase retrieval. We compare the BPNN to conventional deep neural networks (NNs) on several phase retrieval problems, comprising both synthetic and contemporary real-world problems (e.g., metamaterials for which data collection requires substantial expertise and is time consuming). On each phase retrieval problem, we compare against a population of conventional NNs of varying size and hyperparameter settings. Even without any hyper-parameter search, we find that BPNNs consistently outperform the population of optimized NNs in scarce data scenarios, and do so despite being much smaller models. The results can in turn be applied to calculate the refractive index of metamaterials, which is an important problem in emerging areas of material science.

AB - Numerous physical systems are described by ordinary or partial differential equations whose solutions are given by holomorphic or meromorphic functions in the complex domain. In many cases, only the magnitude of these functions are observed on various points on the purely imaginary jω-axis since coherent measurement of their phases is often expensive. However, it is desirable to retrieve the lost phases from the magnitudes when possible. To this end, we propose a physics-infused deep neural network based on the Blaschke products for phase retrieval. Inspired by the Helson and Sarason Theorem, we recover coefficients of a rational function of Blaschke products using a Blaschke Product Neural Network (BPNN), based upon the magnitude observations as input. The resulting rational function is then used for phase retrieval. We compare the BPNN to conventional deep neural networks (NNs) on several phase retrieval problems, comprising both synthetic and contemporary real-world problems (e.g., metamaterials for which data collection requires substantial expertise and is time consuming). On each phase retrieval problem, we compare against a population of conventional NNs of varying size and hyperparameter settings. Even without any hyper-parameter search, we find that BPNNs consistently outperform the population of optimized NNs in scarce data scenarios, and do so despite being much smaller models. The results can in turn be applied to calculate the refractive index of metamaterials, which is an important problem in emerging areas of material science.

UR - http://www.scopus.com/inward/record.url?scp=85150393242&partnerID=8YFLogxK

M3 - Paper

AN - SCOPUS:85150393242

Y2 - 25 April 2022 through 29 April 2022

ER -