Math @ Duke
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Publications [#361495] of Vahid Tarokh
Papers Published
- Dong, J; Ren, S; Deng, Y; Khatib, O; Malof, J; Soltani, M; Padilla, W; Tarokh, V, Blaschke Product Neural Networks (BPNN): A Physics-Infused Neural
Network for Phase Retrieval of Meromorphic Functions
(November, 2021)
(last updated on 2023/06/01)
Abstract: 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 jw-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.
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