Blaschke Product Neural Networks (BPNN): A Physics-Infused Neural Network for Phase Retrieval of Meromorphic Functions

by   Juncheng Dong, et al.

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.


Real-time 3D Nanoscale Coherent Imaging via Physics-aware Deep Learning

Phase retrieval, the problem of recovering lost phase information from m...

The Levin approach to the numerical calculation of phase functions

The solutions of scalar ordinary differential equations become more comp...

Physics informed neural networks for continuum micromechanics

Recently, physics informed neural networks have successfully been applie...

NeuPDE: Neural Network Based Ordinary and Partial Differential Equations for Modeling Time-Dependent Data

We propose a neural network based approach for extracting models from dy...

Δ-PINNs: physics-informed neural networks on complex geometries

Physics-informed neural networks (PINNs) have demonstrated promise in so...

Practical Phase Retrieval Using Double Deep Image Priors

Phase retrieval (PR) concerns the recovery of complex phases from comple...

Please sign up or login with your details

Forgot password? Click here to reset