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Willie Padilla

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3 papers

2 author rows

ICLR Conference 2022 Conference Paper

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

Juncheng Dong
Simiao Ren
Yang Deng
Omar Khatib
Jordan M. Malof
Mohammadreza Soltani
Willie Padilla
Vahid Tarokh

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 $j\omega$-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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NeurIPS Conference 2021 Conference Paper

Benchmarking Data-driven Surrogate Simulators for Artificial Electromagnetic Materials

Yang Deng
Juncheng Dong
Simiao Ren
Omar Khatib
Mohammadreza Soltani
Vahid Tarokh
Willie Padilla
Jordan Malof

Artificial electromagnetic materials (AEMs), including metamaterials, derive their electromagnetic properties from geometry rather than chemistry. With the appropriate geometric design, AEMs have achieved exotic properties not realizable with conventional materials (e. g. , cloaking or negative refractive index). However, understanding the relationship between the AEM structure and its properties is often poorly understood. While computational electromagnetic simulation (CEMS) may help design new AEMs, its use is limited due to its long computational time. Recently, it has been shown that deep learning can be an alternative solution to infer the relationship between an AEM geometry and its properties using a (relatively) small pool of CEMS data. However, the limited publicly released datasets and models and no widely-used benchmark for comparison have made using deep learning approaches even more difficult. Furthermore, configuring CEMS for a specific problem requires substantial expertise and time, making reproducibility challenging. Here, we develop a collection of three classes of AEM problems: metamaterials, nanophotonics, and color filter designs. We also publicly release software, allowing other researchers to conduct additional simulations for each system easily. Finally, we conduct experiments on our benchmark datasets with three recent neural network architectures: the multilayer perceptron (MLP), MLP-mixer, and transformer. We identify the methods and models that generalize best over the three problems to establish the best practice and baseline results upon which future research can build.

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NeurIPS Conference 2020 Conference Paper

Benchmarking Deep Inverse Models over time, and the Neural-Adjoint method

Simiao Ren
Willie Padilla
Jordan Malof

We consider the task of solving generic inverse problems, where one wishes to determine the hidden parameters of a natural system that will give rise to a particular set of measurements. Recently many new approaches based upon deep learning have arisen, generating promising results. We conceptualize these models as different schemes for efficiently, but randomly, exploring the space of possible inverse solutions. As a result, the accuracy of each approach should be evaluated as a function of time rather than a single estimated solution, as is often done now. Using this metric, we compare several state-of-the-art inverse modeling approaches on four benchmark tasks: two existing tasks, a new 2-dimensional sinusoid task, and a challenging modern task of meta-material design. Finally, inspired by our conception of the inverse problem, we explore a simple solution that uses a deep neural network as a surrogate (i. e. , approximation) for the forward model, and then uses backpropagation with respect to the model input to search for good inverse solutions. Variations of this approach - which we term the neural adjoint (NA) - have been explored recently on specific problems, and here we evaluate it comprehensively on our benchmark. We find that the addition of a simple novel loss term - which we term the boundary loss - dramatically improves the NA’s performance, and it consequentially achieves the best (or nearly best) performance in all of our benchmark scenarios.

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