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DEEP LEARNING OF FREE BOUNDARY AND STEFAN PROBLEMS

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arXiv 2020年

作者： Wang, Sifan Perdikaris, Paris Graduate Group in Applied Mathematics and Computational Science University of Pennsylvania PhiladelphiaPA19104 United States Department of Mechanichal Engineering and Applied Mechanics University of Pennsylvania PhiladelphiaPA19104 United States

Free boundary problems appear naturally in numerous areas of mathematics, science and engineering. These problems present a great computational challenge because they necessitate numerical methods that can yield an accurate approximation of free boundaries and complex dynamic interfaces. In this work, we propose a multi-network model based on physics-informed neural networks to tackle a general class of forward and inverse free boundary problems called Stefan problems. Specifically, we approximate the unknown solution as well as any moving boundaries by two deep neural networks. Besides, we formulate a new type of inverse Stefan problems that aim to reconstruct the solution and free boundaries directly from sparse and noisy measurements. We demonstrate the effectiveness of our approach in a series of benchmarks spanning different types of Stefan problems, and illustrate how the proposed framework can accurately recover solutions of partial differential equations with moving boundaries and dynamic interfaces. All code and data accompanying this manuscript are publicly available at https://***/PredictiveIntelligenceLab/DeepStefan. Copyright © 2020, The Authors. All rights reserved.

关键词： Numerical methods

On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networks

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arXiv 2020年

作者： Wang, Sifan Wang, Hanwen Perdikaris, Paris Graduate Group in Applied Mathematics and Computational Science University of Pennsylvania PhiladelphiaPA19104 United States Department of Mechanichal Engineering and Applied Mechanics University of Pennsylvania PhiladelphiaPA19104 United States

Physics-informed neural networks (PINNs) are demonstrating remarkable promise in integrating physical models with gappy and noisy observational data, but they still struggle in cases where the target functions to be approximated exhibit high-frequency or multi-scale features. In this work we investigate this limitation through the lens of Neural Tangent Kernel (NTK) theory and elucidate how PINNs are biased towards learning functions along the dominant eigen-directions of their limiting NTK. Using this observation, we construct novel architectures that employ spatio-temporal and multi-scale random Fourier features, and justify how such coordinate embedding layers can lead to robust and accurate PINN models. Numerical examples are presented for several challenging cases where conventional PINN models fail, including wave propagation and reaction-diffusion dynamics, illustrating how the proposed methods can be used to effectively tackle both forward and inverse problems involving partial differential equations with multi-scale behavior. All code an data accompanying this manuscript will be made publicly available at https://***/PredictiveIntelligenceLab/MultiscalePINNs. Copyright © 2020, The Authors. All rights reserved.

关键词： Partial differential equations

Localization and Mitigation of Loss in Niobium Superconducting Circuits

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PRX Quantum 2022年第2期3卷 020312-020312页

作者： M. Virginia P. Altoé Archan Banerjee Cassidy Berk Ahmed Hajr Adam Schwartzberg Chengyu Song Mohammed Alghadeer Shaul Aloni Michael J. Elowson John Mark Kreikebaum Ed K. Wong Sinéad M. Griffin Saleem Rao Alexander Weber-Bargioni Andrew M. Minor David I. Santiago Stefano Cabrini Irfan Siddiqi D. Frank Ogletree Molecular Foundry Division Lawrence Berkeley National Laboratory Berkeley California 94720 USA Quantum Nanoelectronics Laboratory Department of Physics University of California at Berkeley Berkeley California 94720 USA Materials Sciences Division Lawrence Berkeley National Laboratory Berkeley California 94720 USA Computational Research Division Lawrence Berkeley National Laboratory Berkeley California 94720 USA Graduate Group in Applied Science and Technology University of California at Berkeley Berkeley California 94720 USA Department of Physics King Fahd University of Petroleum and Minerals Dhahran 31261 Kingdom of Saudi Arabia Department of Materials Science and Engineering University of California at Berkeley Berkeley California 94720 USA

Materials imperfections in planar superconducting quantum circuits—in particular, two-level-system (TLS) defects—contribute significantly to decoherence, ultimately limiting the performance of quantum computation and sensing. The identification of specific parasitic layers and their associated loss contributions has, however, proven elusive. Using a combination of x-ray photoemission spectroscopy (XPS) and analytical scanning transmission electron microscopy (STEM), we determine the thickness, chemical composition, and location of the oxides present in niobium-on-silicon coplanar-waveguide (CPW) resonators and quantify their respective contributions to the measured single-photon quality factor (Q). Using selective chemical etching, we reduce first the substrate-air oxide then the metal-air oxide thickness, dramatically reducing both TLS (δTLS) and non-TLS (δhi) losses, resulting in a median Q value over 5×106, with individual devices approaching 6×106. We find that silicon surface oxides host 70% of TLS losses, with a δTLS:δhi loss-density ratio near 11:1. In contrast, niobium surface oxides host 77% of non-TLS losses, uniformly distributed within the oxide layer, with a δTLS:δhi loss-density ratio of 3:4 for the superconducting circuits investigated in this work. Only 7% of losses come from other sources, including the niobium-silicon interface, which is sufficiently clean in our devices to allow epitaxial Nb nucleation on Si. As we mitigate surface losses through selective modification of the interface dielectrics, we arrive in a regime where TLS losses are no longer dominant, which will allow other types of losses in superconducting circuits to be investigated in more detail, including nonequilibrium quasiparticles.

关键词： Quantum information with hybrid systems Low-temperature superconductors Surfaces Electron microscopy Microwave techniques Photoemission spectroscopy

Real-time Sampling and Estimation on Random Access Channels: Age of Information and Beyond

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arXiv 2020年

作者： Chen, Xingran Liao, Xinyu Bidokhti, Shirin Saeedi The Department of Electrical and System Engineering University of Pennsylvania PA19104 United States The Graduate Group of Applied Mathematics and Computational Science University of Pennsylvania PA19104 United States

Next generation multiple access channels require to provision for unprecedented massive user access in a plethora of applications in cyber-physical systems. This work proposes decentralized policies for the real-time monitoring and estimation of autoregressive processes over random access channels. Two classes of policies are investigated: (i) oblivious schemes in which sampling and transmission policies are independent of the processes that are monitored, and (ii) non-oblivious schemes in which transmitters causally observe their corresponding processes for decision making. In the class of oblivious policies, we show that minimizing the expected time-average estimation error is equivalent to minimizing the expected age of information. Consequently, we prove lower and upper bounds on the minimum achievable estimation error in this class. Next, we consider non-oblivious policies and design a threshold policy, called error-based thinning, in which each transmitter node becomes active if its instantaneous error has crossed a fixed threshold (which we optimize). Active nodes then transmit stochastically following a slotted ALOHA policy. A closed-form, approximately optimal, solution is found for the threshold as well as the resulting estimation error. It is shown that non-oblivious policies offer a multiplicative gain close to 3 compared to oblivious policies. Moreover, it is shown that oblivious policies that use age of information for decision making improve the state-of-the-art at least by the multiplicative factor 2. The performance of all discussed policies is compared using simulations. Numerical comparison shows that the performance of the proposed decentralized policy is very close to that of centralized greedy scheduling. Finally, we extend our framework to unreliable random access channels. Simulations show that the multiplicative gain offered by non-oblivious policies (compared to oblivious policies) is independent of the channel erasure probability. Copyrigh

关键词： Transmitters

Selecting the number of components in PCA via random signflips

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arXiv 2020年

作者： Hong, David Sheng, Yue Dobriban, Edgar Department of Electrical and Computer Engineering University of Delaware United States Graduate Group in Applied Mathematics and Computational Science University of Pennsylvania United States Department of Statistics and Data Science University of Pennsylvania United States

Principal component analysis (PCA) is a foundational tool in modern data analysis, and a crucial step in PCA is selecting the number of components to keep. However, classical selection methods (e.g., scree plots, parallel analysis, etc.) lack statistical guarantees in the increasingly common setting of large-dimensional data with heterogeneous noise, i.e., where each entry may have a different noise variance. Moreover, it turns out that these methods, which are highly effective for homogeneous noise, can fail dramatically for data with heterogeneous noise. This paper proposes a new method called signflip parallel analysis (FlipPA) for the setting of approximately symmetric noise: it compares the data singular values to those of "empirical null" matrices generated by flipping the sign of each entry randomly with probability one-half. We develop a rigorous theory for FlipPA, showing that it has nonasymptotic type I error control and that it consistently selects the correct rank for signals rising above the noise floor in the large-dimensional limit (even when the noise is heterogeneous). We also rigorously explain why classical permutation-based parallel analysis degrades under heterogeneous noise. Finally, we illustrate that FlipPA compares favorably to state-of-the art methods via numerical simulations and an illustration on data coming from astronomy. Copyright © 2020, The Authors. All rights reserved.

关键词： Principal component analysis

Which Activation Function Works Best for Training Artificial Pancreas: Empirical Fact and Its Theoretical Explanation

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Which Activation Function Works Best for Training Artificial...

IEEE Symposium Series on computational Intelligence (SSCI)

作者： Lehel Dénes-Fazakas László Szilágyi György Eigner Olga Kosheleva Martine Ceberio Vladik Kreinovich Physiological Controls Research Center Appl. Informatics & Appl. Math. Doctoral School Óbuda University Budapest Hungary Computational Intelligence Res. Group Sapientia University Tg. Mures Romania Physiological Controls Res. Center Óbuda University Budapest Hungary Physiological Controls Res. Center John von Neumann Faculty of Informatics Biomatics and Applied AI Institute Obuda University Budapest Hungary Department of Teacher Education University of Texas at El Paso El Paso Texas USA Department of Computer Science University of Texas at El Paso El Paso Texas USA

One of the most effective ways to help patients at the dangerous levels of diabetes is an artificial pancreas, a device that constantly monitors the patient's blood sugar level and injects insulin based on this level. Patient's reaction to insulin is highly individualized, so the artificial pancreas needs to be trained on each patient. It turns out that the best training results are attained when instead of the usual ReLU neurons, we use their minor modification known as Exponential Linear Units (ELU). In this paper, we provide a theoretical explanation for the empirically observed effectiveness of ELUs.

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Leveraging Undecided Cases in Chart-Reviewed Phenotypes to Enhance Ehr-Based Association Studies

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SSRN

SSRN 2024年

作者： Jian, Xinyao Zhang, Dazheng Yu, Zehao Xu, Hua Bian, Jiang Wu, Yonghui Tong, Jiayi Chen, Yong University of Pennsylvania Perelman School of Medicine PhiladelphiaPA United States Department of Biostatistics Epidemiology and Informatics Perelman School of Medicine The University of Pennsylvania PhiladelphiaPA United States Department of Health Outcomes and Biomedical Informatics College of Medicine University of Florida GainesvilleFL United States Section of Biomedical Informatics and Data Science School of Medicine Yale University New HavenCT United States Cancer Informatics Shared Resource University of Florida Health Cancer Center GainesvilleFL United States Department of Biostatistics Johns Hopkins Bloomberg School of Public Health BaltimoreMD United States The Graduate Group in Applied Mathematics and Computational Science School of Arts and Sciences University of Pennsylvania PhiladelphiaPA United States Leonard Davis Institute of Health Economics PhiladelphiaPA United States PhiladelphiaPA United States PhiladelphiaPA United States

Objectives: In electronic health record (EHR)-based association studies, phenotyping algorithms efficiently classify patient clinical outcomes into binary categories but are susceptible to misclassification errors. The gold standard, manual chart review, involves clinicians determining the true disease status based on their assessment of health records. These clinicians-labeled phenotypes are labor-intensive and typically limited to a small subset of patients, potentially introducing a third "undecided" category when phenotypes are indeterminate. We aim to effectively integrate the algorithm-derived and chart-reviewed outcomes when both are available in EHR-based association *** and Methods: We propose an augmented estimation method that combines the binary algorithm-derived phenotypes for the entire cohort with the trinary chart-reviewed phenotypes for a small, selected subset. Additionally, a cost-effective outcome-dependent sampling strategy is used to address the rare disease scenarios. The proposed trinary chart-reviewed phenotype integrated cost-effective augmented estimation (TriCA) was evaluated through a wide range of simulation settings and real-world applications in Alzheimer's disease and related dementias using EHR data from the OneFlorida+ Clinical Research ***: Compared to estimation based on random sampling, our augmented method improved mean square error by up to 43.5% in simulation studies;compared to estimation using trinary chart-reviewed phenotypes only, our method improved efficiency by up to 32.1% in the real-world data ***: Our simulation studies and real-world application demonstrate that, compared to existing methods, the proposed method provides unbiased estimates with higher statistical ***: The proposed method effectively combined binary algorithm-derived phenotypes for the whole cohort with trinary chart-reviewed outcomes for a limited validation set, making it applicable to

关键词： Electronic health record

A computational fluid dynamics model for the small-scale dynamics of wave, ice floe and interstitial grease ice interaction

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arXiv 2021年

作者： Marquart, Rutger Bogaers, Alfred Skatulla, Sebastian Alberello, Alberto Toffoli, Alessandro Nisters, Carina Vichi, Marcello Computational Continuum Mechanics Research Group Department of Civil Engineering University of Cape Town South Africa Ex Mente Technologies Pretoria South Africa School of Computer Science and Applied Mathematics University of Witwatersrand South Africa Graduate School of Frontier Sciences University of Tokyo Japan Department of Infrastructure Engineering University of Melbourne Australia Institute of Mechanics University of Duisburg-Essen Germany Department of Oceanography University of Cape Town South Africa

The marginal ice zone is a highly dynamical region where sea ice and ocean waves interact. Large-scale sea ice models only compute domain-averaged responses. As the majority of the marginal ice zone consists of mobile ice floes surrounded by grease ice, finer-scale modelling is needed to resolve variations of its mechanical properties, wave-induced pressure gradients and drag forces acting on the ice floes. A novel computational fluid dynamics approach is presented, that considers the heterogeneous sea ice material composition and accounts for the wave-ice interaction dynamics. Results show, after comparing three realistic sea ice layouts with similar concentration and floe diameter, that the discrepancy between the domain-averaged temporal stress and strain rate evolutions increases for decreasing wave period. Furthermore, strain rate and viscosity are mostly affected by the variability of ice floe shape and diameter. Copyright © 2021, The Authors. All rights reserved.

关键词： Drag

Testing Biased Randomization Assumptions and Quantifying Imperfect Matching and Residual Confounding in Matched Observational Studies

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arXiv 2021年

作者： Chen, Kan Heng, Siyu Long, Qi Zhang, Bo Graduate Group of Applied Mathematics and Computational Science School of Arts and Sciences University of Pennsylvania PhiladelphiaPA United States Department of Biostatistics School of Global Public Health New York University New York CityNY United States Department of Biostatistics Epidemiology and Informatics Perelman School of Medicine University of Pennsylvania PhiladelphiaPA United States Vaccine and Infectious Disease Division Fred Hutchinson Cancer Center SeattleWA United States

One central goal of design of observational studies is to embed non-experimental data into an approximate randomized controlled trial using statistical matching. Despite empirical researchers' best intention and effort to create high-quality matched samples, residual imbalance due to observed covariates not being well matched often persists. Although statistical tests have been developed to test the randomization assumption and its implications, few provide a means to quantify the level of residual confounding due to observed covariates not being well matched in matched samples. In this article, we develop two generic classes of exact statistical tests for a biased randomization assumption. One important by-product of our testing framework is a quantity called residual sensitivity value (RSV), which provides a means to quantify the level of residual confounding due to imperfect matching of observed covariates in a matched sample. We advocate taking into account RSV in the downstream primary analysis. The proposed methodology is illustrated by re-examining a famous observational study concerning the effect of right heart catheterization (RHC) in the initial care of critically ill patients. Code implementing the method can be found in the supplementary materials. © 2021, CC BY.

关键词： Statistical tests