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Antithetic Multilevel Methods for Elliptic and Hypo-Elliptic Diffusions with Applications

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arXiv 2024年

作者： Iguchi, Yuga Jasra, Ajay Maama, Mohamed Beskos, Alexandros Department of Statistical Science University College London LondonWC1E 6BT United Kingdom School of Data Science The Chinese University of Hong Kong Shenzhen China Applied Mathematics and Computational Science Program Computer Electrical and Mathematical Sciences and Engineering Division King Abdullah University of Science and Technology Thuwal23955-6900 Saudi Arabia

In this paper we present a new antithetic multilevel Monte Carlo (MLMC) method for the estimation of expectations with respect to laws of diffusion processes that can be elliptic or hypo-elliptic. In particular, we consider the case where one has to resort to time discretization of the diffusion and numerical simulation of such schemes. Motivated by recent developments, we introduce a new MLMC estimator of expectations, which does not require simulation of intractable Lévy areas but has a weak error of order 2 and achieves the optimal computational complexity. We then show how this approach can be used in the context of the filtering problem associated to partially observed diffusions with discrete time observations. We illustrate with numerical simulations that our new approaches provide efficiency gains for several problems relative to some existing methods. © 2024, CC BY.

关键词： Diffusion

SoK: On Finding Common Ground in Loss Landscapes Using Deep Model Merging Techniques

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arXiv 2024年

作者： Khan, Arham Nief, Todd Hudson, Nathaniel Sakarvadia, Mansi Grzenda, Daniel Ajith, Aswathy Pettyjohn, Jordan Chard, Kyle Foster, Ian Department of Computer Science University of Chicago ChicagoIL United States Data Science & Learning Division Argonne National Laboratory LemontIL United States Department of Applied Mathematics and Statistics Colorado School of Mines GoldenCO United States

Understanding neural networks is crucial to creating reliable and trustworthy deep learning models. Most contemporary research in interpretability analyzes just one model at a time via causal intervention or activation analysis. Yet despite successes, these methods leave significant gaps in our understanding of the training behaviors of neural networks, how their inner representations emerge, and how we can predictably associate model components with task-specific behaviors. Seeking new insights from work in related fields, here we survey literature in the field of model merging, a field that aims to combine the abilities of various neural networks by merging their parameters and identifying task-specific model components in the process. We analyze the model merging literature through the lens of loss landscape geometry, an approach that enables us to connect observations from empirical studies on interpretability, security, model merging, and loss landscape analysis to phenomena that govern neural network training and the emergence of their inner representations. To systematize knowledge in this area, we present a novel taxonomy of model merging techniques organized by their core algorithmic principles. Additionally, we distill repeated empirical observations from the literature in these fields into characterizations of four major aspects of loss landscape geometry: mode convexity, determinism, directedness, and connectivity. We argue that by improving our understanding of the principles underlying model merging and loss landscape geometry, this work contributes to the goal of ensuring secure and trustworthy machine learning in practice. © 2024, CC BY.

关键词： Federated learning

Bias-Variance Trade-off in Physics-Informed Neural Networks with Randomized Smoothing for High-Dimensional PDEs

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arXiv 2023年

作者： Hu, Zheyuan Yang, Zhouhao Wang, Yezhen Karniadakis, George Em Kawaguchi, Kenji Division of Applied Mathematics Brown University ProvidenceRI02912 United States Department of Computer Science National University of Singapore Singapore119077 Singapore Advanced Computing Mathematics and Data Division Pacific Northwest National Laboratory RichlandWA United States

Physics-Informed Neural Networks (PINNs) have triggered a paradigm shift in scientific computing, leveraging mesh-free properties and robust approximation capabilities. While proving effective for low-dimensional partial differential equations (PDEs), the computational cost of PINNs remains a hurdle in high-dimensional scenarios. This is particularly pronounced when computing high-order and high-dimensional derivatives in the physics-informed loss. Randomized Smoothing PINN (RS-PINN) introduces Gaussian noise for stochastic smoothing of the original neural net model, enabling the use of Monte Carlo methods for derivative approximation, which eliminates the need for costly automatic differentiation. Despite its computational efficiency, especially in the approximation of high-dimensional derivatives, RS-PINN introduces biases in both loss and gradients, negatively impacting convergence, especially when coupled with stochastic gradient descent (SGD) algorithms. We present a comprehensive analysis of biases in RS-PINN, attributing them to the nonlinearity of the Mean Squared Error (MSE) loss as well as the intrinsic nonlinearity of the PDE itself. We propose tailored bias correction techniques, delineating their application based on the order of PDE nonlinearity. The derivation of an unbiased RS-PINN allows for a detailed examination of its advantages and disadvantages compared to the biased version. Specifically, the biased version has a lower variance and runs faster than the unbiased version, but it is less accurate due to the bias. To optimize the bias-variance trade-off, we combine the two approaches in a hybrid method that balances the rapid convergence of the biased version with the high accuracy of the unbiased version. In addition to methodological contributions, we present an enhanced implementation of RS-PINN. Extensive experiments on diverse high-dimensional PDEs, including Fokker-Planck, Hamilton-Jacobi-Bellman (HJB), viscous Burgers’, Allen-Cahn, and Sine

关键词： Stochastic systems

Butterfly-Accelerated Solution of Coupled Volume Integral and Hydrodynamic Equations

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Butterfly-Accelerated Solution of Coupled Volume Integral an...

USNC-URSI Radio science Meeting (Joint with AP-S Symposium)

作者： Doolos Aibek Uulu Yang Liu Abdulkadir C. Yucel Hakan Bagci Department of Mathematics and Natural Sciences University of Central Asia (UCA) Naryn Kyrgyzstan Applied Mathematics & Computational Research Division Lawrence Berkeley National Laboratory Berkeley USA School of Electrical and Electronic Engineering Nanyang Technological University (NTU) Singapore Computer Electrical and Mathematical Science and Engineering Division King Abdullah University of Science and Technology (KAUST) Thuwal Saudi Arabia

ISBN: (数字)9789463968119

ISBN: (纸本)9798350359497

In recent years, advancements in plasmonic devices have enabled the localization of electromagnetic fields at subwavelength scales, contributing to breakthroughs in various scientific and engineering fields (J. C. Ndukaife, et al., ACS Nano, 2014,8,9,9035-9043). These structures are often made of metals and their electric properties are typically represented by the classical Drude model (S. A. Maier, 2007, Springer, New York). As the size of the plasmonic structures approaches the nanoscale, the classical Drude model falls short in explaining the collective movement of free electrons in metals. The interaction between moving charges and the electromagnetic fields can be described by a coupled system of the Maxwell and hydrodynamic equations (X. Zheng, et al., IEEE Trans. Antennas Propag., 2018,66,9, 4759-4771).

关键词： Location awareness Conferences Integral equations Metals Plasmons Hydrodynamics Mathematical models

Hutchinson Trace Estimation for High-Dimensional and High-Order Physics-Informed Neural Networks

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arXiv 2023年

作者： Hu, Zheyuan Shi, Zekun Karniadakis, George Em Kawaguchi, Kenji Department of Computer Science National University of Singapore Singapore119077 Singapore Division of Applied Mathematics Brown University ProvidenceRI02912 United States Advanced Computing Mathematics and Data Division Pacific Northwest National Laboratory RichlandWA United States

Physics-Informed Neural Networks (PINNs) have proven effective in solving partial differential equations (PDEs), especially when some data are available by seamlessly blending data and physics. However, extending PINNs to high-dimensional and even high-order PDEs encounters significant challenges due to the computational cost associated with automatic differentiation in the residual loss function calculation. Herein, we address the limitations of PINNs in handling high-dimensional and high-order PDEs by introducing the Hutchinson Trace Estimation (HTE) method. Starting with the second-order high-dimensional PDEs, which are ubiquitous in scientific computing, HTE is applied to transform the calculation of the entire Hessian matrix into a Hessian vector product (HVP). This approach not only alleviates the computational bottleneck via Taylor-mode automatic differentiation but also significantly reduces memory consumption from the Hessian matrix to an HVP's scalar output. We further showcase HTE's convergence to the original PINN loss and its unbiased behavior under specific conditions. Comparisons with the Stochastic Dimension Gradient Descent (SDGD) highlight the distinct advantages of HTE, particularly in scenarios with significant variability and variance among dimensions. We further extend the application of HTE to higher-order and higher-dimensional PDEs, specifically addressing the biharmonic equation. By employing tensor-vector products (TVP), HTE efficiently computes the colossal tensor associated with the fourth-order high-dimensional biharmonic equation, saving memory and enabling rapid computation. The effectiveness of HTE is illustrated through experimental setups, demonstrating comparable convergence rates with SDGD under memory and speed constraints. Additionally, HTE proves valuable in accelerating the Gradient-Enhanced PINN (gPINN) version as well as the Biharmonic equation. Overall, HTE opens up a new capability in scientific machine learning for tackl

关键词： Stochastic systems

Importance of kernel bandwidth in quantum machine learning

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Physical Review A 2022年第4期106卷 042407-042407页

作者： Ruslan Shaydulin Stefan M. Wild Global Technology Applied Research JPMorgan Chase New York New York 10017 USA Mathematics and Computer Science Division Argonne National Laboratory Lemont Illinois 60439 USA

Quantum kernel methods are considered a promising avenue for applying quantum computers to machine learning problems. Identifying hyperparameters controlling the inductive bias of quantum machine learning models is expected to be crucial given the central role hyperparameters play in determining the performance of classical machine learning methods. In this work we introduce the hyperparameter controlling the bandwidth of a quantum kernel and show that it controls the expressivity of the resulting model. We use extensive numerical experiments with multiple quantum kernels and classical data sets to show consistent change in the model behavior from underfitting (bandwidth too large) to overfitting (bandwidth too small), with optimal generalization in between. We draw a connection between the bandwidth of classical and quantum kernels and show analogous behavior in both cases. Furthermore, we show that optimizing the bandwidth can help mitigate the exponential decay of kernel values with qubit count, which is the cause behind recent observations that the performance of quantum kernel methods decreases with qubit count. We reproduce these negative results and show that if the kernel bandwidth is optimized, the performance instead improves with growing qubit count and becomes competitive with the best classical methods.

关键词： Machine learning Quantum algorithms

Cross-system biological image quality enhancement based on the generative adversarial network as a foundation for establishing a multi-institute microscopy cooperative network

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arXiv 2024年

作者： Panek, Dominik Rząca, Carina Szczypior, Maksymilian Sorysz, Joanna Misztal, Krzysztof Baster, Zbigniew Rajfur, Zenon Doctoral School of Exact and Natural Sciences Jagiellonian University ul. Lojasiewicza 11 Kraków30-348 Poland Department of Molecular and Interfacial Biophysics Faculty of Physics Astronomy and Applied Computer Science Jagiellonian University ul. Lojasiewicza 11 Kraków30‑348 Poland Undergraduate Program in Biophysics Faculty of Physics Astronomy and Applied Computer Science Jagiellonian University ul. Lojasiewicza 11 Kraków30‑348 Poland Undergraduate program in Biomedical Engineering Faculty of Electrical Engineering Automatics Computer Science and Biomedical Engineering AGH University of Science and Technology Al. A. Mickiewicza 30 Building B-1 Kraków30-059 Poland Division of Computational Mathematics Faculty of Mathematics and Computer Science Jagiellonian University 6 Lojasiewicza St. Kraków Poland

High-quality fluorescence imaging of biological systems is limited by processes like photobleaching and phototoxicity, and also in many cases, by limited access to the latest generations of microscopes. Moreover, low temporal resolution can lead to a motion blur effect in living systems. Our work presents a deep learning (DL) generative-adversarial approach to the problem of obtaining high-quality (HQ) images based on their low-quality (LQ) equivalents. We propose a generative-adversarial network (GAN) for contrast transfer between two different separate microscopy systems: a confocal microscope (producing HQ images) and a wide-field fluorescence microscope (producing LQ images). Our model proves that such transfer is possible, allowing us to receive HQ-generated images characterized by low mean squared error (MSE) values, high structural similarity index (SSIM), and high peak signal-to-noise ratio (PSNR) values. For our best model in the case of comparing HQ-generated images and HQ-ground truth images, the median values of the metrics are 6·10-4, 0.9413, and 31.87, for MSE, SSIM, and PSNR, respectively. In contrast, in the case of comparison between LQ and HQ ground truth median values of the metrics are equal to 0.0071, 0.8304, and 21.48 for MSE, SSIM, and PSNR respectively. Therefore, we observe a significant increase ranging from 14% to 49% for SSIM and PSNR respectively. These results, together with other single-system cross-modality studies, provide proof of concept for further implementation of a cross-system biological image quality enhancement. © 2024, CC BY-SA.

关键词： Fluorescence imaging

Temperature-anisotropy conjugate magnon squeezing in antiferromagnets

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arXiv 2023年

作者： Shiranzaei, Mahroo Fransson, Jonas Mousolou, Vahid Azimi Division of Materials Theory Department of Physics and Astronomy Uppsala University Box 516 UppsalaSE-75120 Sweden Department of Applied Mathematics and Computer Science Faculty of Mathematics and Statistics University of Isfahan Isfahan81746-73441 Iran

Quantum squeezing is an essential asset in the field of quantum science and technology. In this study, we investigate the impact of temperature and anisotropy on squeezing of quantum fluctuations in two-mode magnon states within uniaxial antiferromagnetic materials. Through our analysis, we discover that the inherent nonlinearity in these bipartite magnon systems gives rise to a conjugate magnon squeezing effect across all energy eigenbasis states, driven by temperature and anisotropy. We show that temperature induces amplitude squeezing, whereas anisotropy leads to phase squeezing. In addition, we observe that the two-mode squeezing characteristic of magnon eigenenergy states is associated with amplitude squeezing. This highlights the constructive impact of temperature and the destructive impact of anisotropy on two-mode magnon squeezing. Nonetheless, our analysis shows that the destructive effect of anisotropy is bounded. We demonstrate this by showing that, at a given temperature, the squeezing of the momentum (phase) quadrature (or equivalently, the stretching of the position (amplitude) quadrature) approaches a constant function of anisotropy after a finite value of anisotropy. Moreover, our study demonstrates that higher magnon squeeze factors can be achieved at higher temperatures, smaller levels of anisotropy, and closer to the Brillouin zone center. All these characteristics are specific to low-energy magnons in the uniaxial antiferromagnetic materials that we examine here. © 2023, CC0.

关键词： Anisotropy

Tensor neural networks for high-dimensional Fokker-Planck equations

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arXiv 2024年

作者： Wang, Taorui Hu, Zheyuan Kawaguchi, Kenji Zhang, Zhongqiang Karniadakis, George Em Department of Mathematical Sciences Worcester Polytechnic Institute WorcesterMA United States Department of Computer Science National University of Singapore 119077 Singapore Division of Applied Mathematics Brown University ProvidenceRI02912 United States Advanced Computing Mathematics and Data Division Pacific Northwest National Laboratory RichlandWA United States

We solve high-dimensional steady-state Fokker-Planck equations on the whole space by applying tensor neural networks. The tensor networks are a linear combination of tensor products of one-dimensional feedforward networks or a linear combination of several selected radial basis functions. The use of tensor feedforward networks allows us to efficiently exploit auto-differentiation (in physical variables) in major Python packages while using radial basis functions can fully avoid auto-differentiation, which is rather expensive in high dimensions. We then use the physics-informed neural networks and stochastic gradient descent methods to learn the tensor networks. One essential step is to determine a proper bounded domain or numerical support for the Fokker-Planck equation. To better train the tensor radial basis function networks, we impose some constraints on parameters, which lead to relatively high accuracy. We demonstrate numerically that the tensor neural networks in physics-informed machine learning are efficient for steady-state Fokker-Planck equations from two to ten dimensions. Copyright © 2024, The Authors. All rights reserved.

关键词： Radial basis function networks