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Tackling the Curse of Dimensionality in Fractional and Tempered Fractional PDEs with Physics-Informed Neural Networks

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arXiv 2024年

作者： Hu, Zheyuan Kawaguchi, Kenji Zhang, Zhongqiang Em Karniadakis, George Department of Computer Science National University of Singapore 119077 Singapore Department of Mathematical Sciences Worcester Polytechnic Institute WorcesterMA01609 United States Division of Applied Mathematics Brown University ProvidenceRI02912 United States Advanced Computing Mathematics and Data Division Pacific Northwest National Laboratory RichlandWA United States

Fractional and tempered fractional partial differential equations (PDEs) are effective models of long-range interactions, anomalous diffusion, and non-local effects. Traditional numerical methods for these problems are mesh-based, thus struggling with the curse of dimensionality (CoD). Physics-informed neural networks (PINNs) offer a promising solution due to their universal approximation, generalization ability, and mesh-free training. In principle, Monte Carlo fractional PINN (MC-fPINN) estimates fractional derivatives using Monte Carlo methods and thus could lift CoD. However, this may cause significant variance and errors, hence affecting convergence;in addition, MC-fPINN is sensitive to hyperparameters. In general, numerical methods and specifically PINNs for tempered fractional PDEs are under-developed. Herein, we extend MC-fPINN to tempered fractional PDEs to address these issues, resulting in the Monte Carlo tempered fractional PINN (MC-tfPINN). To reduce possible high variance and errors from Monte Carlo sampling, we replace the one-dimensional (1D) Monte Carlo with 1D Gaussian quadrature, applicable to both MC-fPINN and MC-tfPINN. We validate our methods on various forward and inverse problems of fractional and tempered fractional PDEs, scaling up to 100,000 dimensions. Our improved MC-fPINN/MC-tfPINN using quadrature consistently outperforms the original versions in accuracy and convergence speed in very high dimensions. © 2024, CC BY.

关键词： Monte Carlo methods

Score-fPINN: Fractional Score-Based Physics-Informed Neural Networks for High-Dimensional Fokker-Planck-Lévy Equations

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arXiv 2024年

作者： Hu, Zheyuan Zhang, Zhongqiang Karniadakis, George Em Kawaguchi, Kenji Department of Computer Science National University of Singapore Singapore119077 Singapore Department of Mathematical Sciences Worcester Polytechnic Institute WorcesterMA01609 United States Division of Applied Mathematics Brown University ProvidenceRI02912 United States Advanced Computing Mathematics and Data Division Pacific Northwest National Laboratory RichlandWA United States

We introduce an innovative approach for solving high-dimensional Fokker-Planck-Lévy (FPL) equations in modeling non-Brownian processes across disciplines such as physics, finance, and ecology. We utilize a fractional score function and Physical-informed neural networks (PINN) to lift the curse of dimensionality (CoD) and alleviate numerical overflow from exponentially decaying solutions with dimensions. The introduction of a fractional score function allows us to transform the FPL equation into a second-order partial differential equation without fractional Laplacian and thus can be readily solved with standard physics-informed neural networks (PINNs). We propose two methods to obtain a fractional score function: fractional score matching (FSM) and score-fPINN for fitting the fractional score function. While FSM is more cost-effective, it relies on known conditional distributions. On the other hand, score-fPINN is independent of specific stochastic differential equations (SDEs) but requires evaluating the PINN model's derivatives, which may be more costly. We conduct our experiments on various SDEs and demonstrate numerical stability and effectiveness of our method in dealing with high-dimensional problems, marking a significant advancement in addressing the CoD in FPL equations. © 2024, CC BY.

关键词： Cost effectiveness

Designing a Framework for Solving Multiobjective Simulation Optimization Problems

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arXiv 2023年

作者： Chang, Tyler H. Wild, Stefan M. Mathematics and Computer Science Division Argonne National Laboratory 9700 S Cass Ave Bldg 240 LemontIL60439 United States Applied Mathematics and Computational Research Division Lawrence Berkeley National Laboratory 1 Cyclotron Rd BerkeleyCA94720 United States

Multiobjective simulation optimization (MOSO) problems are optimization problems with multiple conflicting objectives, where evaluation of at least one of the objectives depends on a black-box numerical code or real-world experiment, which we refer to as a simulation. While an extensive body of research is dedicated to developing new algorithms and methods for solving these and related problems, it is challenging and time consuming to integrate these techniques into real world production-ready solvers. This is partly due to the diversity and complexity of modern state-of-the-art MOSO algorithms and methods and partly due to the complexity and specificity of many real-world problems and their corresponding computing environments. The complexity of this problem is only compounded when introducing potentially complex and/or domain-specific surrogate modeling techniques, problem formulations, design spaces, and data acquisition functions. This paper carefully surveys the current state-of-the-art in MOSO algorithms, techniques, and solvers;as well as problem types and computational environments where MOSO is commonly applied. We then present several key challenges in the design of a Parallel Multiobjective Simulation Optimization framework (ParMOO) and how they have been addressed. Finally, we provide two case studies demonstrating how customized ParMOO solvers can be quickly built and deployed to solve real-world MOSO problems. Copyright © 2023, The Authors. All rights reserved.

关键词： Software design

Bayesian Inference for Non-Synchronously Observed Diffusions

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arXiv 2025年

作者： Jasra, Ajay Kamatani, Kengo Wu, Amin School of Data Science The Chinese University of Hong Kong Shenzhen Shenzhen China Institute of Statistical Mathematics Tokyo190-0014 Japan Statistics Program Computer Electrical and Mathematical Sciences and Engineering Division King Abdullah University of Science and Technology Thuwal23955-6900 Saudi Arabia

We consider the problem of Bayesian inference for bi-variate data observed in time but with observation times which occur non-synchronously. In particular, this occurs in a wide variety of applications in finance, such as high-frequency trading or crude oil futures trading. We adopt a diffusion model for the data and formulate a Bayesian model with priors on unknown parameters along with a latent representation for the the so-called missing data. We then consider computational methodology to fit the model using Markov chain Monte Carlo (MCMC). We have to resort to time-discretization methods as the complete data likelihood is intractable and this can cause considerable issues for MCMC when the data are observed in low frequencies. In a high frequency observation frequencies we present a simple particle MCMC method based on an Euler–Maruyama time discretization, which can be enhanced using multilevel Monte Carlo (MLMC). In the low frequency observation regime we introduce a novel bridging representation of the posterior in continuous time to deal with the issues of MCMC in this case. This representation is discretized and fitted using MCMC and MLMC. We apply our methodology to real and simulated data to establish the efficacy of our methodology. © 2025, CC BY.

关键词： Crude oil

CONVERGENCE SPEED AND APPROXIMATION ACCURACY OF NUMERICAL MCMC

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arXiv 2022年

作者： Cui, Tiangang Dong, Jing Jasra, Ajay Tong, Xin T. School of Mathematics Monash University Australia Graduate School of Business Columbia University United States Applied Mathematics and Computational Science Program Computer Electrical Mathematical Sciences and Engineering Division King Abdullah University of Science and Technology Saudi Arabia Department of Mathematics National University of Singapore Singapore

When implementing Markov Chain Monte Carlo (MCMC) algorithms, perturbation caused by numerical errors is sometimes inevitable. This paper studies how perturbation of MCMC affects the convergence speed and Monte Carlo estimation accuracy. Our results show that when the original Markov chain converges to stationarity fast enough and the perturbed transition kernel is a good approximation to the original transition kernel, the corresponding perturbed sampler has similar convergence speed and high approximation accuracy as well. We discuss two different analysis frameworks: ergodicity and spectral gap, both are widely used in the literature. Our results can be easily extended to obtain non-asymptotic error bounds for MCMC estimators. We also demonstrate how to apply our convergence and approximation results to the analysis of specific sampling algorithms, including Random walk Metropolis and Metropolis adjusted Langevin algorithm with perturbed target densities, and parallel tempering Monte Carlo with perturbed densities. Finally we present some simple numerical examples to verify our theoretical claims. Copyright © 2022, The Authors. All rights reserved.

关键词： Inverse problems

On Time Uniform Wong-Zakai Approximation Theorems

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arXiv 2023年

作者： Del Moral, Pierre Hu, Shulan Jasra, Ajay Ruzayqat, Hamza Wang, Xinyu Institut de Mathematiques de Bordeaux Bordeaux33405 France School of Statistics and Mathematics Zhongnan University of Economics and Law 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 Wenlan School of Business Zhongnan University of Economics and Law China

We consider the long time behavior of Wong-Zakai approximations of stochastic differential equations. These piecewise smooth diffusion approximations are of great importance in many areas, such as those with ordinary differential equations associated to random smooth fluctuations;e.g. robust filtering problems. In many examples, the mean error estimate bounds that have been derived in the literature can grow exponentially with respect to the time horizon. We show in a simple example that indeed mean error estimates do explode exponentially in the time parameter, i.e. in that case a Wong-Zakai approximation is only useful for extremely short time intervals. Under spectral conditions, we present some quantitative time-uniform convergence theorems, i.e. time-uniform mean error bounds, yielding what seems to be the first results of this type for Wong-Zakai diffusion *** Codes 60H35(Primary), 65C30(Secondary) Copyright © 2023, The Authors. All rights reserved.

关键词： Brownian movement

Nested Instrumental Variables Design: Switcher Average Treatment Effect, Identification, Efficient Estimation and Generalizability

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arXiv 2024年

作者： Wang, Rui Zhao, Ying-Qi Dukes, Oliver Zhang, Bo Department of Biostatistics University of Washington United States Public Health Sciences Division Fred Hutchinson Cancer Center United States Department of Applied Mathematics Computer Science and Statistics Ghent University Belgium Vaccine and Infectious Disease Division Fred Hutchinson Cancer Center United States

Instrumental variables (IV) are a commonly used tool to estimate causal effects from nonrandomized data. An archetype of an IV is a randomized trial with non-compliance where the randomized treatment assignment serves as an IV for the non-ignorable treatment received. Under a monotonicity assumption, a valid IV non-parametrically identifies the average treatment effect among a non-identified, latent complier subgroup, whose generalizability is often under debate. In many studies, there could exist multiple versions of an IV, for instance, different nudges to take the same treatment in different study sites in a multicentre clinical trial. These different versions of an IV may result in different compliance rates and offer a unique opportunity to study IV estimates’ generalizability. In this article, we introduce a novel nested IV assumption and study identification of the average treatment effect among two latent subgroups: always-compliers and switchers, who are defined based on the joint potential treatment received under two versions of a binary IV. We derive the efficient influence function for the SWitcher Average Treatment Effect (SWATE) under a non-parametric model and propose efficient estimators. We then propose formal statistical tests of the generalizability of IV estimates under the nested IV framework. We apply the proposed method to the Prostate, Lung, Colorectal and Ovarian (PLCO) Cancer Screening Trial and study the causal effect of colorectal cancer screening and its generalizability. Copyright © 2024, The Authors. All rights reserved.

关键词： Lung cancer

Anisotropic, multiband, and strong-coupling superconductivity of the Pb0.64Bi0.36 alloy

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Physical Review B 2024年第21期110卷 214510-214510页

作者： Sylwia Gutowska Karolina Górnicka Paweł Wójcik Tomasz Klimczuk Bartlomiej Wiendlocha University of Vienna Faculty of Physics and Center for Computational Materials Science Kolingasse 14-16 A-1090 Vienna Austria AGH University of Krakow Faculty of Physics and Applied Computer Science Aleja Mickiewicza 30 30-059 Krakow Poland Materials Science and Technology Division Oak Ridge National Laboratory Oak Ridge Tennessee 37831 USA Faculty of Applied Physics and Mathematics and Advanced Materials Centre Gdansk University of Technology ul. Narutowicza 11/12 80-233 Gdańsk Poland

This paper presents theoretical and experimental studies on the superconductivity of Pb0.64Bi0.36 alloy, which is a prototype of strongly coupled superconductors and exhibits one of the strongest coupling under ambient pressure among the materials studied so far. The critical temperature, the specific heat in the superconducting state, and the magnetic critical fields are experimentally determined. Deviations from the single-gap s-wave BCS-like behavior are observed. The electronic structure, phonons, and electron-phonon interactions are analyzed in relation to the metallic Pb, explaining why the Pb-Bi alloy exhibits such a large value of the electron-phonon coupling parameter λ≃2. Superconductivity is studied using the isotropic Eliashberg formalism as well as the anisotropic density functional theory for superconductors. We find that while Pb is a two-gap superconductor with well-defined separate superconducting gaps, in the Pb-Bi alloy an overlapped three-gap-like structure is formed with a strong anisotropy. Furthermore, the chemical disorder, inherent to this alloy, leads to strong electron scattering, which is found to reduce the critical temperature.

关键词： Anisotropy

Relative magnetic field and slipping effect on Casson dusty fluid of two phase fluctuating flow over inclined parallel plate

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South African Journal of Chemical Engineering 2023年 44卷 135-146页

作者： khan, Dolat Kumam, Poom Kumam, Wiyada Suttiarporn, Panawan Rehman, Ali Fixed Point Research Laboratory Fixed Point Theory and Applications Research Group Center of Excellence in Theoretical and Computational Science (TaCS-CoE) Faculty of Science King Mongkut's University of Technology Thonburi (KMUTT) 126 Pracha Uthit Rd. Bang Mod Bangkok Thung Khru 10140 Thailand Center of Excellence in Theoretical and Computational Science (TaCS-CoE) Faculty of Science King Mongkut's University of Technology Thonburi (KMUTT) 126 Pracha Uthit Rd. Bang Mod Bangkok Thung Khru 10140 Thailand Applied Mathematics for Science and Engineering Research Unit (AMSERU) Program in Applied Statistics Department of Mathematics and Computer Science Faculty of Science and Technology Rajamangala University of Technology Thanyaburi (RMUTT) Pathum Thani 12110 Thailand Faculty of Science Energy and Environment King Mongkut's University of Technology North Bangkok Rayong Campus Rayong 21120 Thailand Special Interest Group for Modelling and Data Analytics Faculty of Ocean Engineering Technology and Informatics Universiti Malaysia Terengganu Terengganu Kuala Nerus 21030 Malaysia

Due to its significant applications in magnetic devices for cell separation, magnetic drugs for cancer tumor treatment, blood flow adjustment during surgery, magnetic endoscopy, and fluid pumping in industrial and engineering processes, magnetohydrodynamics (MHD) has a wide range of applications. Therefore, the aim of this study to investigate the Relative Magnetic Field and slipping effect on Casson dusty fluid of two-phase fluctuating flow over inclined parallel plate. More exactly, the relative magnetic phenomena are fixed relatively to the plate (MFFRP) or fixed relatively to the fluid (MFFRF). The mathematical modeling was established as the set of partial differential equations for dusty Casson fluid. Buckingham's pi theorem is used to find out the dimensionless variables, to make our system dimensionless. The solution in perturb form is to find out by incorporating Poincare–Lighthill perturbation technique (PLPT). To know in-depth different parameters the graphical results for both velocities are plotted with the help of software Mathcad- 15. It is clearly detected that the relative magnetic field performs a vital part in the fluid as well as particle motion. Moreover, the relative magnetic field play a vital role in the decreasing of skin friction of both plates. © 2023

关键词： Brinkman fluid Dust particles Fluctuating vertical plate Inclined relative magnetic field PLPT