检索结果-内蒙古大学图书馆

Monte Carlo PINNs: deep learning approach for forward and inverse problems involving high dimensional fractional partial differential equations

学校读者我要写书评

暂无评论

arXiv 2022年

作者： Guo, Ling Wu, Hao Yu, Xiaochen Zhou, Tao Department of Mathematics Shanghai Normal University Shanghai China School of Mathematical sciences Tongji University Shanghai China Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing China

We introduce a sampling based machine learning approach, Monte Carlo physics informed neural networks (MC-PINNs), for solving forward and inverse fractional partial differential equations (FPDEs). As a generalization of physics informed neural networks (PINNs), our method relies on deep neural network surrogates in addition to a stochastic approximation strategy for computing the fractional derivatives of the DNN outputs. A key ingredient in our MC-PINNs is to construct an unbiased estimation of the physical soft constraints in the loss function. Our directly sampling approach can yield less overall computational cost compared to fPINNs proposed in [1] and thus provide an opportunity for solving high dimensional fractional PDEs. We validate the performance of MC-PINNs method via several examples that include high dimensional integral fractional Laplacian equations, parametric identification of time-space fractional PDEs, and fractional diffusion equation with random inputs. The results show that MC-PINNs is flexible and promising to tackle high-dimensional FPDEs. © 2022, CC BY.

关键词： Deep neural networks

Lyapunov exponents and Lagrangian chaos suppression in compressible homogeneous isotropic turbulence

学校读者我要写书评

暂无评论

arXiv 2023年

作者： Yu, Haijun Fouxon, Itzhak Wang, Jianchun Li, Xiangru Yuan, Li Mao, Shipeng Mond, Michael NCMIS & LSEC Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Beijing100190 China School of Mathematical Sciences University of Chinese Academy of Sciences Beijing100049 China Department of Mechanical Engineering Ben-Gurion University of the Negev Beer-Sheva84105 Israel Department of Computational Science and Engineering Yonsei University Seoul03722 Korea Republic of Department of Mechanics and Aerospace Engineering Southern University of Science and Technology Shenzhen518055 China

We study Lyapunov exponents of tracers in compressible homogeneous isotropic turbulence at different turbulent Mach number Mt and Taylor-scale Reynolds number Reλ. We demonstrate that statistics of finite-time Lyapunov exponents have the same form as in incompressible flow due to density-velocity coupling. The modulus of the smallest Lyapunov exponent λ3 provides the principal Lyapunov exponent of the time-reversed flow, which is usually wrong in a compressible flow. This exponent, along with the principal Lyapunov exponent λ1, determines all the exponents due to vanishing of the sum of all Lyapunov exponents. Numerical results by high-order schemes for solving the Navier-Stokes equations and tracking particles verify these theoretical predictions. We found that: 1) The largest normalized Lyapunov exponent λ1τη, where τη is the Kolmogorov timescale, is a decreasing function of Mt. Its dependence on Reλ is weak when the driving force is solenoidal, while it is an increasing function of Reλ when the solenoidal and compressible forces are comparable. Similar facts hold for |λ3|, in contrast with well-studied short-correlated model;2) The ratio of the first two Lyapunov exponents λ1/λ2 decreases with Reλ, and is virtually independent of Mt for Mt ≤ 1 in the case of solenoidal force but decreases as Mt increases when solenoidal and compressible forces are comparable;3) For purely solenoidal force, λ1 : λ2 : λ3 ≈ 4: 1: −5 for Reλ > 80, which is consistent with incompressible turbulence studies;4) The ratio of dilation-to-vorticity is a more suitable parameter to characterize Lyapunov exponents than Mt © 2023, CC BY-NC-ND.

关键词： Navier Stokes equations

Three Kinds of Novel Multi-Symplectic Methods for Stochastic Hamiltonian Partial Differential Equations

学校读者我要写书评

暂无评论

SSRN

SSRN 2022年

作者： Hong, Jialin Hou, Baohui Li, Qiang Sun, Liying Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China School of Mathematical Sciences University of Chinese Academy of Sciences Beijing100049 China

Stochastic Hamiltonian partial differential equations, which possess the multi-symplectic conservation law, are an important and fairly large class of systems. The multi-symplectic methods inheriting the geometric features of stochastic Hamiltonian partial differential equations provide numerical approximations with better numerical stability, and are of vital significance for obtaining correct numerical results. In this paper, we propose three novel multi-symplectic methods for stochastic Hamiltonian partial differential equations based on the local radial basis function collocation method, the splitting technique and the partitioned Runge-Kutta method. Concrete numerical methods are presented for nonlinear stochastic wave equations, stochastic nonlinear Schr\"odinger equations, stochastic Korteweg-de Vries equations and stochastic Maxwell equations. We take stochastic wave equations as examples to perform numerical experiments, which indicate the validity of the proposed methods. © 2022, The Authors. All rights reserved.

关键词： Stochastic systems

ADAPTIVE DEEP DENSITY APPROXIMATION FOR FRACTIONAL FOKKER-PLANCK EQUATIONS

学校读者我要写书评

暂无评论

arXiv 2022年

作者： Zeng, Li Wan, Xiaoliang Zhou, Tao LSEC Institute of Computational Mathematics and Scientific/Engineering Computing AMSS Chinese Academy of Sciences Beijing China Department of Mathematics Center for Computation and Technology Louisiana State University Baton Rouge70803 United States

In this work, we propose adaptive deep learning approaches based on normalizing flows for solving fractional Fokker-Planck equations (FPEs). The solution of a FPE is a probability density function (PDF). Traditional mesh-based methods are ineffective because of the unbounded computation domain, a large number of dimensions and the nonlocal fractional operator. To this end, we represent the solution with an explicit PDF model induced by a flow-based deep generative model, simplified KRnet, which constructs a transport map from a simple distribution to the target distribution. We consider two methods to approximate the fractional Laplacian. One method is the Monte Carlo approximation. The other method is to construct an auxiliary model with Gaussian radial basis functions (GRBFs) to approximate the solution such that we may take advantage of the fact that the fractional Laplacian of a Gaussian is known analytically. Based on these two different ways for the approximation of the fractional Laplacian, we propose two models, MCNF and GRBFNF, to approximate stationary FPEs and MCTNF to approximate time-dependent FPEs. To further improve the accuracy, we refine the training set and the approximate solution alternately. A variety of numerical examples is presented to demonstrate the effectiveness of our adaptive deep density approaches. Copyright © 2022, The Authors. All rights reserved.

关键词： Fokker Planck equation

Three kinds of novel multi-symplectic methods for stochastic Hamiltonian partial differential equations

学校读者我要写书评

暂无评论

arXiv 2022年

Stochastic Hamiltonian partial differential equations, which possess the multi-symplectic conservation law, are an important and fairly large class of systems. The multi-symplectic methods inheriting the geometric features of stochastic Hamiltonian partial differential equations provide numerical approximations with better numerical stability, and are of vital significance for obtaining correct numerical results. In this paper, we propose three novel multi-symplectic methods for stochastic Hamiltonian partial differential equations based on the local radial basis function collocation method, the splitting technique, and the partitioned Runge–Kutta method. Concrete numerical methods are presented for nonlinear stochastic wave equations, stochastic nonlinear Schrödinger equations, stochastic Korteweg-de Vries equations and stochastic Maxwell equations. We take stochastic wave equations as examples to perform numerical experiments, which indicate the validity of the proposed methods. © 2022, CC BY.

关键词： Stochastic systems

Rapid quantum ground state preparation via dissipative dynamics

学校读者我要写书评

暂无评论

arXiv 2025年

作者： Zhan, Yongtao Ding, Zhiyan Huhn, Jakob Gray, Johnnie Preskill, John Chan, Garnet Kin-Lic Lin, Lin Institute for Quantum Information and Matter California Institute of Technology United States Division of Physics Mathematics and Astronomy California Institute of Technology United States Department of Mathematics University of California Berkeley United States Department of Physics Arnold Sommerfeld Center for Theoretical Physics Ludwig-Maximilians-Universität München Germany Division of Chemistry and Chemical Engineering California Institute of Technology United States Department of Physics California Institute of Technology United States AWS Center for Quantum Computing Applied Mathematics and Computational Research Division Lawrence Berkeley National Laboratory United States

Inspired by natural cooling processes, dissipation has become a promising approach for preparing low-energy states of quantum systems. However, the potential of dissipative protocols remains unclear beyond certain commuting Hamiltonians. This work provides significant analytical and numerical insights into the power of dissipation for preparing the ground state of non-commuting Hamiltonians. For quasi-free dissipative dynamics, including certain 1D spin systems with boundary dissipation, our results reveal a new connection between the mixing time in trace distance and the spectral properties of a non-Hermitian Hamiltonian, leading to an explicit and sharp bound on the mixing time that scales polynomially with system size. For more general spin systems, we develop a tensor network-based algorithm for constructing the Lindblad jump operator and for simulating the dynamics. Using this algorithm, we demonstrate numerically that dissipative ground state preparation protocols can achieve rapid mixing for certain 1D local Hamiltonians under bulk dissipation, with a mixing time that scales logarithmically with the system size. We then prove the rapid mixing result for certain weakly interacting spin and fermionic systems in arbitrary dimensions, extending recent results for high-temperature quantum Gibbs samplers to the zero-temperature regime. Our theoretical approaches are applicable to systems with singular stationary states, and are thus expected to have applications beyond the specific systems considered in this study. © 2025, CC BY.

关键词： Dynamics

DPA-2:a large atomic model as a multitask learner

学校读者我要写书评

暂无评论

npj computational Materials 2024年第1期10卷 185-199页

作者： Duo Zhang Xinzijian Liu Xiangyu Zhang Chengqian Zhang Chun Cai Hangrui Bi Yiming Du Xuejian Qin Anyang Peng Jiameng Huang Bowen Li Yifan Shan Jinzhe Zeng Yuzhi Zhang Siyuan Liu Yifan Li Junhan Chang Xinyan Wang Shuo Zhou Jianchuan Liu Xiaoshan Luo Zhenyu Wang Wanrun Jiang Jing Wu Yudi Yang Jiyuan Yang Manyi Yang Fu-Qiang Gong Linshuang Zhang Mengchao Shi Fu-Zhi Dai Darrin M.York Shi Liu Tong Zhu Zhicheng Zhong Jian Lv Jun Cheng Weile Jia Mohan Chen Guolin Ke Weinan E Linfeng Zhang Han Wang AI for Science Institute BeijingP.R.China DP Technology BeijingP.R.China Academy for Advanced Interdisciplinary Studies Peking UniversityBeijingP.R.China State Key Lab of Processors Institute of Computing TechnologyChinese Academy of SciencesBeijingP.R.China University of Chinese Academy of Sciences BeijingP.R.China HEDPS CAPTCollege of EngineeringPeking UniversityBeijingP.R.China Ningbo Institute of Materials Technology and Engineering Chinese Academy of SciencesNingboP.R.China CAS Key Laboratory of Magnetic Materials and Devices and Zhejiang Province Key Laboratory of Magnetic Materials and Application Technology Chinese Academy of SciencesNingboP.R.China School of Electronics Engineering and Computer Science Peking UniversityBeijingP.R.China Shanghai Engineering Research Center of Molecular Therapeutics&New Drug Development School of Chemistry and Molecular EngineeringEast China Normal UniversityShanghaiP.R.China Laboratory for Biomolecular Simulation Research Institute for Quantitative Biomedicine and Department of Chemistry and Chemical BiologyRutgers UniversityPiscatawayNJUSA Department of Chemistry Princeton UniversityPrincetonNJUSA College of Chemistry and Molecular Engineering Peking UniversityBeijingP.R.China Yuanpei College Peking UniversityBeijingP.R.China School of Electrical Engineering and Electronic Information Xihua UniversityChengduP.R.China State Key Laboratory of Superhard Materials College of PhysicsJilin UniversityChangchunP.R.China Key Laboratory of Material Simulation Methods&Software of Ministry of Education College of PhysicsJilin UniversityChangchunP.R.China International Center of Future Science Jilin UniversityChangchunP.R.China Key Laboratory for Quantum Materialsof Zhejiang Province Department of PhysicsSchool of ScienceWestlake UniversityHangzhouP.R.China Atomistic Simulations Italian Institute of TechnologyGenovaItaly State Key Laboratory of Physical Chemistry of Solid Surface iChEMCollege of Chemistry and Chemical EngineeringXiame

The rapid advancements in artificial intelligence(AI)are catalyzing transformative changes in atomic modeling,simulation,and ***-driven potential energy models havedemonstrated the capability to conduct large-scale,long-duration simulations with the accuracy of ab initio electronic structure ***,the model generation process remains a bottleneck for large-scale *** propose a shift towards a model-centric ecosystem,wherein a large atomic model(LAM),pretrained across multiple disciplines,can be efficiently fine-tuned and distilled for various downstream tasks,thereby establishing a new framework for molecular *** this study,we introduce the DPA-2 architecture as a prototype for ***-trained on a diverse array of chemical and materials systemsusing a multi-task approach,DPA-2demonstrates superior generalization capabilities across multiple downstream tasks compared to the traditional single-task pre-training and fine-tuning *** approach sets the stage for the development and broad application of LAMs in molecular and materials simulation research.

关键词： DPA establishing thereby

Shape of my heart: Cardiac models through learned signed distance functions

学校读者我要写书评

暂无评论

arXiv 2023年

作者： Verhülsdonk, Jan Grandits, Thomas Costabal, Francisco Sahli Pinetz, Thomas Krause, Rolf Auricchio, Angelo Haase, Gundolf Pezzuto, Simone Effland, Alexander Institute for Applied Mathematics University of Bonn Germany Department of Mathematics and Scientific Computing University of Graz Austria Institute for Biological and Medical Engineering Pontificia Universidad Católica de Chile Chile Center for Computational Medicine in Cardiology Università della Svizzera italiana Switzerland FernUni Schweiz Brig Switzerland Instituto Cardiocentro Ticino EOC Switzerland Department of Mathematics University of Trento Italy

The efficient construction of anatomical models is one of the major challenges of patient-specific in-silico models of the human heart. Current methods frequently rely on linear statistical models, allowing no advanced topological changes, or requiring medical image segmentation followed by a meshing pipeline, which strongly depends on image resolution, quality, and modality. These approaches are therefore limited in their transferability to other imaging domains. In this work, the cardiac shape is reconstructed by means of three-dimensional deep signed distance functions with Lipschitz regularity. For this purpose, the shapes of cardiac MRI reconstructions are learned to model the spatial relation of multiple chambers. We demonstrate that this approach is also capable of reconstructing anatomical models from partial data, such as point clouds from a single ventricle, or modalities different from the trained MRI, such as the electroanatomical mapping (EAM). Copyright © 2023, The Authors. All rights reserved.

关键词： Image resolution

A Deep Learning Method for computing Eigenvalues of the Fractional Schrödinger Operator

学校读者我要写书评

暂无评论

arXiv 2023年

作者： Yixiao, Guo Pingbing, Ming LSEC Institute of Computational Mathematics and Scientific/Engineering Computing AMSS Chinese Academy of Sciences No. 55 East Road Zhong-Guan-Cun Beijing100190 China School of Mathematical Sciences University of Chinese Academy of Sciences Beijing100049 China

We present a novel deep learning method for computing eigenvalues of the fractional Schrödinger operator. Our approach combines a newly developed loss function with an innovative neural network architecture that incorporates prior knowledge of the problem. These improvements enable our method to handle both high-dimensional problems and problems posed on irregular bounded domains. We successfully compute up to the first 30 eigenvalues for various fractional Schrödinger operators. As an application, we share a conjecture to the fractional order isospectral problem that has not yet been studied. Copyright © 2023, The Authors. All rights reserved.

关键词： Eigenvalues and eigenfunctions