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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

Mechanisms of Formation and Modeling of the Refractory Compound Film Structures

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Mechanisms of Formation and Modeling of the Refractory Compo...

International Conference on Nanomaterials: Application & Properties (NAP)

作者： Alexander Goncharov Ivan Kolinko Dmytro Shyrokorad Grygoriy Kornich Andrii Yunda Svetlana Goncharova Department of Applied Mathematics and Complex Systems Modelling Sumy State University Sumy Ukraine Institute of Materials Science Slovak University of Technology in Bratislava Slovak Republic Computer Sciences and Technologies Faculty National University “Zaporizhzhia Polytechnic” Zaporizhzhia Ukraine Institute of Applied Physics National Academy of Sciences of Ukraine Sumy Ukraine Department of Biophysics Biochemistry Pharmacology and Biomolecular Engineering Sumy State University Sumy Ukraine

ISBN: (数字)9798350380125

ISBN: (纸本)9798350380132

In this work, the molecular dynamics method was used for modeling the formation of critical nuclei at the initial stage. The deposition of a titanium nitride film on an iron substrate was chosen as a model object. The 2NN MEAM multiatomic potential was used to calculate the interaction between atoms. Simulations of N and Ti atom deposition on an Fe(100) substrate were carried out using the MPI version of the open-source program LAMMPS (Molecular Dynamics Simulator). The substrate was modeled as a semi-infinite crystal with a frozen lower atomic layer, periodic boundary conditions at the lateral crystal sides, and a free deposition surface.

关键词： Atomic layer deposition Crystals Atoms Titanium nitride Boundary conditions Iron Compounds Substrates Surface treatment

Zero Mach Number Limit of the Compressible Primitive Equations: Ill-prepared Initial data

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arXiv 2022年

作者： Liu, Xin Titi, Edriss S. Department of Mathematics Texas A&M University College StationTX77840 United States Department of Applied Mathematics and Theoretical Physics University of Cambridge CambridgeCB3 0WA United Kingdom Department of Computer Science and Applied Mathematics Weizmann Institute of Science Rehovot76100 Israel

In the work, we consider the zero Mach number limit of compressible primitive equations in the domain R2 × 2T or T2 × 2T. We identify the limit equations to be the primitive equations with the incompressible condition. The convergence behaviors are studied in both R2 × 2T and T2 × 2T, respectively. This paper takes into account the high oscillating acoustic waves and is an extension of our previous work in [29].MSC Codes 35B25, 35B40, 76N10 © 2022, CC BY.

关键词： Mach number

A Multi-Technique Study of C2H4 Adsorption on Fe3O4(001)

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arXiv 2023年

作者： Puntscher, Lena Sombut, Panukorn Wang, Chunlei Ulreich, Manuel Pavelec, Jiri Rafsanjani-Abbasi, Ali Meier, Matthias Lagin, Adam Setvin, Martin Diebold, Ulrike Franchini, Cesare Schmid, Michael Parkinson, Gareth S. Institute of Applied Physics TU Wien Vienna Austria Faculty of Physics Center for Computational Materials Science University of Vienna Vienna Austria Department of Surface and Plasma Science Faculty of Mathematics and Physics Charles University Prague Czech Republic Dipartimento di Fisica e Astronomia Università di Bologna Bologna Italy

The adsorption/desorption of ethene (C2H4), also commonly known as ethylene, on Fe3O4(001) was studied under ultrahigh vacuum conditions using temperature programmed desorption (TPD), scanning tunneling microscopy, x-ray photoelectron spectroscopy, and density functional theory (DFT) based computations. To interpret the TPD data, we have employed a new analysis method based on equilibrium thermodynamics. C2H4 adsorbs intact at all coverages and interacts most strongly with surface defects such as antiphase domain boundaries and Fe adatoms. On the regular surface, C2H4 binds atop surface Fe sites up to a coverage of 2 molecules per (√2×√2)R45° unit cell, with every second Fe occupied. A desorption energy of 0.36 eV is determined by analysis of the TPD spectra at this coverage, which is approximately 0.1-0.2 eV lower than the value calculated by DFT+U with van der Waals corrections. Additional molecules are accommodated in between the Fe rows. These are stabilized by attractive interactions with the molecules adsorbed at Fe sites. The total capacity of the surface for C2H4 adsorption is found to be close to 4 molecules per (√2×√2)R45° unit cell. © 2023, CC BY.

关键词： Ethylene

Supervised, semi-supervised, and unsupervised learning of the Domany-Kinzel model

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arXiv 2023年

作者： Tuo, Kui Li, Wei Deng, Shengfeng Zhu, Yueying Institute of Particle Physics Central China Normal University Wuhan430079 China School of Physics and Information Technology Shannxi Normal University Xi’an710061 China Research Center of Applied Mathematics and Interdisciplinary Science Wuhan Textile University Wuhan430073 China

The Domany Kinzel (DK) model encompasses several types of non-equilibrium phase transitions, depending on the selected parameters. We apply supervised, semi-supervised, and unsupervised learning methods to studying the phase transitions and critical behaviors of the (1+1)-dimensional DK model. The supervised and the semi-supervised learning methods permit the estimations of the critical points, the spatial and temporal correlation exponents, concerning labelled and unlabelled DK configurations, respectively. Furthermore, Principal Component Analysis (PCA) and autoencoder can classify the DK phases. Copyright © 2023, The Authors. All rights reserved.

关键词： Principal component analysis

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

Comparative study on atomic ionization in bicircular laser fields by length and velocity gauges S-matrix theory

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Chinese physics B 2020年第2期29卷 254-261页

作者： Hong Xia Xin-Yan Jia Xiao-Lei Hao Li Guo Dai-He Fan Gen-Bai Chu Jing Chen School of Physical Science and Technology Southwest Jiaotong UniversityChengdu 610031China Institute of Theoretical Physics and Department of Physics Shanxi UniversityTaiyuan 030006China State Key Laboratory for Quantum Optics and Center for Cold Atom Physics Shanghai Institute of Optics and Fine MechanicsChinese Academy of SciencesShanghai 201800China Science and Technology on Plasma Physics Laboratory Laser Fusion Research CenterChina Academy of Engineering PhysicsMianyang 621900China HEDPS Center for Applied Physics and TechnologyPeking UniversityBeijing 100871China Institute of Applied Physics and Computational Mathematics Beijing 100088China

Ionization of atoms in counter-rotating and co-rotating bicircular laser fields is studied using the S-matrix theory in both length and velocity *** show that for both the bicircular fields,ionization rates are enhanced when the two circularly polarized lights have comparable *** addition,the curves of ionization rate versus the field amplitude ratio of the two colors for counter-rotating and co-rotating fields coincide with each other in the length gauge case at the total laser intensity 5×10^14 W/cm^2,which agrees with the experimental ***,the degree of the coincidence between the ionization rate curves of the two bicircular fields decreases with the increasing field amplitude ratio and decreasing total laser *** the help of the ADK theory,the above characteristics of the ionization rate curves can be well interpreted,which is related to the transition from the tunneling to multiphoton ionization mechanism.

关键词： above-threshold ionization(ATI) bicircular laser fields length gauge velocity gauge S-matrix theory

Machine learning interatomic potential for molecular dynamics simulation of the ferroelectric KNbO3 perovskite

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Physical Review B 2023年第1期107卷 014101-014101页

作者： Hao-Cheng Thong XiaoYang Wang Jian Han Linfeng Zhang Bei Li Ke Wang Ben Xu State Key Laboratory of New Ceramics and Fine Processing School of Materials Science and Engineering Tsinghua University Beijing 100084 People's Republic of China Graduate School China Academy of Engineering Physics Beijing 100193 People's Republic of China Laboratory of Computational Physics Institute of Applied Physics and Computational Mathematics Beijing 100088 People's Republic of China Beijing Institute of Big Data Research Beijing 100871 People's Republic of China School of Materials Science and Engineering Research Center for Materials Genome Engineering Wuhan University of Technology Wuhan 430070 People's Republic of China

Ferroelectric perovskites have been ubiquitously applied in piezoelectric devices for decades, among which ecofriendly lead-free (K,Na)NbO3−based materials have been recently demonstrated to be an excellent candidate for sustainable development. Molecular dynamics is a versatile theoretical calculation approach for the investigation of the dynamical properties of ferroelectric perovskites. However, molecular dynamics simulation of ferroelectric perovskites has been limited to simple systems, since the conventional construction of interatomic potential is rather difficult and inefficient. In the present study, we construct a machine-learning interatomic potential of KNbO3 [as a representative system of (K,Na)NbO3] by using a deep neural network model. Including first-principles calculation data into the training data set ensures the quantum-mechanics accuracy of the interatomic potential. The molecular dynamics based on machine-learning interatomic potential shows good agreement with the first-principles calculations, which can accurately predict multiple fundamental properties, e.g., atomic force, energy, elastic properties, and phonon dispersion. In addition, the interatomic potential exhibits satisfactory performance in the simulation of domain wall and temperature-dependent phase transition. The construction of interatomic potential based on machine learning could potentially be transferred to other ferroelectric perovskites and consequently benefit the theoretical study of ferroelectrics.

关键词： Domain walls Elasticity Ferroelectricity First-principles calculations Phase transitions Phonons Piezoelectricity Machine learning Molecular dynamics

How Graph Neural Networks Learn: Lessons from Training Dynamics

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arXiv 2023年

作者： Yang, Chenxiao Wu, Qitian Wipf, David Sun, Ruoyu Yan, Junchi School of Artificial Intelligence Department of Computer Science and Engineering MoE Lab of AI Shanghai Jiao Tong University China Amazon Web Services United States School of Data Science The Chinese University of Hong Kong Shenzhen China Shenzhen International Center for Industrial and Applied Mathematics Shenzhen Research Institute of Big Data China

A long-standing goal in deep learning has been to characterize the learning behavior of black-box models in a more interpretable manner. For graph neural networks (GNNs), considerable advances have been made in formalizing what functions they can represent, but whether GNNs will learn desired functions during the optimization process remains less clear. To fill this gap, we study their training dynamics in function space. In particular, we find that the gradient descent optimization of GNNs implicitly leverages the graph structure to update the learned function, as can be quantified by a phenomenon which we call kernel-graph alignment. We provide theoretical explanations for the emergence of this phenomenon in the overparameterized regime and empirically validate it on real-world GNNs. This finding offers new interpretable insights into when and why the learned GNN functions generalize, highlighting their limitations in heterophilic graphs. Practically, we propose a parameter-free algorithm that directly uses a sparse matrix (i.e. graph adjacency) to update the learned function. We demonstrate that this embarrassingly simple approach can be as effective as GNNs while being orders-of-magnitude faster. Copyright © 2023, The Authors. All rights reserved.

关键词： Graph neural networks