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arXiv 2018年

作者： Torquato, Salvatore Department of Chemistry Department of Physics Princeton Institute for the Science and Technology of Materials Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

Hyperuniform states of matter are correlated systems that are characterized by an anomalous suppression of long-wavelength (i.e., large-length-scale) density fluctuations compared to those found in garden-variety disordered systems, such as ordinary fluids and amorphous solids. All perfect crystals, perfect quasicrystals and special disordered systems are hyperuniform. Thus, the hyperuniformity concept enables a unified framework to classify and structurally characterize crystals, quasicrystals and the exotic disordered varieties. While disordered hyperuniform systems were largely unknown in the scientific community over a decade ago, now there is a realization that such systems arise in a host of contexts across the physical, materials, chemical, mathematical, engineering, and biological sciences, including disordered ground states, glass formation, jamming, Coulomb systems, spin systems, photonic and electronic band structure, localization of waves and excitations, self-organization, fluid dynamics, number theory, stochastic point processes, integral and stochastic geometry, the immune system, and photoreceptor cells. Such unusual amorphous states can be obtained via equilibrium or nonequilibrium routes, and come in both quantum-mechanical and classical varieties. The connections of hyperuniform states of matter to many different areas of fundamental science appear to be profound and yet our theoretical understanding of these unusual systems is only in its infancy. The purpose of this review article is to introduce the reader to the theoretical foundations of hyperuniform ordered and disordered systems. Special focus will be placed on fundamental and practical aspects of the disordered kinds, including our current state of knowledge of these exotic amorphous systems as well as their formation and novel physical properties. Copyright © 2018, The Authors. All rights reserved.

关键词： Ground state

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Floquet-Weyl semimetals generated by an optically resonant interband transition

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Physical Review B 2022年第8期106卷 085206-085206页

作者： Runnan Zhang Ken-ichi Hino Nobuya Maeshima Doctoral Program in Materials Science Graduate School of Pure and Applied Sciences University of Tsukuba Tsukuba Ibaraki 305-8573 Japan Division of Materials Science Faculty of Pure and Applied Sciences University of Tsukuba Tsukuba 305-8573 Japan Center for Computational Sciences University of Tsukuba Tsukuba 305-8577 Japan

Floquet-Weyl semimetals (FWSMs) generated by irradiation of a continuous-wave laser with left-hand circular polarization (rotating in counterclockwise sense with time) on the group II–V narrow-gap semiconductor Zn3As2 are theoretically investigated, where the frequency of the laser is set nearly resonant with the band gap of the crystal. It is found that the excitation of the crystal by such a laser induces two types of FWSM phases that differ absolutely in character. To be specific, the associated two pairs of Weyl points are stably formed by band touching between Floquet sidebands ascribable to a valence band labeled Jz=±3/2 and a conduction band labeled Jz=±1/2, where Jz represents the z component of total angular-momentum quantum number of the Γ point and a double sign corresponds. Here, one FWSM state composed of the up-spin Floquet sidebands relevant to Jz=3/2 and 1/2 shows almost quadratic band-touching in the vicinity of the associated pair of Weyl points, while the other FWSM state composed of the down-spin Floquet sidebands relevant to Jz=−3/2 and −1/2 shows linear band-touching. Furthermore, it is revealed that both up-spin and down-spin sidebands host nontrivial two-dimensional surface states that are pinned to the respective pairs of the Weyl points. Both surface states also show different energy dispersions and physical properties. A more detailed discussion is presented in the text on the origin of the above findings, the chirality of the FWSM phases, the alteration of topological order, laser-induced magnetic properties, and so on.

关键词： Optical pumping Topological materials Weyl semimetal

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Deep Potential Molecular Dynamics: A Scalable Model with the Accuracy of Quantum Mechanics

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Physical Review Letters 2018年第14期120卷 143001-143001页

作者： Linfeng Zhang Jiequn Han Han Wang Roberto Car Weinan E Department of Mathematics and Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA and Center for Data Science Beijing International Center for Mathematical Research Peking University Beijing Institute of Big Data Research Beijing 100871 People’s Republic of China

We introduce a scheme for molecular simulations, the deep potential molecular dynamics (DPMD) method, based on a many-body potential and interatomic forces generated by a carefully crafted deep neural network trained with ab initio data. The neural network model preserves all the natural symmetries in the problem. It is first-principles based in the sense that there are no ad hoc components aside from the network model. We show that the proposed scheme provides an efficient and accurate protocol in a variety of systems, including bulk materials and molecules. In all these cases, DPMD gives results that are essentially indistinguishable from the original data, at a cost that scales linearly with system size.

关键词： Classical statistical mechanics Density functional theory First-principles calculations Interatomic & molecular potentials Learning Neuronal networks Potential energy surfaces Solid-state chemistry Thermal properties Van der Waals interaction Molecular dynamics

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A statistical mechanics framework for constructing non-equilibrium thermodynamic models

arXiv

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arXiv 2023年

作者： Leadbetter, Travis Purohit, Prashant K. Reina, Celia Graduate Group in Applied Mathematics and Computational Science University of Pennsylvania PhiladelphiaPA19104 United States Department of Mechanical Engineering and Applied Mechanics University of Pennsylvania PhiladelphiaPA19104 United States

Far-from-equilibrium phenomena are critical to all natural and engineered systems, and essential to biological processes responsible for life. For over a century and a half, since Carnot, Clausius, Maxwell, Boltzmann, and Gibbs, among many others, laid the foundation for our understanding of equilibrium processes, scientists and engineers have dreamed of an analogous treatment of non-equilibrium systems. But despite tremendous efforts, a universal theory of non-equilibrium behavior akin to equilibrium statistical mechanics and thermodynamics has evaded description. Several methodologies have proved their ability to accurately describe complex non-equilibrium systems at the macroscopic scale, but their accuracy and predictive capacity is predicated on either phenomenological kinetic equations fit to microscopic data, or on running concurrent simulations at the particle level. Instead, we provide a framework for deriving stand-alone macroscopic thermodynamics models directly from microscopic physics without fitting in overdamped Langevin systems. The only necessary ingredient is a functional form for a parameterized, approximate density of states, in analogy to the assumption of a uniform density of states in the equilibrium microcanonical ensemble. We highlight this framework's effectiveness by deriving analytical approximations for evolving mechanical and thermodynamic quantities in a model of coiled-coil proteins and double stranded DNA, thus producing, to the authors' knowledge, the first derivation of the governing equations for a phase propagating system under general loading conditions without appeal to phenomenology. The generality of our treatment allows for application to any system described by Langevin dynamics with arbitrary interaction energies and external driving, including colloidal macromolecules, hydrogels, and biopolymers. © 2023, CC BY.

关键词： Statistical mechanics

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Learning the solution operator of parametric partial differential equations with physics-informed deeponets

arXiv

引用

arXiv 2021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

Deep operator networks (DeepONets) are receiving increased attention thanks to their demonstrated capability to approximate nonlinear operators between infinite-dimensional Banach spaces. However, despite their remarkable early promise, they typically require large training data-sets consisting of paired input-output observations which may be expensive to obtain, while their predictions may not be consistent with the underlying physical principles that generated the observed data. In this work, we propose a novel model class coined as physics-informed DeepONets, which introduces an effective regularization mechanism for biasing the outputs of DeepOnet models towards ensuring physical consistency. This is accomplished by leveraging automatic differentiation to impose the underlying physical laws via soft penalty constraints during model training. We demonstrate that this simple, yet remarkably effective extension can not only yield a significant improvement in the predictive accuracy of DeepOnets, but also greatly reduce the need for large training data-sets. To this end, a remarkable observation is that physics-informed DeepONets are capable of solving parametric partial differential equations (PDEs) without any paired input-output observations, except for a set of given initial or boundary conditions. We illustrate the effectiveness of the proposed framework through a series of comprehensive numerical studies across various types of PDEs. Strikingly, a trained physics informed DeepOnet model can predict the solution of O(103) time-dependent PDEs in a fraction of a second - up to three orders of magnitude faster compared a conventional PDE solver. The data and code accompanying this manuscript are publicly available at https://***/PredictiveIntelligenceLab/Physics-informed-DeepONets. Copyright © 2021, The Authors. All rights reserved.

关键词： Mathematical operators

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Current-driven homogenization and effective medium parameters for finite samples

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Physical Review B 2013年第12期88卷 125131-125131页

作者： Vadim A. Markel Igor Tsukerman Departments of Radiology and Bioengineering and the Graduate Group in Applied Mathematics and Computational Science University of Pennsylvania Philadelphia Pennsylvania 19104 USA Department of Electrical and Computer Engineering The University of Akron Ohio 44325-3904 USA

Reflection and refraction of electromagnetic waves by artificial periodic composites (metamaterials) can be accurately modeled by an effective medium theory only if the boundary of the medium is explicitly taken into account and the two effective parameters of the medium (the index of refraction and the impedance) are correctly determined. Theories that consider infinite periodic composites do not satisfy the above condition. As a result, they can not model reflection and transmission by finite samples with the desired accuracy and are not useful for design of metamaterial-based devices. As an instructive case in point, we consider the “current-driven” homogenization theory, which has recently gained popularity. We apply this theory to the case of one-dimensional periodic medium wherein both exact and homogenization results can be obtained analytically in closed form. We show that, beyond the well-understood zero-cell limit, the current-driven homogenization result is inconsistent with the exact reflection and transmission characteristics of the slab.

关键词： ELECTROMAGNETISM MAGNETIC fields METAMATERIALS MAGNETOELECTRIC effect REFRACTION (Optics)

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Long-time integration of parametric evolution equations with physics-informed deeponets

arXiv

引用

arXiv 2021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

Ordinary and partial differential equations (ODEs/PDEs) play a paramount role in analyzing and simulating complex dynamic processes across all corners of science and engineering. In recent years machine learning tools are aspiring to introduce new effective ways of simulating PDEs, however existing approaches are not able to reliably return stable and accurate predictions across long temporal horizons. We aim to address this challenge by introducing an effective framework for learning infinite-dimensional operators that map random initial conditions to associated PDE solutions within a short time interval. Such latent operators can be parametrized by deep neural networks that are trained in an entirely self-supervised manner without requiring any paired input-output observations. Global long-time predictions across a range of initial conditions can be then obtained by iteratively evaluating the trained model using each prediction as the initial condition for the next evaluation step. This introduces a new approach to temporal domain decomposition that is shown to be effective in performing accurate long-time simulations for a wide range of parametric ODE and PDE systems, from wave propagation, to reaction-diffusion dynamics and stiff chemical kinetics, all at a fraction of the computational cost needed by classical numerical solvers. © 2021, CC BY-NC-SA.

关键词： Domain decomposition methods

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THE EXTENT OF SATURATION OF INDUCED IDEALS

arXiv

引用

arXiv 2022年

作者： Tsukuura, Kenta Doctoral Program in Mathematics Degree Programs in Pure and Applied Sciences Graduate School of Science and Technology University of Tsukuba Tsukuba305-8571 Japan

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UNDERSTANDING AND MITIGATING GRADIENT PATHOLOGIES IN PHYSICS-INFORMED NEURAL NETWORKS

arXiv

引用

arXiv 2020年

作者： Wang, Sifan Teng, Yujun Perdikaris, Paris Graduate Group in Applied Mathematics and Computational Science University of Pennsylvania PhiladelphiaPA 19104 United States Department of Mechanichal Engineering and Applied Mechanics University of Pennsylvania PhiladelphiaPA 19104 United States

The widespread use of neural networks across different scientific domains often involves constraining them to satisfy certain symmetries, conservation laws, or other domain knowledge. Such constraints are often imposed as soft penalties during model training and effectively act as domain-specific regularizers of the empirical risk loss. Physics-informed neural networks is an example of this philosophy in which the outputs of deep neural networks are constrained to approximately satisfy a given set of partial differential equations. In this work we review recent advances in scientific machine learning with a specific focus on the effectiveness of physics-informed neural networks in predicting outcomes of physical systems and discovering hidden physics from noisy data. We will also identify and analyze a fundamental mode of failure of such approaches that is related to numerical stiffness leading to unbalanced back-propagated gradients during model training. To address this limitation we present a learning rate annealing algorithm that utilizes gradient statistics during model training to balance the interplay between different terms in composite loss functions. We also propose a novel neural network architecture that is more resilient to such gradient pathologies. Taken together, our developments provide new insights into the training of constrained neural networks and consistently improve the predictive accuracy of physics-informed neural networks by a factor of 50-100x across a range of problems in computational physics. All code and data accompanying this manuscript are publicly available at https://***/PredictiveIntelligenceLab/ GradientPathologiesPINNs. Copyright © 2020, The Authors. All rights reserved.

关键词： Network architecture

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LEARNING ONLY ON BOUNDARIES: A PHYSICS-INFORMED NEURAL OPERATOR FOR SOLVING PARAMETRIC PARTIAL DIFFERENTIAL EQUATIONS IN COMPLEX GEOMETRIES

arXiv

引用

arXiv 2023年

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

Recently deep learning surrogates and neural operators have shown promise in solving partial differential equations (PDEs). However, they often require a large amount of training data and are limited to bounded domains. In this work, we present a novel physics-informed neural operator method to solve parametrized boundary value problems without labeled data. By reformulating the PDEs into boundary integral equations (BIEs), we can train the operator network solely on the boundary of the domain. This approach reduces the number of required sample points from O(Nd) to O(Nd−1), where d is the domain's dimension, leading to a significant acceleration of the training process. Additionally, our method can handle unbounded problems, which are unattainable for existing physics-informed neural networks (PINNs) and neural operators. Our numerical experiments show the effectiveness of parametrized complex geometries and unbounded problems. Copyright © 2023, The Authors. All rights reserved.

关键词： Boundary element method

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