Pub Date : 2020-07-16DOI: 10.3997/2214-4609.202011041
U. Waheed, E. Haghighat, T. Alkhalifah, Chao Song, Q. Hao
The eikonal equation is utilized across a wide spectrum of science and engineering disciplines. In seismology, it regulates seismic wave traveltimes needed for applications like source localization, imaging, and inversion. Several numerical algorithms have been developed over the years to solve the eikonal equation. However, they suffer from computational bottleneck when repeated computations are needed for perturbations in the velocity model and/or the source location, particularly in large 3D models. Here, we employ the emerging paradigm of physics-informed neural networks (PINNs) to solve the eikonal equation. By minimizing a loss function formed by imposing the validity of the eikonal equation, we train a neural network to produce traveltimes that are consistent with the underlying partial differential equation. More specifically, to tackle point-source singularity, we use the factored eikonal equation. We observe sufficiently high traveltime accuracy for most applications of interest. We also demonstrate how machine learning techniques like transfer learning and surrogate modeling can be used to massively speed up traveltime computations for updated velocity models and source locations. These properties of the PINN eikonal solver are highly desirable in obtaining an efficient forward modeling engine for seismic inversion applications.
{"title":"Eikonal Solution Using Physics-Informed Neural Networks","authors":"U. Waheed, E. Haghighat, T. Alkhalifah, Chao Song, Q. Hao","doi":"10.3997/2214-4609.202011041","DOIUrl":"https://doi.org/10.3997/2214-4609.202011041","url":null,"abstract":"The eikonal equation is utilized across a wide spectrum of science and engineering disciplines. In seismology, it regulates seismic wave traveltimes needed for applications like source localization, imaging, and inversion. Several numerical algorithms have been developed over the years to solve the eikonal equation. However, they suffer from computational bottleneck when repeated computations are needed for perturbations in the velocity model and/or the source location, particularly in large 3D models. Here, we employ the emerging paradigm of physics-informed neural networks (PINNs) to solve the eikonal equation. By minimizing a loss function formed by imposing the validity of the eikonal equation, we train a neural network to produce traveltimes that are consistent with the underlying partial differential equation. More specifically, to tackle point-source singularity, we use the factored eikonal equation. We observe sufficiently high traveltime accuracy for most applications of interest. We also demonstrate how machine learning techniques like transfer learning and surrogate modeling can be used to massively speed up traveltime computations for updated velocity models and source locations. These properties of the PINN eikonal solver are highly desirable in obtaining an efficient forward modeling engine for seismic inversion applications.","PeriodicalId":8424,"journal":{"name":"arXiv: Computational Physics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88701816","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-07-06DOI: 10.1103/physrevx.10.041034
Michele Invernizzi, P. Piaggi, M. Parrinello
The sampling problem lies at the heart of atomistic simulations and over the years many different enhanced sampling methods have been suggested towards its solution. These methods are often grouped into two broad families. On the one hand methods such as umbrella sampling and metadynamics that build a bias potential based on few order parameters or collective variables. On the other hand, tempering methods such as replica exchange that combine different thermodynamic ensembles in one single expanded ensemble. We instead adopt a unifying perspective, focusing on the target probability distribution sampled by the different methods. This allows us to introduce a new class of collective-variables-based bias potentials that can be used to sample any of the expanded ensembles normally sampled via replica exchange. We also provide a practical implementation, by properly adapting the iterative scheme of the recently developed on-the-fly probability enhanced sampling method [Invernizzi and Parrinello, J. Phys. Chem. Lett. 11.7 (2020)], which was originally introduced for metadynamics-like sampling. The resulting method is very general and can be used to achieve different types of enhanced sampling. It is also reliable and simple to use, since it presents only few and robust external parameters and has a straightforward reweighting scheme. Furthermore, it can be used with any number of parallel replicas. We show the versatility of our approach with applications to multicanonical and multithermal-multibaric simulations, thermodynamic integration, umbrella sampling, and combinations thereof.
采样问题是原子模拟的核心，多年来，人们提出了许多不同的增强采样方法来解决这个问题。这些方法通常分为两大类。一方面，伞式采样和元动力学等方法基于少量序参数或集体变量建立了偏置势。另一方面，回火方法，如副本交换，将不同的热力学集成在一个单一的扩展集成中。相反，我们采用统一的观点，关注不同方法抽样的目标概率分布。这使我们能够引入一类新的基于集体变量的偏置势，可用于对通常通过副本交换采样的任何扩展集合进行采样。我们还提供了一个实际的实现，通过适当地采用最近开发的动态概率增强采样方法的迭代方案[Invernizzi和Parrinello, J. Phys]。化学。Lett. 11.7(2020)]，最初是为类似元动力学的采样而引入的。所得到的方法是非常通用的，可用于实现不同类型的增强采样。它还可靠且使用简单，因为它只提供少量且健壮的外部参数，并且具有直接的重加权方案。此外，它可以用于任意数量的并行副本。我们展示了我们的方法的多功能性，应用于多声道和多热-多压模拟，热力学集成，伞式采样及其组合。
{"title":"Unified Approach to Enhanced Sampling","authors":"Michele Invernizzi, P. Piaggi, M. Parrinello","doi":"10.1103/physrevx.10.041034","DOIUrl":"https://doi.org/10.1103/physrevx.10.041034","url":null,"abstract":"The sampling problem lies at the heart of atomistic simulations and over the years many different enhanced sampling methods have been suggested towards its solution. These methods are often grouped into two broad families. On the one hand methods such as umbrella sampling and metadynamics that build a bias potential based on few order parameters or collective variables. On the other hand, tempering methods such as replica exchange that combine different thermodynamic ensembles in one single expanded ensemble. We instead adopt a unifying perspective, focusing on the target probability distribution sampled by the different methods. This allows us to introduce a new class of collective-variables-based bias potentials that can be used to sample any of the expanded ensembles normally sampled via replica exchange. We also provide a practical implementation, by properly adapting the iterative scheme of the recently developed on-the-fly probability enhanced sampling method [Invernizzi and Parrinello, J. Phys. Chem. Lett. 11.7 (2020)], which was originally introduced for metadynamics-like sampling. The resulting method is very general and can be used to achieve different types of enhanced sampling. It is also reliable and simple to use, since it presents only few and robust external parameters and has a straightforward reweighting scheme. Furthermore, it can be used with any number of parallel replicas. We show the versatility of our approach with applications to multicanonical and multithermal-multibaric simulations, thermodynamic integration, umbrella sampling, and combinations thereof.","PeriodicalId":8424,"journal":{"name":"arXiv: Computational Physics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86612143","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The analytical solution of neutron transport equation has fascinated mathematicians and physicists alike since the Milne half-space problem was introduce in 1921 [1]. Numerous numerical solutions exist, but understandably, there are only a few analytical solutions, with the prominent one being the singular eigenfunction expansion (SEE) introduced by Case [2] in 1960. For the half-space, the method, though yielding, an elegant analytical form resulting from half-range completeness, requires numerical evaluation of complicated integrals. In addition, one finds closed form analytical expressions only for the infinite medium and half-space cases. One can find the flux in a slab only iteratively. That is to say, in general one must expend a considerable numerical effort to get highly precise benchmarks from SEE. As a result, investigators have devised alternative methods, such as the CN [3], FN [4] and Greens Function Method (GFM) [5] based on the SEE have been devised. These methods take the SEE at their core and construct a numerical method around the analytical form. The FN method in particular has been most successful in generating highly precise benchmarks. No method yielding a precise numerical solution has yet been based solely on a fundamental discretization until now. Here, we show for the albedo problem with a source on the vacuum boundary of a homogeneous medium, a precise numerical solution is possible via Lagrange interpolation over a discrete set of directions.
{"title":"Solution of the Monoenergetic Neutron Transport Equation in a Half Space via Singular Eigenfunction Expansion","authors":"B. Ganapol","doi":"10.13182/T123-33541","DOIUrl":"https://doi.org/10.13182/T123-33541","url":null,"abstract":"The analytical solution of neutron transport equation has fascinated mathematicians and physicists alike since the Milne half-space problem was introduce in 1921 [1]. Numerous numerical solutions exist, but understandably, there are only a few analytical solutions, with the prominent one being the singular eigenfunction expansion (SEE) introduced by Case [2] in 1960. For the half-space, the method, though yielding, an elegant analytical form resulting from half-range completeness, requires numerical evaluation of complicated integrals. In addition, one finds closed form analytical expressions only for the infinite medium and half-space cases. One can find the flux in a slab only iteratively. That is to say, in general one must expend a considerable numerical effort to get highly precise benchmarks from SEE. As a result, investigators have devised alternative methods, such as the CN [3], FN [4] and Greens Function Method (GFM) [5] based on the SEE have been devised. These methods take the SEE at their core and construct a numerical method around the analytical form. The FN method in particular has been most successful in generating highly precise benchmarks. No method yielding a precise numerical solution has yet been based solely on a fundamental discretization until now. Here, we show for the albedo problem with a source on the vacuum boundary of a homogeneous medium, a precise numerical solution is possible via Lagrange interpolation over a discrete set of directions.","PeriodicalId":8424,"journal":{"name":"arXiv: Computational Physics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91475386","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-06-16DOI: 10.1103/PHYSREVFLUIDS.6.050501
Suraj Pawar, O. San
In the past couple of years, there is a proliferation in the use of machine learning approaches to represent subgrid scale processes in geophysical flows with an aim to improve the forecasting capability and to accelerate numerical simulations of these flows. Despite its success for different types of flow, the online deployment of a data-driven closure model can cause instabilities and biases in modeling the overall effect of subgrid scale processes, which in turn leads to inaccurate prediction. To tackle this issue, we exploit the data assimilation technique to correct the physics-based model coupled with the neural network as a surrogate for unresolved flow dynamics in multiscale systems. In particular, we use a set of neural network architectures to learn the correlation between resolved flow variables and the parameterizations of unresolved flow dynamics and formulate a data assimilation approach to correct the hybrid model during their online deployment. We illustrate our framework in a application of the multiscale Lorenz 96 system for which the parameterization model for unresolved scales is exactly known. Our analysis, therefore, comprises a predictive dynamical core empowered by (i) a data-driven closure model for subgrid scale processes, (ii) a data assimilation approach for forecast error correction, and (iii) both data-driven closure and data assimilation procedures. We show significant improvement in the long-term perdition of the underlying chaotic dynamics with our framework compared to using only neural network parameterizations for future prediction. Moreover, we demonstrate that these data-driven parameterization models can handle the non-Gaussian statistics of subgrid scale processes, and effectively improve the accuracy of outer data assimilation workflow loops in a modular non-intrusive way.
{"title":"Data assimilation empowered neural network parametrizations for subgrid processes in geophysical flows","authors":"Suraj Pawar, O. San","doi":"10.1103/PHYSREVFLUIDS.6.050501","DOIUrl":"https://doi.org/10.1103/PHYSREVFLUIDS.6.050501","url":null,"abstract":"In the past couple of years, there is a proliferation in the use of machine learning approaches to represent subgrid scale processes in geophysical flows with an aim to improve the forecasting capability and to accelerate numerical simulations of these flows. Despite its success for different types of flow, the online deployment of a data-driven closure model can cause instabilities and biases in modeling the overall effect of subgrid scale processes, which in turn leads to inaccurate prediction. To tackle this issue, we exploit the data assimilation technique to correct the physics-based model coupled with the neural network as a surrogate for unresolved flow dynamics in multiscale systems. In particular, we use a set of neural network architectures to learn the correlation between resolved flow variables and the parameterizations of unresolved flow dynamics and formulate a data assimilation approach to correct the hybrid model during their online deployment. We illustrate our framework in a application of the multiscale Lorenz 96 system for which the parameterization model for unresolved scales is exactly known. Our analysis, therefore, comprises a predictive dynamical core empowered by (i) a data-driven closure model for subgrid scale processes, (ii) a data assimilation approach for forecast error correction, and (iii) both data-driven closure and data assimilation procedures. We show significant improvement in the long-term perdition of the underlying chaotic dynamics with our framework compared to using only neural network parameterizations for future prediction. Moreover, we demonstrate that these data-driven parameterization models can handle the non-Gaussian statistics of subgrid scale processes, and effectively improve the accuracy of outer data assimilation workflow loops in a modular non-intrusive way.","PeriodicalId":8424,"journal":{"name":"arXiv: Computational Physics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84844259","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The viscosity and thermal conductivity of binary, ternary and quaternary mixtures of helium, neon, argon, and krypton at low density are computed for wide ranges of temperature and molar fractions, applying the Chapman-Enskog method. Ab initio interatomic potentials are employed in order to calculate the omega-integrals. The relative numerical errors of the viscosity and thermal conductivity do not exceed 1.e-6 and 1.e-5, respectively. The relative uncertainty related to the interatomic potential is about 0.1%. A comparison of the present data with results reported in other papers available in the literature shows a significant improvement of accuracy of the transport coefficients considered here.
{"title":"Transport coefficients of multi-component mixtures of noble gases based on ab initio potentials: Viscosity and thermal conductivity","authors":"F. Sharipov, V. J. Benites","doi":"10.1063/5.0016261","DOIUrl":"https://doi.org/10.1063/5.0016261","url":null,"abstract":"The viscosity and thermal conductivity of binary, ternary and quaternary mixtures of helium, neon, argon, and krypton at low density are computed for wide ranges of temperature and molar fractions, applying the Chapman-Enskog method. Ab initio interatomic potentials are employed in order to calculate the omega-integrals. The relative numerical errors of the viscosity and thermal conductivity do not exceed 1.e-6 and 1.e-5, respectively. The relative uncertainty related to the interatomic potential is about 0.1%. A comparison of the present data with results reported in other papers available in the literature shows a significant improvement of accuracy of the transport coefficients considered here.","PeriodicalId":8424,"journal":{"name":"arXiv: Computational Physics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73817924","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-06-11DOI: 10.1103/physrevb.102.134209
J. Lian, Hong-Yu Wu, Wei‐Qing Huang, Wangyu Hu, Gui‐Fang Huang
Generating irreducible site-occupancy configurations by taking advantage of crystal symmetry is a ubiquitous method for accelerating of disordered structure prediction, which plays an important role in condensed matter physics and material science. Here, we present a new algorithm for generating irreducible site-occupancy configurations, that works for arbitrary parent cell with any supercell expansion matrix, and for any number of atom types with arbitrary stoichiometry. The new algorithm identifies the symmetrically equivalent configurations by searching the space group operations of underlying lattice and building the equivalent atomic matrix based on it. Importantly, an integer representation of configurations can greatly accelerate the speed of elimination of duplicate configurations, resulting into a linear scale of run time with the number of irreducible configurations that finally found. Moreover, based on our new algorithm, we write the corresponding code named as disorder in FORTRAN programming language, and the performance test results show that the time efficiency of our disorder code is superior to that of other related codes (supercell, enumlib and SOD).
{"title":"Algorithm for generating irreducible site-occupancy configurations","authors":"J. Lian, Hong-Yu Wu, Wei‐Qing Huang, Wangyu Hu, Gui‐Fang Huang","doi":"10.1103/physrevb.102.134209","DOIUrl":"https://doi.org/10.1103/physrevb.102.134209","url":null,"abstract":"Generating irreducible site-occupancy configurations by taking advantage of crystal symmetry is a ubiquitous method for accelerating of disordered structure prediction, which plays an important role in condensed matter physics and material science. Here, we present a new algorithm for generating irreducible site-occupancy configurations, that works for arbitrary parent cell with any supercell expansion matrix, and for any number of atom types with arbitrary stoichiometry. The new algorithm identifies the symmetrically equivalent configurations by searching the space group operations of underlying lattice and building the equivalent atomic matrix based on it. Importantly, an integer representation of configurations can greatly accelerate the speed of elimination of duplicate configurations, resulting into a linear scale of run time with the number of irreducible configurations that finally found. Moreover, based on our new algorithm, we write the corresponding code named as disorder in FORTRAN programming language, and the performance test results show that the time efficiency of our disorder code is superior to that of other related codes (supercell, enumlib and SOD).","PeriodicalId":8424,"journal":{"name":"arXiv: Computational Physics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89935600","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Justin S. Smith, N. Lubbers, A. Thompson, K. Barros
Abstract Machine learning models, trained on data from ab initio quantum simulations, are yielding molecular dynamics potentials with unprecedented accuracy. One limiting factor is the quantity of available training data, which can be expensive to obtain. A quantum simulation often provides all atomic forces, in addition to the total energy of the system. These forces provide much more information than the energy alone. It may appear that training a model to this large quantity of force data would introduce significant computational costs. Actually, training to all available force data should only be a few times more expensive than training to energies alone. Here, we present a new algorithm for efficient force training, and benchmark its accuracy by training to forces from real-world datasets for organic chemistry and bulk aluminum.
{"title":"Simple and efficient algorithms for training machine learning potentials to force data.","authors":"Justin S. Smith, N. Lubbers, A. Thompson, K. Barros","doi":"10.2172/1763572","DOIUrl":"https://doi.org/10.2172/1763572","url":null,"abstract":"Abstract Machine learning models, trained on data from ab initio quantum simulations, are yielding molecular dynamics potentials with unprecedented accuracy. One limiting factor is the quantity of available training data, which can be expensive to obtain. A quantum simulation often provides all atomic forces, in addition to the total energy of the system. These forces provide much more information than the energy alone. It may appear that training a model to this large quantity of force data would introduce significant computational costs. Actually, training to all available force data should only be a few times more expensive than training to energies alone. Here, we present a new algorithm for efficient force training, and benchmark its accuracy by training to forces from real-world datasets for organic chemistry and bulk aluminum.","PeriodicalId":8424,"journal":{"name":"arXiv: Computational Physics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76657317","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-05-20DOI: 10.1016/j.mechrescom.2020.103614
David Sondak, Thomas M. Smith, R. Pawlowski, S. Conde, J. Shadid
{"title":"High Rayleigh number variational multiscale large eddy simulations of Rayleigh-Bénard convection","authors":"David Sondak, Thomas M. Smith, R. Pawlowski, S. Conde, J. Shadid","doi":"10.1016/j.mechrescom.2020.103614","DOIUrl":"https://doi.org/10.1016/j.mechrescom.2020.103614","url":null,"abstract":"","PeriodicalId":8424,"journal":{"name":"arXiv: Computational Physics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74703500","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this work, we present a numerical implementation to compute the atom centered descriptors introduced by Bartok et al (Phys. Rev. B, 87, 184115, 2013) based on the harmonic analysis of the atomic neighbor density function. Specifically, we focus on two types of descriptors, the smooth SO(3) power spectrum with the explicit inclusion of a radial basis and the SO(4) bispectrum obtained through mapping the radial component onto a polar angle of a four dimensional hypersphere. With these descriptors, various interatomic potentials for binary Ni-Mo alloys are obtained based on linear and neural network regression models. Numerical experiments suggest that both descriptors produce similar results in terms of accuracy. For linear regression, the smooth SO(3) power spectrum is superior to the SO(4) bispectrum when a large band limit is used. In neural network regression, a better accuracy can be achieved with even less number of expansion components for both descriptors. As such, we demonstrate that spectral neural network potentials are feasible choices for large scale atomistic simulation.
{"title":"Spectral neural network potentials for binary alloys","authors":"David Zagaceta, Howard Yanxon, Q. Zhu","doi":"10.1063/5.0013208","DOIUrl":"https://doi.org/10.1063/5.0013208","url":null,"abstract":"In this work, we present a numerical implementation to compute the atom centered descriptors introduced by Bartok et al (Phys. Rev. B, 87, 184115, 2013) based on the harmonic analysis of the atomic neighbor density function. Specifically, we focus on two types of descriptors, the smooth SO(3) power spectrum with the explicit inclusion of a radial basis and the SO(4) bispectrum obtained through mapping the radial component onto a polar angle of a four dimensional hypersphere. With these descriptors, various interatomic potentials for binary Ni-Mo alloys are obtained based on linear and neural network regression models. Numerical experiments suggest that both descriptors produce similar results in terms of accuracy. For linear regression, the smooth SO(3) power spectrum is superior to the SO(4) bispectrum when a large band limit is used. In neural network regression, a better accuracy can be achieved with even less number of expansion components for both descriptors. As such, we demonstrate that spectral neural network potentials are feasible choices for large scale atomistic simulation.","PeriodicalId":8424,"journal":{"name":"arXiv: Computational Physics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79497415","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The distribution of eigenvalues of the wave equation in a bounded domain is known as Weyl's problem. We describe several computational projects related to the cumulative state number, defined as the number of states having wavenumber up to a maximum value. This quantity and its derivative, the density of states, have important applications in nuclear physics, degenerate Fermi gases, blackbody radiation, Bose-Einstein condensation and the Casimir effect. Weyl's theorem states that, in the limit of large wavenumbers, the cumulative state number depends only on the volume of the bounding domain and not on its shape. Corrections to this behavior are well known and depend on the surface area of the bounding domain, its curvature and other features. We describe several projects that allow readers to investigate this dependence for three bounding domains - a rectangular box, a sphere, and a circular cylinder. Quasi-one- and two-dimensional systems can be analyzed by considering various limits. The projects have applications in statistical mechanics, but can also be integrated into quantum mechanics, nuclear physics, or computational physics courses.
{"title":"Weyl's problem: A computational approach","authors":"Isaac Bowser, Ken Kiers, E. Mitchell, J. Kiers","doi":"10.1119/10.0001657","DOIUrl":"https://doi.org/10.1119/10.0001657","url":null,"abstract":"The distribution of eigenvalues of the wave equation in a bounded domain is known as Weyl's problem. We describe several computational projects related to the cumulative state number, defined as the number of states having wavenumber up to a maximum value. This quantity and its derivative, the density of states, have important applications in nuclear physics, degenerate Fermi gases, blackbody radiation, Bose-Einstein condensation and the Casimir effect. Weyl's theorem states that, in the limit of large wavenumbers, the cumulative state number depends only on the volume of the bounding domain and not on its shape. Corrections to this behavior are well known and depend on the surface area of the bounding domain, its curvature and other features. We describe several projects that allow readers to investigate this dependence for three bounding domains - a rectangular box, a sphere, and a circular cylinder. Quasi-one- and two-dimensional systems can be analyzed by considering various limits. The projects have applications in statistical mechanics, but can also be integrated into quantum mechanics, nuclear physics, or computational physics courses.","PeriodicalId":8424,"journal":{"name":"arXiv: Computational Physics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73007675","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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