Pub Date : 2020-04-25DOI: 10.1103/PHYSREVRESEARCH.2.033110
M. Magill, Andrew M. Nagel, H. D. de Haan
The neural network method of solving differential equations is used to approximate the electric potential and corresponding electric field in the slit-well microfluidic device. The device's geometry is non-convex, making this a challenging problem to solve using the neural network method. To validate the method, the neural network solutions are compared to a reference solution obtained using the finite element method. Additional metrics are presented that measure how well the neural networks recover important physical invariants that are not explicitly enforced during training: spatial symmetries and conservation of electric flux. Finally, as an application-specific test of validity, neural network electric fields are incorporated into particle simulations. Conveniently, the same loss functional used to train the neural networks also seems to provide a reliable estimator of the networks' true errors, as measured by any of the metrics considered here. In all metrics, deep neural networks significantly outperform shallow neural networks, even when normalized by computational cost. Altogether, the results suggest that the neural network method can reliably produce solutions of acceptable accuracy for use in subsequent physical computations, such as particle simulations.
{"title":"Neural network solutions to differential equations in nonconvex domains: Solving the electric field in the slit-well microfluidic device","authors":"M. Magill, Andrew M. Nagel, H. D. de Haan","doi":"10.1103/PHYSREVRESEARCH.2.033110","DOIUrl":"https://doi.org/10.1103/PHYSREVRESEARCH.2.033110","url":null,"abstract":"The neural network method of solving differential equations is used to approximate the electric potential and corresponding electric field in the slit-well microfluidic device. The device's geometry is non-convex, making this a challenging problem to solve using the neural network method. To validate the method, the neural network solutions are compared to a reference solution obtained using the finite element method. Additional metrics are presented that measure how well the neural networks recover important physical invariants that are not explicitly enforced during training: spatial symmetries and conservation of electric flux. Finally, as an application-specific test of validity, neural network electric fields are incorporated into particle simulations. Conveniently, the same loss functional used to train the neural networks also seems to provide a reliable estimator of the networks' true errors, as measured by any of the metrics considered here. In all metrics, deep neural networks significantly outperform shallow neural networks, even when normalized by computational cost. Altogether, the results suggest that the neural network method can reliably produce solutions of acceptable accuracy for use in subsequent physical computations, such as particle simulations.","PeriodicalId":8424,"journal":{"name":"arXiv: Computational Physics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81252994","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 : 2020-04-23DOI: 10.1007/978-3-030-69784-6_2
B. Anwasia
{"title":"The Maxwell–Stefan Diffusion Limit of a Hard-Sphere Kinetic Model for Mixtures","authors":"B. Anwasia","doi":"10.1007/978-3-030-69784-6_2","DOIUrl":"https://doi.org/10.1007/978-3-030-69784-6_2","url":null,"abstract":"","PeriodicalId":8424,"journal":{"name":"arXiv: Computational Physics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75636578","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 : 2020-03-11DOI: 10.21468/SCIPOSTPHYSCORE.2.2.011
C. Bauer
We assess numerical stabilization methods employed in fermion many-body quantum Monte Carlo simulations. In particular, we empirically compare various matrix decomposition and inversion schemes to gain control over numerical instabilities arising in the computation of equal-time and time-displaced Green's functions within the determinant quantum Monte Carlo (DQMC) framework. Based on this comparison, we identify a procedure based on pivoted QR decompositions which is both efficient and accurate to machine precision. The Julia programming language is used for the assessment and implementations of all discussed algorithms are provided in the open-source software library StableDQMC.jl [this http URL].
{"title":"Fast and stable determinant quantum Monte Carlo","authors":"C. Bauer","doi":"10.21468/SCIPOSTPHYSCORE.2.2.011","DOIUrl":"https://doi.org/10.21468/SCIPOSTPHYSCORE.2.2.011","url":null,"abstract":"We assess numerical stabilization methods employed in fermion many-body quantum Monte Carlo simulations. In particular, we empirically compare various matrix decomposition and inversion schemes to gain control over numerical instabilities arising in the computation of equal-time and time-displaced Green's functions within the determinant quantum Monte Carlo (DQMC) framework. Based on this comparison, we identify a procedure based on pivoted QR decompositions which is both efficient and accurate to machine precision. The Julia programming language is used for the assessment and implementations of all discussed algorithms are provided in the open-source software library StableDQMC.jl [this http URL].","PeriodicalId":8424,"journal":{"name":"arXiv: Computational Physics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88502779","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}
We present a reversible and symplectic algorithm called ROLL, for integrating the equations of motion in molecular dynamics simulations of simple fluids on a hypersphere $mathcal{S}^d$ of arbitrary dimension $d$. It is derived in the framework of Geometric Algebra and shown to be mathematically equivalent to algorithm RATTLE. An application to molecular dynamics simulation of the one component plasma is briefly discussed
{"title":"A symplectic integrator for molecular dynamics on a hypersphere","authors":"J. Caillol","doi":"10.5488/cmp.23.23603","DOIUrl":"https://doi.org/10.5488/cmp.23.23603","url":null,"abstract":"We present a reversible and symplectic algorithm called ROLL, for integrating the equations of motion in molecular dynamics simulations of simple fluids on a hypersphere $mathcal{S}^d$ of arbitrary dimension $d$. It is derived in the framework of Geometric Algebra and shown to be mathematically equivalent to algorithm RATTLE. An application to molecular dynamics simulation of the one component plasma is briefly discussed","PeriodicalId":8424,"journal":{"name":"arXiv: Computational Physics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86066596","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 : 2020-03-01DOI: 10.4208/CICP.OA-2020-0118
Ruo Li, Q. Du, Lei Zhang
Phase field methods have been widely used to study phase transitions and polarization switching in ferroelectric thin films. In this paper, we develop an efficient numerical scheme for the variational phase field model based on variational forms of the electrostatic energy and the relaxation dynamics of the polarization vector. The spatial discretization combines the Fourier spectral method with the finite difference method to handle three-dimensional mixed boundary conditions. It allows for an efficient semi-implicit discretization for the time integration of the relaxation dynamics. This method avoids explicitly solving the electrostatic equilibrium equation (a Poisson equation) and eliminates the use of associated Lagrange multipliers. We present several numerical examples including phase transitions and polarization switching processes to demonstrate the effectiveness of the proposed method.
{"title":"Numerical Discretization of Variational Phase Field Model for Phase Transitions in Ferroelectric Thin Films","authors":"Ruo Li, Q. Du, Lei Zhang","doi":"10.4208/CICP.OA-2020-0118","DOIUrl":"https://doi.org/10.4208/CICP.OA-2020-0118","url":null,"abstract":"Phase field methods have been widely used to study phase transitions and polarization switching in ferroelectric thin films. In this paper, we develop an efficient numerical scheme for the variational phase field model based on variational forms of the electrostatic energy and the relaxation dynamics of the polarization vector. The spatial discretization combines the Fourier spectral method with the finite difference method to handle three-dimensional mixed boundary conditions. It allows for an efficient semi-implicit discretization for the time integration of the relaxation dynamics. This method avoids explicitly solving the electrostatic equilibrium equation (a Poisson equation) and eliminates the use of associated Lagrange multipliers. We present several numerical examples including phase transitions and polarization switching processes to demonstrate the effectiveness of the proposed method.","PeriodicalId":8424,"journal":{"name":"arXiv: Computational Physics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89981621","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}
D. Toscano, I. A. Santece, R. Guedes, H. S. Assis, A. Miranda, C. D. de Araújo, F. Sato, P. Coura, S. A. Leonel
Micromagnetic simulations have been performed to investigate the controllability of the skyrmion position in antiferromagnetic nanotracks with their magnetic properties modified spatially. In this study we have modeled magnetic defects as local variations on the material parameters, such as the exchange stiffness, saturation magnetization, perpendicular magnetocrystalline anisotropy and Dzyaloshinskii-Moriya constant. Thus, we have observed not only pinning (potential well) but also scattering (potential barrier) of antiferromagnetic skyrmions, when adjusting either a local increase or a local reduction for each material parameter. In order to control of the skyrmion motion it is very important to impose certain positions along the nanotrack where the skyrmion can stop. Magnetic defects incorporated intentionally in antiferromagnetic racetracks can be useful for such purpose. In order to provide guidelines for experimental studies, we vary both material parameters and size of the modified region. The found results show that the efficiency of skyrmion trap depends on a suitable combination of magnetic defect parameters. Furthermore, we discuss the reason why skyrmions are either attracted or repelled by a region magnetically modified.
{"title":"Traps for pinning and scattering of antiferromagnetic skyrmions via magnetic properties engineering","authors":"D. Toscano, I. A. Santece, R. Guedes, H. S. Assis, A. Miranda, C. D. de Araújo, F. Sato, P. Coura, S. A. Leonel","doi":"10.1063/5.0006219","DOIUrl":"https://doi.org/10.1063/5.0006219","url":null,"abstract":"Micromagnetic simulations have been performed to investigate the controllability of the skyrmion position in antiferromagnetic nanotracks with their magnetic properties modified spatially. In this study we have modeled magnetic defects as local variations on the material parameters, such as the exchange stiffness, saturation magnetization, perpendicular magnetocrystalline anisotropy and Dzyaloshinskii-Moriya constant. Thus, we have observed not only pinning (potential well) but also scattering (potential barrier) of antiferromagnetic skyrmions, when adjusting either a local increase or a local reduction for each material parameter. In order to control of the skyrmion motion it is very important to impose certain positions along the nanotrack where the skyrmion can stop. Magnetic defects incorporated intentionally in antiferromagnetic racetracks can be useful for such purpose. In order to provide guidelines for experimental studies, we vary both material parameters and size of the modified region. The found results show that the efficiency of skyrmion trap depends on a suitable combination of magnetic defect parameters. Furthermore, we discuss the reason why skyrmions are either attracted or repelled by a region magnetically modified.","PeriodicalId":8424,"journal":{"name":"arXiv: Computational Physics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75323856","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}
R. Vasudevan, K. Kelley, E. Eliseev, S. Jesse, H. Funakubo, A. Morozovska, Sergei V. Kalinin
The universal tendency in scanning probe microscopy (SPM) over the last two decades is to transition from simple 2D imaging to complex detection and spectroscopic imaging modes. The emergence of complex SPM engines brings forth the challenge of reliable data interpretation, i.e. conversion from detected signal to descriptors specific to tip-surface interactions and subsequently to materials properties. Here, we implemented a Bayesian inference approach for the analysis of the image formation mechanisms in band excitation (BE) SPM. Compared to the point estimates in classical functional fit approaches, Bayesian inference allows for the incorporation of extant knowledge of materials and probe behavior in the form of corresponding prior distribution and return the information on the material functionality in the form of readily interpretable posterior distributions. We note that in application of Bayesian methods, special care should be made for proper setting on the problem as model selection vs. establishing practical parameter equivalence. We further explore the non-linear mechanical behaviors at topological defects in a classical ferroelectric material, PbTiO3. We observe the non-trivial evolution of Duffing resonance frequency and the nonlinearity of the sample surface, suggesting the presence of the hidden elements of domain structure. These observations suggest that the spectrum of anomalous behaviors at the ferroelectric domain walls can be significantly broader than previously believed and can extend to non-conventional mechanical properties in addition to static and microwave conductance.
{"title":"Bayesian inference in band excitation scanning probe microscopy for optimal dynamic model selection in imaging","authors":"R. Vasudevan, K. Kelley, E. Eliseev, S. Jesse, H. Funakubo, A. Morozovska, Sergei V. Kalinin","doi":"10.1063/5.0005323","DOIUrl":"https://doi.org/10.1063/5.0005323","url":null,"abstract":"The universal tendency in scanning probe microscopy (SPM) over the last two decades is to transition from simple 2D imaging to complex detection and spectroscopic imaging modes. The emergence of complex SPM engines brings forth the challenge of reliable data interpretation, i.e. conversion from detected signal to descriptors specific to tip-surface interactions and subsequently to materials properties. Here, we implemented a Bayesian inference approach for the analysis of the image formation mechanisms in band excitation (BE) SPM. Compared to the point estimates in classical functional fit approaches, Bayesian inference allows for the incorporation of extant knowledge of materials and probe behavior in the form of corresponding prior distribution and return the information on the material functionality in the form of readily interpretable posterior distributions. We note that in application of Bayesian methods, special care should be made for proper setting on the problem as model selection vs. establishing practical parameter equivalence. We further explore the non-linear mechanical behaviors at topological defects in a classical ferroelectric material, PbTiO3. We observe the non-trivial evolution of Duffing resonance frequency and the nonlinearity of the sample surface, suggesting the presence of the hidden elements of domain structure. These observations suggest that the spectrum of anomalous behaviors at the ferroelectric domain walls can be significantly broader than previously believed and can extend to non-conventional mechanical properties in addition to static and microwave conductance.","PeriodicalId":8424,"journal":{"name":"arXiv: Computational Physics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81256914","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}
R. Maulik, N. Garland, Xianzhu Tang, Prasanna Balaprakash
The closure problem in fluid modeling is a well-known challenge to modelers aiming to accurately describe their system of interest. Over many years, analytic formulations in a wide range of regimes have been presented but a practical, generalized fluid closure for magnetized plasmas remains an elusive goal. In this study, as a first step towards constructing a novel data based approach to this problem, we apply ever-maturing machine learning methods to assess the capability of neural network architectures to reproduce crucial physics inherent in popular magnetized plasma closures. We find encouraging results, indicating the applicability of neural networks to closure physics but also arrive at recommendations on how one should choose appropriate network architectures for given locality properties dictated by underlying physics of the plasma.
{"title":"Neural network representability of fully ionized plasma fluid model closures","authors":"R. Maulik, N. Garland, Xianzhu Tang, Prasanna Balaprakash","doi":"10.1063/5.0006457","DOIUrl":"https://doi.org/10.1063/5.0006457","url":null,"abstract":"The closure problem in fluid modeling is a well-known challenge to modelers aiming to accurately describe their system of interest. Over many years, analytic formulations in a wide range of regimes have been presented but a practical, generalized fluid closure for magnetized plasmas remains an elusive goal. In this study, as a first step towards constructing a novel data based approach to this problem, we apply ever-maturing machine learning methods to assess the capability of neural network architectures to reproduce crucial physics inherent in popular magnetized plasma closures. We find encouraging results, indicating the applicability of neural networks to closure physics but also arrive at recommendations on how one should choose appropriate network architectures for given locality properties dictated by underlying physics of the plasma.","PeriodicalId":8424,"journal":{"name":"arXiv: Computational Physics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87321441","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 : 2020-02-06DOI: 10.1103/physrevb.102.134213
Yanming Che, C. Gneiting, Tao Liu, F. Nori
The discovery of topological features of quantum states plays an important role in modern condensed matter physics and various artificial systems. Due to the absence of local order parameters, the detection of topological quantum phase transitions remains a challenge. Machine learning may provide effective methods for identifying topological features. In this work, we show that the unsupervised manifold learning can successfully retrieve topological quantum phase transitions in momentum and real space. Our results show that the Chebyshev distance between two data points sharpens the characteristic features of topological quantum phase transitions in momentum space, while the widely used Euclidean distance is in general suboptimal. Then a diffusion map or isometric map can be applied to implement the dimensionality reduction, and to learn about topological quantum phase transitions in an unsupervised manner. We demonstrate this method on the prototypical Su-Schrieffer-Heeger (SSH) model, the Qi-Wu-Zhang (QWZ) model, and the quenched SSH model in momentum space, and further provide implications and demonstrations for learning in real space, where the topological invariants could be unknown or hard to compute. The interpretable good performance of our approach shows the capability of manifold learning, when equipped with a suitable distance metric, in exploring topological quantum phase transitions.
{"title":"Topological quantum phase transitions retrieved through unsupervised machine learning","authors":"Yanming Che, C. Gneiting, Tao Liu, F. Nori","doi":"10.1103/physrevb.102.134213","DOIUrl":"https://doi.org/10.1103/physrevb.102.134213","url":null,"abstract":"The discovery of topological features of quantum states plays an important role in modern condensed matter physics and various artificial systems. Due to the absence of local order parameters, the detection of topological quantum phase transitions remains a challenge. Machine learning may provide effective methods for identifying topological features. In this work, we show that the unsupervised manifold learning can successfully retrieve topological quantum phase transitions in momentum and real space. Our results show that the Chebyshev distance between two data points sharpens the characteristic features of topological quantum phase transitions in momentum space, while the widely used Euclidean distance is in general suboptimal. Then a diffusion map or isometric map can be applied to implement the dimensionality reduction, and to learn about topological quantum phase transitions in an unsupervised manner. We demonstrate this method on the prototypical Su-Schrieffer-Heeger (SSH) model, the Qi-Wu-Zhang (QWZ) model, and the quenched SSH model in momentum space, and further provide implications and demonstrations for learning in real space, where the topological invariants could be unknown or hard to compute. The interpretable good performance of our approach shows the capability of manifold learning, when equipped with a suitable distance metric, in exploring topological quantum phase transitions.","PeriodicalId":8424,"journal":{"name":"arXiv: Computational Physics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83115464","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 : 2020-01-01DOI: 10.1007/978-3-030-40245-7_9
R. A. Vargas-Hernández, R. Krems
{"title":"Physical Extrapolation of Quantum Observables by Generalization with Gaussian Processes","authors":"R. A. Vargas-Hernández, R. Krems","doi":"10.1007/978-3-030-40245-7_9","DOIUrl":"https://doi.org/10.1007/978-3-030-40245-7_9","url":null,"abstract":"","PeriodicalId":8424,"journal":{"name":"arXiv: Computational Physics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77510983","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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