Stochastic Variational Method Research Articles

Variational formulation of compressible hydrodynamics in curved spacetime and symmetry of stress tensor

Hydrodynamics of the non-relativistic compressible fluid in the curved spacetime is derived using the generalized framework of the stochastic variational method (SVM) for continuum medium. The fluid-stress tensor of the resultant equation becomes asymmetric for the exchange of the indices, different from the standard Euclidean one. Its incompressible limit suggests that the viscous term should be represented with the Bochner Laplacian. Moreover the modified Navier–Stokes–Fourier equation proposed by Brenner can be considered even in the curved spacetime. To confirm the compatibility with the symmetry principle, SVM is applied to the gauge-invariant Lagrangian of a charged compressible fluid and then the Lorentz force is reproduced as the interaction between the Abelian gauge fields and the viscous charged fluid.

Variational Metric Scaling for Metric-Based Meta-Learning

Metric-based meta-learning has attracted a lot of attention due to its effectiveness and efficiency in few-shot learning. Recent studies show that metric scaling plays a crucial role in the performance of metric-based meta-learning algorithms. However, there still lacks a principled method for learning the metric scaling parameter automatically. In this paper, we recast metric-based meta-learning from a Bayesian perspective and develop a variational metric scaling framework for learning a proper metric scaling parameter. Firstly, we propose a stochastic variational method to learn a single global scaling parameter. To better fit the embedding space to a given data distribution, we extend our method to learn a dimensional scaling vector to transform the embedding space. Furthermore, to learn task-specific embeddings, we generate task-dependent dimensional scaling vectors with amortized variational inference. Our method is end-to-end without any pre-training and can be used as a simple plug-and-play module for existing metric-based meta-algorithms. Experiments on miniImageNet show that our methods can be used to consistently improve the performance of existing metric-based meta-algorithms including prototypical networks and TADAM.

Uncertainty relation for angle from a quantum-hydrodynamical perspective

We revisit the problem of the uncertainty relation for angle by using quantum hydrodynamics formulated in terms of the stochastic variational method (SVM), where we need not define the angle operator. We derive both the Kennard and Robertson–Schrödinger inequalities for canonical variables in polar coordinates. The inequalities have state-dependent minimum values which can be smaller than ħ∕2 and then permit a finite uncertainty of angle for the eigenstate of the angular momentum. The present approach provides a useful methodology to study quantum behaviors in arbitrary coordinates.

A stochastic microscopic approach to the $^{10}{\rm Be}$ and $^{11}{\rm Be}$ nuclei

Abstract We use a microscopic multicluster model to investigate the structure of $^{10}{\rm Be}$ and of $^{11}{\rm Be}$. These nuclei are described by $\alpha +\alpha+n+n$ and $\alpha +\alpha+n+n+n$ configurations, respectively, within the Generator Coordinate Method (GCM). The 4- and 5-body models raise the problem of a large number of generator coordinates (6 for $^{10}{\rm Be}$ and 9 for $^{11}{\rm Be}$), which requires specific treatment. We address this issue by using the Stochastic Variational Method (SVM), which is based on an optimal choice of the basis functions, generated randomly. The model provides good energy spectra for low-lying states of both nuclei. We also compute rms radii and densities, as well as electromagnetic transition probabilities. We analyze the structure of $^{10}{\rm Be}$ and of $^{11}{\rm Be}$ by considering energy curves, where one of the generator coordinates is fixed during the minimization procedure.

Weakly bound states of dtμ muonic molecular ion in quantum electrodynamics

The energy spectrum of ground and excited bound states of muonic molecule dtμ is calculated on the basis of stochastic variational method. The basis wave functions of the muonic molecule are taken in the Gaussian form. The matrix elements of the Hamiltonian are calculated analytically. For numerical calculation, a computer code is written in the MATLAB system. As a result, the numerical energy levels of ground and excited states of the muonic molecule dtμ are obtained.

Using Social Network Information in Community-Based Bayesian Truth Discovery

We investigate the problem of truth discovery based on opinions from multiple agents (who may be unreliable or biased) that form a social network. We consider the case where agents' reliabilities or biases are correlated if they belong to the same community, which defines a group of agents with similar opinions regarding a particular event. An agent can belong to different communities for different events, and these communities are unknown a priori. We incorporate knowledge of the agents' social network in our truth discovery framework and develop Laplace variational inference methods to estimate agents' reliabilities, communities, and the event states. We also develop a stochastic variational inference method to scale our model to large social networks. Simulations and experiments on real data suggest that when observations are sparse, our proposed methods perform better than several other inference methods, including majority voting, TruthFinder, AccuSim, the Confidence-Aware Truth Discovery method, the Bayesian classifier combination (BCC) method, and the community BCC method.

Emergence of nuclear clustering in electric-dipole excitations ofLi6

Nuclear clustering plays an important role, especially in the dynamics of light nuclei. The importance of the emergence of the nuclear clustering was discussed in the recent measurement of the photoabsorption cross sections as it offered the possibility of the coexistence of various excitation modes which are closely related to the nuclear clustering. To understand the excitation mechanism, we study the electric-dipole ($E1$) responses of $^6$Li with a fully microscopic six-body calculation. The ground-state wave function is accurately described with a superposition of correlated Gaussian (CG) functions with the aid of the stochastic variational method. The final-state wave functions are also expressed by a number of the CG functions including important configurations to describe the six-body continuum states excited by the $E1$ field. We found that the out-of-phase transitions occur due to the oscillations of the valence nucleons against the $^4$He cluster at the low energies around 10 MeV indicating ``soft'' giant-dipole-resonance(GDR)-type excitations, which are very unique in the $^6$Li system but could be found in other nuclear systems. At the high energies beyond $\sim 30$ MeV typical GDR-type transitions occur. The $^3$He-$^3$H clustering plays an important role to the GDR phenomena in the intermediate energy regions around 20 MeV.

The onset of ΛΛ hypernuclear binding

Binding energies of light, A≤6, ΛΛ hypernuclei are calculated using the stochastic variational method in a pionless effective field theory (π̸EFT) approach at leading order with the purpose of assessing critically the onset of binding in the strangeness S=−2 hadronic sector. The π̸EFT input in this sector consists of (i) a ΛΛ contact term constrained by the ΛΛ scattering length aΛΛ, using a range of values compatible with ΛΛ correlations observed in relativistic heavy ion collisions, and (ii) a ΛΛN contact term constrained by the only available A≤6 ΛΛ hypernucler binding energy datum of ΛΛ6He. The recently debated neutral three-body and four-body systems ΛΛ3n and ΛΛ4n are found unbound by a wide margin. A relatively large value of |aΛΛ|≳1.5 fm is needed to bind ΛΛ4H, thereby questioning its particle stability. In contrast, the particle stability of the A=5 ΛΛ hypernuclear isodoublet ΛΛ5H–ΛΛ5He is robust, with Λ separation energy of order 1 MeV.

Novel effect induced by spacetime curvature in quantum hydrodynamics

The interplay between quantum fluctuation and spacetime curvature is shown to induce an additional quantum-curvature (QC) term in the energy-momentum tensor of fluid using the generalized framework of the stochastic variational method (SVM). The QC term is necessary to satisfy the momentum conservation but the corresponding quantum hydrodynamics is not necessarily cast into the form of the Schrödinger equation, differently from the case of the Euclidean spacetime. This seems to suggest that the existence of the Hilbert space is not a priori requirement in the quantization of curved spacetime systems. As an example, we apply the Friedmann-Robertson-Walker (FRW) metric and show that this term contributes to the cosmological acceleration although it is too small in the present non-relativistic toy model.

Energy Levels of Mesomolecular Ions of Hydrogen in Variational Approach

Energies of the ground and excited states of mesomolecular ions tdμ, dpμ, and tpμ, are calculated based on the stochastic variational method of quantum electrodynamics. Basis wave functions are chosen in the Gaussian form. Hamiltonian matrix elements are calculated analytically. A computer code in the MATLAB environment is developed for numerical calculations. The values of energies of the ground and excited states are obtained.

Calculations ofη-nuclear quasi-bound states in few-body systems

We report on our Stochastic Variational Method (SVM) calculations ofη-nuclear quasi-bound states in s-shell nuclei as well as the very recent calculation of the p-shell nucleus6Li. TheηNpotentials used were constructed fromηNscattering amplitudes obtained within coupled-channel models that incorporateN*(1535) resonance. We found thatη6Li is bound in theηNinteraction models that yield ReaηN≥ 0.67 fm. Additional repulsion caused by the imaginary part ofηNpotentials shifts the onset ofη-nuclear binding toη4He, yielding very likely no quasi-bound state inη3He.

Energy levels in muonic helium

The energy spectrum of bound states and hyperfine structure of muonic helium is calculated on the basis of stochastic variational method. The basis wave functions of muonic helium are taken in the Gaussian form. The matrix elements of the Hamiltonian are calculated analytically. For numerical calculation a computer code is written in the MATLAB system. As a result, numerical values of bound state energies and hyperfine structure are obtained. We calculate also correction to the structure of the nucleus, vacuum polarization and relativistic correction.

Hyperfine structure of muonic mesomolecules tdµ, tpµ, dpµ in variational method

The hyperfine structure of energy levels of muonic molecules tdµ, tpµ and dpµ is calculated on the basis of stochastic variational method. The basis wave functions are taken in the Gaussian form. The matrix elements of the Hamiltonian are calculated analytically. Vacuum polarization, relativistic and nuclear structure corrections are taken into account to increase the accuracy. For numerical calculation, a computer code is written in the MATLAB system. Numerical values of energy levels of hyperfine structure in muonic molecules tdµ, tpµ and dpµ are obtained.

Bound states of dtμ, pdμ and t pμ mesomolecules

The energy spectrum of excited bound states of muonic molecules ptμ, pdμ, and dtμ is calculated on the basis of the stochastic variational method. The basis wave functions of the muonic molecule are taken in the Gaussian form. The matrix elements of the Hamiltonian are calculated analytically. For numerical calculation, a computer code was written in the MATLAB system. As a result, the numerical values of bound state energies for excited P-states of muonic molecules ptμ, pdμ and dtμ were obtained.

Excitonic complexes in anisotropic atomically thin two-dimensional materials: Black phosphorus and TiS3

The effect of anisotropy in the energy spectrum on the binding energy and structural properties of excitons, trions, and biexcitons is investigated. To this end we employ the stochastic variational method with a correlated Gaussian basis. We present results for the binding energy of different excitonic complexes in black phosphorus (bP) and TiS$_3$ and compare them with recent results in the literature when available, for which we find good agreement. The binding energies of excitonic complexes in bP are larger than those in TiS$_3$. We calculate the different average interparticle distances in bP and TiS$_3$ and show that excitonic complexes in bP are strongly anisotropic whereas in TiS$_3$ they are almost isotropic, even though the constituent particles have an anisotropic energy spectrum. This is also confirmed by the correlation functions.

Finding Convincing Arguments Using Scalable Bayesian Preference Learning

We introduce a scalable Bayesian preference learning method for identifying convincing arguments in the absence of gold-standard ratings or rankings. In contrast to previous work, we avoid the need for separate methods to perform quality control on training data, predict rankings and perform pairwise classification. Bayesian approaches are an effective solution when faced with sparse or noisy training data, but have not previously been used to identify convincing arguments. One issue is scalability, which we address by developing a stochastic variational inference method for Gaussian process (GP) preference learning. We show how our method can be applied to predict argument convincingness from crowdsourced data, outperforming the previous state-of-the-art, particularly when trained with small amounts of unreliable data. We demonstrate how the Bayesian approach enables more effective active learning, thereby reducing the amount of data required to identify convincing arguments for new users and domains. While word embeddings are principally used with neural networks, our results show that word embeddings in combination with linguistic features also benefit GPs when predicting argument convincingness.

Non-adiabatic mass correction to the rovibrational states of molecules: Numerical application for the molecular ion.

General transformation expressions of the second-order non-adiabatic Hamiltonian of the atomic nuclei, including the kinetic-energy correction terms, are derived upon the change from laboratory-fixed Cartesian coordinates to general curvilinear coordinate systems commonly used in rovibrational computations. The kinetic-energy or so-called "mass-correction" tensor elements are computed with the stochastic variational method and floating explicitly correlated Gaussian functions for the molecular ion in its ground electronic state. {Further numerical applications for the 4 molecular ion are presented in the forthcoming paper, Paper II [E. Mátyus, J. Chem. Phys. 149, 194112 (2018)]}. The general, curvilinear non-adiabatic kinetic energy operator expressions are used in the examples, and non-adiabatic rovibrational energies and corrections are determined by solving the rovibrational Schrödinger equation including the diagonal Born-Oppenheimer as well as the mass-tensor corrections.

Resolving the Λ Hypernuclear Overbinding Problem in Pionless Effective Field Theory.

We address the Λ hypernuclear "overbinding problem" in light hypernuclei which stands for a 1-3MeV excessive Λ separation energy calculated in _{Λ}^{5}He. This problem arises in most few-body calculations that reproduce ground-state Λ separation energies in the lighter Λ hypernuclei within various hyperon-nucleon interaction models. Recent pionless effective field theory (πEFT) nuclear few-body calculations are extended in this work to Λ hypernuclei. At leading order, the ΛN low-energy constants are associated with ΛN scattering lengths, and the ΛNN low-energy constants are fitted to Λ separation energies (B_{Λ}^{exp}) for A≤4. The resulting πEFT interaction reproduces in few-body stochastic variational method calculations the reported value B_{Λ}^{exp}(_{Λ}^{5}He)=3.12±0.02 MeV within a fraction of MeV over a broad range of πEFT cutoff parameters. Possible consequences and extensions to heavier hypernuclei and to neutron-star matter are discussed.

Excitons, trions, and biexcitons in transition-metal dichalcogenides: Magnetic-field dependence

The influence of a perpendicular magnetic field on the binding energy and structural properties of excitons, trions, and biexcitons in monolayers of semiconducting transition metal dichalcogenides (TMDs) is investigated. The stochastic variational method (SVM) with a correlated Gaussian basis is used to calculate the different properties of these few-particle systems. In addition, we present a simplified variational approach which supports the SVM results for excitons as a function of magnetic field. The exciton diamagnetic shift is compared with recent experimental results and we extend this concept to trions and biexcitons. The effect of a local potential fluctuation, which we model by a circular potential well, on the binding energy of trions and biexcitons is investigated and found to significantly increase the binding of those excitonic complexes.

Generalization of uncertainty relation for quantum and stochastic systems

The generalized uncertainty relation applicable to quantum and stochastic systems is derived within the stochastic variational method. This relation not only reproduces the well-known inequality in quantum mechanics but also is applicable to the Gross–Pitaevskii equation and the Navier–Stokes–Fourier equation, showing that the finite minimum uncertainty between the position and the momentum is not an inherent property of quantum mechanics but a common feature of stochastic systems. We further discuss the possible implication of the present study in discussing the application of the hydrodynamic picture to microscopic systems, like relativistic heavy-ion collisions.