Variational Monte Carlo Method Research Articles

Distilling the Essential Elements of Nuclear Binding via Neural-Network Quantum States.

To distill the essential elements of nuclear binding, we seek the simplest Hamiltonian capable of modeling atomic nuclei with percent-level accuracy. A critical aspect of this endeavor consists of accurately solving the quantum many-body problem without incurring an exponential computing cost with the number of nucleons. We address this challenge by leveraging a variational MonteCarlo method based on a highly expressive neural-network quantum state ansatz. In addition to computing binding energies and charge radii of nuclei with up to A=20 nucleons, by evaluating their magnetic moments, we demonstrate that neural-network quantum states are able to correctly capture the self-emerging nuclear shell structure. To this end, we introduce a novel computational protocol based on adding an external magnetic field to the nuclear Hamiltonian, which allows the neural network to learn the preferred polarization of the nucleus within the given magnetic field.

A Kaczmarz-inspired approach to accelerate the optimization of neural network wavefunctions

Neural network wavefunctions optimized using the variational Monte Carlo method have been shown to produce highly accurate results for the electronic structure of atoms and small molecules, but the high cost of optimizing such wavefunctions prevents their application to larger systems. We propose the Subsampled Projected-Increment Natural Gradient Descent (SPRING) optimizer to reduce this bottleneck. SPRING combines ideas from the recently introduced minimum-step stochastic reconfiguration optimizer (MinSR) and the classical randomized Kaczmarz method for solving linear least-squares problems. We demonstrate that SPRING outperforms both MinSR and the popular Kronecker-Factored Approximate Curvature method (KFAC) across a number of small atoms and molecules, given that the learning rates of all methods are optimally tuned. For example, on the oxygen atom, SPRING attains chemical accuracy after forty thousand training iterations, whereas both MinSR and KFAC fail to do so even after one hundred thousand iterations.

Second-order optimization strategies for neural network quantum states.

The Variational Monte Carlo (VMC) method has recently seen important advances through the use of neural network quantum states. While more and more sophisticated ansatze have been designed to tackle a wide variety of quantum many-body problems, modest progress has been made on the associated optimization algorithms. In this work, we revisit the Kronecker-Factored Approximate Curvature (KFAC), an optimizer that has been used extensively in a variety of simulations. We suggest improvements in the scaling and the direction of this optimizer and find that they substantially increase its performance at a negligible additional cost. We also reformulate the VMC approach in a game theory framework, to propose a novel optimizer based on decision geometry. We find that on a practical test case for continuous systems, this new optimizer consistently outperforms any of the KFAC improvements in terms of stability, accuracy and speed of convergence. Beyond VMC, the versatility of this approach suggests that decision geometry could provide a solid foundation for accelerating a broad class of machine learning algorithms. This article is part of the theme issue 'The liminal position of Nuclear Physics: from hadrons to neutron stars'.

Neural-network quantum-state study of the long-range antiferromagnetic Ising chain.

We investigate quantum phase transitions in the transverse field Ising chain with algebraically decaying long-range (LR) antiferromagnetic interactions using the variational Monte Carlo method with the restricted Boltzmann machine employed as a trial wave function ansatz. First, we measure the critical exponents and the central charge through the finite-size scaling analysis, verifying the contrasting observations in the previous tensor network studies. The correlation function exponent and the central charge deviate from the short-range (SR) Ising values at a small decay exponent α_{LR}, while the other critical exponents examined are very close to the SR Ising exponents regardless of α_{LR} examined. However, in the further test of the critical Binder ratio, we find that the universal ratio of the SR limit does not hold for α_{LR}<2, implying a deviation in the criticality. On the other hand, we find evidence of the conformal invariance breakdown in the conformal field theory (CFT) test of the correlation function. The deviation from the CFT description becomes more pronounced as α_{LR} decreases, although a precise breakdown threshold is yet to be determined.

Performance assessment of the effective core potentials under the fermionic neural network: First and second row elements.

The rapid development of deep learning techniques has driven the emergence of a neural network-based variational Monte Carlo (VMC) method (referred to as FermiNet), which has manifested high accuracy and strong predictive power in the electronic structure calculations of atoms, molecules, and some periodic systems. Recently, the implementation of the effective core potential (ECP) scheme has further facilitated more efficient calculations in practice. However, there is still a lack of comprehensive assessments of the ECP's performance under the FermiNet. In this work, we set sail to fill this gap by conducting extensive tests on the first two row elements regarding their atomic, spectral, and molecular properties. Our major finding is that, in general, the qualities of ECPs have been correctly reflected under FermiNet. Two recently built ECP tables, namely, correlation consistent ECP (ccECP) and energy consistent correlated electron pseudopotential (eCEPP), seem to prevail in terms of overall performance. In particular, ccECP performs slightly better on spectral precision and covers more elements, while eCEPP is more systematically built from both shape and energy consistency and better treats the core polarization. On the other hand, the high accuracy of the all-electron calculations is hindered by the absence of relativistic effects as well as the numerical instabilities in some heavier elements. Finally, with further in-depth discussions, we generate possible directions for developing and improving FermiNet in the near future.

Ensemble variational Monte Carlo for optimization of correlated excited state wave functions

Variational Monte Carlo methods have recently been applied to the calculation of excited states; however, it is still an open question what objective function is most effective. A promising approach is to optimize excited states using a penalty to minimize overlap with lower eigenstates, which has the drawback that states must be computed one at a time. We derive a general framework for constructing objective functions with minima at the the lowest N eigenstates of a many-body Hamiltonian. The objective function uses a weighted average of the energies and an overlap penalty, which must satisfy several conditions. We show this objective function has a minimum at the exact eigenstates for a finite penalty, and provide a few strategies to minimize the objective function. The method is demonstrated using ab initio variational Monte Carlo to calculate the degenerate first excited state of a CO molecule.

Towards a transferable fermionic neural wavefunction for molecules.

Deep neural networks have become a highly accurate and powerful wavefunction ansatz in combination with variational Monte Carlo methods for solving the electronic Schrödinger equation. However, despite their success and favorable scaling, these methods are still computationally too costly for wide adoption. A significant obstacle is the requirement to optimize the wavefunction from scratch for each new system, thus requiring long optimization. In this work, we propose a neural network ansatz, which effectively maps uncorrelated, computationally cheap Hartree-Fock orbitals, to correlated, high-accuracy neural network orbitals. This ansatz is inherently capable of learning a single wavefunction across multiple compounds and geometries, as we demonstrate by successfully transferring a wavefunction model pre-trained on smaller fragments to larger compounds. Furthermore, we provide ample experimental evidence to support the idea that extensive pre-training of such a generalized wavefunction model across different compounds and geometries could lead to a foundation wavefunction model. Such a model could yield high-accuracy ab-initio energies using only minimal computational effort for fine-tuning and evaluation of observables.

Force and stress calculations with a neural-network wave function for solids.

Accurate ab initio calculations of real solids are of fundamental importance in fields such as chemistry, phases and materials science. Recently, variational Monte Carlo (VMC) based on neural-network wave functions has been developed as a promising option to solve the existing challenges in ab initio calculations. In this study, we discuss the calculation of interatomic forces and stress tensors of real solids with a neural-network-based VMC method. A new scheme for computing forces is proposed based on the space-warp coordination transformation method, which achieves better accuracy, efficiency and robustness than existing methods. In addition, we also designed new periodic features of the neural network to further improve the robustness of force calculations for different lattices. This work paves the way for further extending the application of machine-learning quantum Monte Carlo methods in materials modelling.

A DOMAIN-DECOMPOSED VAE METHOD FOR BAYESIAN INVERSE PROBLEMS

Bayesian inverse problems are often computationally challenging when the forward model is governed by complex partial differential equations (PDEs). This is typically caused by expensive forward model evaluations and highdimensional parameterization of priors. This paper proposes a domain-decomposed variational autoencoder Markov chain Monte Carlo (DD-VAE-MCMC) method to tackle these challenges simultaneously. Through partitioning the global physical domain into small subdomains, the proposed method first constructs local deterministic generative models based on local historical data, which provide efficient local prior representations. Gaussian process models with active learning address the domain decomposition interface conditions. Then inversions are conducted on each subdomain independently in parallel and in low-dimensional latent parameter spaces. The local inference solutions are postprocessed through the Poisson image blending procedure to result in an efficient global inference result. Numerical examples are provided to demonstrate the performance of the proposed method.

Neural network quantum states analysis of the Shastry-Sutherland model

We utilize neural network quantum states (NQS) to investigate the ground state properties of the Heisenberg model on a Shastry-Sutherland lattice using the variational Monte Carlo method. We show that already relatively simple NQSs can be used to approximate the ground state of this model in its different phases and regimes. We first compare several types of NQSs with each other on small lattices and benchmark their variational energies against the exact diagonalization results. We argue that when precision, generality, and computational costs are taken into account, a good choice for addressing larger systems is a shallow restricted Boltzmann machine NQS. We then show that such NQS can describe the main phases of the model in zero magnetic field. Moreover, NQS based on a restricted Boltzmann machine correctly describes the intriguing plateaus forming in magnetization of the model as a function of increasing magnetic field.

Compressed lithium atom under Debye screening

AbstractIn the present paper, we introduced two approximation methods to investigate the spherically compressed lithium atom under the influence of Debye plasma. Specifically, the diffusion and the variational Monte Carlo methods are used for the first time to study this case. We incorporated an exponential screening into the nuclear Coulomb potential to account for the plasma influence. We focused our investigation on analyzing the combined effect of compression and plasma environment on the behavior of the lithium atom in its ground state. The resulting outcomes exhibited consistent pattern across various confinement radii and Debye plasma lengths . Many of the results are novel contributions, as they have yet to be explored. Moreover, the findings from this study demonstrated reasonable agreement with the limited existing theoretical results.

The beryllium atom, ions, and iso-electronic ions in magnetic field using the variational Monte Carlo method

The variational Monte Carlo method is applied to investigate several properties of the beryllium atom, ions, and its iso-electronic ions. For the ground and the excited states of the beryllium atom and ions, the energy eigenvalues were evaluated freely and under the influence of magnetic field. Furthermore, the iso-electronic ions (B+ and C2+) are also investigated under the influence of the external magnetic field. Suitable trial wave functions including the spin and the correlated parts are used in these investigation. Some new excited states were included in the present work, such as the low-lying states (1s22s3s and 1s22s3p) and the core states (1s2s23s and 1s2s3s2). For the spin functions of these states, we used two different functions for the singlet and the triplet excited states. Moreover, the energies of the beryllium ions (Be+ and Be2+) were evaluated freely and in the presence of magnetic field. The obtained results are in good agreement with the corresponding results of other works.Graphical abstract

Comprehensive AbInitio Investigation of the Phase Diagram of Quasi-One-Dimensional Molecular Solids.

An abinitio investigation of the family of molecular compounds TM_{2}X is conducted, where TM is either TMTSF or TMTTF and X takes centrosymmetric monovalent anions. By deriving the extended Hubbard-type Hamiltonians from first-principles band calculations and evaluating not only the intermolecular transfer integrals but also the Coulomb parameters, we discuss their material dependence in the unified phase diagram. Furthermore, we apply the many-variable variational MonteCarlo method to accurately determine the symmetry-breaking phase transitions, and show the development of the charge and spin orderings. We show that the material-dependent parameter can be taken as the correlation effect, represented by the value of the screened on-site Coulomb interaction U relative to the intrachain transfer integrals, for the comprehensive understanding of the spin and charge ordering in this system.

Wigner crystallization in quasi-one-dimensional quantum wire

We study the ground-state properties of quasi-one-dimensional paramagnetic electron fluid using variational quantum Monte Carlo method. The electrons are transversally confined using the harmonically regularized Coulomb potential with long range order. In this work we have investigated the crossover from liquid to crystalline phase in the finite width paramagnetic quantum wire. The calculated pair correlation function shows the oscillations of period 2r_text {s} a_0, where a_0 is the Bohr’s radius and r_text {s} is Wigner Seitz radius or electron-density parameter which corresponds to a k=4k_text {F} peak with k_text {F} being Fermi wave vector in the static structure factor at finite electron density. It is a signature of the onset of Wigner crystallization. It is found that the cross-over from liquid to a quasi-Wigner phase is due to enhancement of the electron correlation effects as r_text {s} increases or as the wire width b decreases.

Adaptive Nonintrusive Reconstruction of Solutions to High-Dimensional Parametric PDEs

Numerical methods for random parametric PDEs can greatly benefit from adaptive refinement schemes, in particular when functional approximations are computed as in stochastic Galerkin and stochastic collocations methods. This work is concerned with a nonintrusive generalization of the adaptive Galerkin finite element method with residual-based error estimation. It combines the nonintrusive character of a randomized least squares method with the a posteriori error analysis of stochastic Galerkin methods. The proposed approach uses the variational Monte Carlo method to obtain a quasi-optimal low-rank approximation of the Galerkin projection in a highly efficient hierarchical tensor format. We derive an adaptive refinement algorithm which is steered by a reliable error estimator. Opposite to stochastic Galerkin methods, the approach is easily applicable to a wide range of problems, enabling a fully automated adjustment of all discretization parameters. Benchmark examples with affine and (unbounded) lognormal coefficient fields illustrate the performance of the nonintrusive adaptive algorithm, showing the expected convergence rates of single-level strategies.

Correlation energy of the paramagnetic electron gas at the thermodynamic limit

The variational and diffusion quantum Monte Carlo methods are used to calculate the correlation energy of the paramagnetic three-dimensional homogeneous electron gas at intermediate to high density. Ground-state energies in finite cells are determined using Slater-Jastrow-backflow trial wave functions, and finite-size errors are removed using twist-averaged boundary conditions and extrapolation of the energy per particle to the thermodynamic limit of infinite system size. Our correlation energies in the thermodynamic limit are more accurate than previous results. The present diffusion quantum Monte Carlo energies together with our recently reported [Phys. Rev. B 105, 245135 (2022)] results at low density, are used to parametrize the correlation energy of the electron gas using a functional form that satisfies the exact asymptotic behavior at high density.

Wire-width and electron-density dependence of the crossover in the peak of the static structure factor from 2kF → 4kF in one-dimensional paramagnetic electron gases

We use the variational quantum Monte Carlo (VMC) method to study the wire-width $(b)$ and electron-density $({r}_{\text{s}})$ dependences of the ground-state properties of quasi-one-dimensional paramagnetic electron fluids. The onset of a quasi-Wigner crystal phase is known to depend on electron density and the crossover occurs in the low density regime. We study the effect of wire width on the crossover of the dominant peak in the static structure factor from $k=2{k}_{\text{F}}$ to $k=4{k}_{\text{F}}$. It is found that, for a fixed electron density, in the charge structure factor the crossover from the dominant peak occurring at $2{k}_{\text{F}}$ to $4{k}_{\text{F}}$ occurs as the wire width decreases. Our study suggests that the crossover is due to the interplay of both ${r}_{\text{s}}$ and $b&lt;{r}_{\text{s}}$. The finite wire-width correlation effect is reflected in the peak height of the charge and spin structure factors. We fit the dominant peaks of the charge and spin structure factors assuming fit functions based on our finite wire-width theory and clues from bosonization, resulting in a good fit of the VMC data. The pronounced peaks in the charge and spin structure factors at $4{k}_{\text{F}}$ and $2{k}_{\text{F}}$, respectively, indicate the complete decoupling of the charge and spin degrees of freedom. Furthermore, the wire-width dependence of the electron correlation energy and the Tomonaga-Luttinger parameter ${K}_{\ensuremath{\rho}}$ is found to be significant.

Monte Carlo study of cuprate superconductors in a four-band model: role of orbital degrees of freedom

Understanding the various competing phases in cuprate superconductors is a long-standing challenging problem. Recent studies have shown that orbital degrees of freedom, both Cu e g orbitals and O p orbitals, are a key ingredient for a unified understanding of cuprate superconductors, including the material dependence. Here we investigate a four-band model derived from the first-principles calculations with the variational Monte Carlo method, which allows us to elucidate competing phases on an equal footing. The obtained results can consistently explain the doping dependence of superconductivity, antiferromagnetic and stripe phases, phase separation in the underdoped region, and also novel magnetism in the heavily-overdoped region. The presence of p orbitals is critical to the charge-stripe features, which induce two types of stripe phases with s)-wave and d-wave bond stripe. On the other hand, the presence of orbital is indispensable to material dependence of the superconducting transition temperature (), and enhances local magnetic moment as a source of novel magnetism in the heavily-overdoped region as well. These findings beyond one-band description could provide a major step toward a full explanation of unconventional normal state and high in cuprate supercondutors.

Solving the nuclear pairing model with neural network quantum states.

We present a variational Monte Carlo method that solves the nuclear many-body problem in the occupation number formalism exploiting an artificial neural network representation of the ground-state wave function. A memory-efficient version of the stochastic reconfiguration algorithm is developed to train the network by minimizing the expectation value of the Hamiltonian. We benchmark this approach against widely used nuclear many-body methods by solving a model used to describe pairing in nuclei for different types of interaction and different values of the interaction strength. Despite its polynomial computational cost, our method outperforms coupled-cluster and provides energies that are in excellent agreement with the numerically exact full configuration-interaction values.

Bayesian geoacoustic parameter inversion based on dispersion characteristics of normal mode water wave and ground wave

Most of shallow water geoacoustic inversions based on modal dispersion cannot reliably estimate the deep geoacoustic parameters. Because these studies focused on the dispersions of water waves but ignored the dispersions of ground waves. Therefore, in this paper a Bayesian geoacoustic inversion is studied based on wideband modal dispersions of water waves and ground waves. Firstly, the modal dispersion curves with Airy phase components are discussed. Secondly, the Bayesian inversion theory and a novel sample-efficient inference algorithm, namely Variational Bayesian Monte Carlo, are introduced briefly. In the Bayesian inversion, the posterior probability densities of unknown parameters are inferred, which can provide the prediction closest to the observation data and the uncertainty of the prediction. Considering that the forward acoustic model is computationally intensive, the posterior analysis is carried out by using the Variational Bayesian Monte Carlo method. It is performed by finding the variational distribution closest to the target distribution and requires less computation time than the Markov chain Monto Carlo method. In the simulation study, a range-independent two-layer seabed, including the sediment layer and basement layer, is modeled, on the assumption that the water column is homogeneous. The function of shear wave in waveguide is ignored. The compressional sound speed of the sediment layer varies linearly from &lt;i&gt;c&lt;/i&gt;&lt;sub&gt;1U&lt;/sub&gt; to &lt;i&gt;c&lt;/i&gt;&lt;sub&gt;1L&lt;/sub&gt; between 0 and &lt;i&gt;h&lt;/i&gt;&lt;sub&gt;1&lt;/sub&gt;, while other geoacoustic parameters are constant. By comparing the inversion results with and without the information of ground waves for different signal-to-noise ratios, it can be concluded that the deep geoacoustic parameters are more sensitive to the dispersions of ground waves. And then, a shallow-water experimental study is carried out in the Bohai Sea of China. The average water depth is about 20 m. The wideband pulse signals are recorded by a hydrophone with a sensitivity of –170 dB re 1 V/μPa. The received signals include well-defined Airy phase components, and the modal dispersion curves of water waves and ground waves are extracted accurately. The experimental results indicate that the Bayesian inversion combining water and ground wave dispersions can not only estimate the deep geoacoustic parameters reliably, but also reduce the inversion uncertainties of other model parameters, such as the shallow geoacoustic parameters, water depth, etc. The estimated source-receiver range and water sound speed are close to their measured values. The modal dispersion curves predicted by the posterior mean samples are in good consistence with those extracted from the experimental data in different ranges. In addition, the good forecast of transmission loss also demonstrates the reliability of the joint inversion.