Artificial Neural Networks as Trial Wave Functions for Quantum Monte Carlo

Keßler, Jan Henning; Calcavecchia, Francesco; Kühne, Thomas D.

doi:10.1002/adts.202000269

Cited by 23 publications

(15 citation statements)

References 55 publications

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“…While there has been some progress [20][21][22][23][24] in analyzing the expressiveness of the permutation equivariant mappings used in the backflow construction [25], the understanding of the effectiveness of the antisymmetric neural network layers remains limited [20,23,26]. Interestingly, Refs.…”

Section: Introductionmentioning

confidence: 99%

Explicitly antisymmetrized neural network layers for variational Monte Carlo simulation

Lin¹,

Goldshlager²,

Lin³

2021

Preprint

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The combination of neural networks and quantum Monte Carlo methods has arisen as a promising path forward for highly accurate electronic structure calculations. Previous proposals have combined equivariant neural network layers with a final antisymmetric layer in order to satisfy the antisymmetry requirements of the electronic wavefunction. However, to date it is unclear if one can represent antisymmetric functions of physical interest, and it is difficult to precisely measure the expressiveness of the antisymmetric layer. This work attempts to address this problem by introducing explicitly antisymmetrized universal neural network layers. This approach has a computational cost which increases factorially with respect to the system size, but we are nonetheless able to apply it to small systems to better understand how the structure of the antisymmetric layer affects its performance. We first introduce a generic antisymmetric (GA) neural network layer, which we use to replace the entire antisymmetric layer of the highly accurate ansatz known as the FermiNet. We demonstrate that the resulting FermiNet-GA architecture can yield effectively the exact ground state energy for small atoms and molecules. We then consider a factorized antisymmetric (FA) layer which more directly generalizes the FermiNet by replacing the products of determinants with products of antisymmetrized neural networks. We find, interestingly, that the resulting FermiNet-FA architecture does not outperform the FermiNet. This strongly suggests that the sum of products of antisymmetries is a key limiting aspect of the FermiNet architecture. To explore this further, we investigate a slight modification of the FermiNet, called the full determinant mode, which replaces each product of determinants with a single combined determinant. We find that the full single-determinant FermiNet closes a large part of the gap between the standard single-determinant FermiNet and FermiNet-GA on small atomic and molecular problems. Surprisingly, on the nitrogen molecule at a dissociating bond length of 4.0 Bohr, the full single-determinant FermiNet can significantly outperform the largest standard FermiNet calculation with 64 determinants, yielding an energy within 0.4 kcal/mol of the best available computational benchmark.

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Section: Introductionmentioning

confidence: 99%

Explicitly antisymmetrized neural network layers for variational Monte Carlo simulation

Lin¹,

Goldshlager²,

Lin³

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…Research into neural network Ansätze has covered Boltzmann machines and feedforward neural networks over a wide range of systems [23,[32][33][34], including more recent Slater determinant Ansätze [35,36]. Other more recent examples incorporate physics based structure [3], or large and deep networks [2,24,26] on molecular sys-tems.…”

Section: Related Workmentioning

confidence: 99%

Wave function Ansatz (but Periodic) Networks and the Homogeneous Electron Gas

Wilson¹,

Moroni²,

Holzmann³

et al. 2022

Preprint

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We design a neural network Ansatz for variationally finding the ground-state wave function of the Homogeneous Electron Gas, a fundamental model in the physics of extended systems of interacting fermions. We study the spin-polarised and paramagnetic phases with 7, 14 and 19 electrons over a broad range of densities from rs = 1 to rs = 100, obtaining similar or higher accuracy compared to a state-of-the-art iterative backflow baseline even in the challenging regime of very strong correlation. Our work extends previous applications of neural network Ansätze to molecular systems with methods for handling periodic boundary conditions, and makes two notable changes to improve performance: splitting the pairwise streams by spin alignment and generating backflow coordinates for the orbitals from the network. This contribution establishes neural network models as flexible and high precision Ansätze for periodic electronic systems, an important step towards applications to crystalline solids.

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“…[ 14 ] As a general purpose trial wave function, feed‐forward NNs can simulate quantum Monte Carlo for continuous many‐body systems. [ 15 ] Deep neural networks are combined with a partial differential equation, which provides understanding of the relationship between the pore‐scale electrode structure reaction and device‐scale electrochemical reaction uniformity. [ 16 ] Besides, to overcome major healthcare challenges in real world, such as COVID‐19 and skin cancer, people focus on the uncertain ANNs and make them interpretable to detect and diagnose of medical images, which has achieved competitive accuracy and gained clinicians trust.…”

Section: Introductionmentioning

confidence: 99%

Interpretability of Neural Networks with Probability Density Functions

Pan

Pedrycz

Cui

et al. 2022

Advcd Theory and Sims

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It is an interesting topic to interpret artificial neural networks (ANNs) by considering some change various approaches. This paper explores the relationship between the input and output units of the simplest ANN, a single layer perceptron for the binary classification problem, from the probability point of view. If the feature variables of datasets follow independent normal distribution and outputs are activated by sigmoid function or smooth Relu function, we advocate that the probability density function (pdf) of the output variable is an exponential family distribution. Furthermore, by introducing an intermediate variable, the pdf of the output variable can be written as a linear combination of three normal distributions with same spread but different centers. Based on these results, the probability of the predicted class label can be written as a standard normal cumulative distribution function (cdf). The originality of this paper comes with interesting theoretical results to provide ANNs with a new description of the relationship between input variables to output variables, which can enable ANNs to be understood from a new perspective. Extensive experiments based on one artificial synthesized dataset and ten real‐world benchmark datasets validate the reasonability of those results.

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Artificial Neural Networks as Trial Wave Functions for Quantum Monte Carlo

Cited by 23 publications

References 55 publications

Explicitly antisymmetrized neural network layers for variational Monte Carlo simulation

Explicitly antisymmetrized neural network layers for variational Monte Carlo simulation

Wave function Ansatz (but Periodic) Networks and the Homogeneous Electron Gas

Interpretability of Neural Networks with Probability Density Functions

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