Comparison between tensor methods and neural networks in electronic structure calculations

TL;DR

This paper compares tensor and neural network methods for electronic structure calculations, demonstrating their convergence properties and proposing a new neural network approach based on wavefunction training on a simplex.

Contribution

It provides a comparative analysis of DMRG, FermiNet, and PauliNet methods and introduces a novel neural network architecture for solving the Schrödinger equation.

Findings

Tensor and neural network methods show convergence with parameters.

Energy calculations are compared for several atoms and molecules.

A new neural network approach based on wavefunction training on a simplex is proposed.

Abstract

This article compares the tensor method density matrix renormalization group (DMRG) with two neural network based methods -namely FermiNet and PauliNet) for determining the ground state wavefunction of the many-body electronic Schr{\"o}dinger problem. We provide numerical simulations illustrating the main features of the methods and showing convergence with respect to some parameter, such as the rank for DMRG, and number of pretraining iterations for neural networks. We then compare the obtained energy with the methods for a few atoms and molecules, for some of which the exact value of the energy is known for the sake of comparison. In the last part of the article, we propose a new kind of neural network to solve the Schr{\"o}dinger problem based on the training of the wavefunction on a simplex, and an explicit permutation for evaluating the wavefunction on the whole space. We provide numerical results on a toy problem for the sake of illustration.