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arXiv:2412.03319 · ESAIM Proceedings and Surveys

Comparison between tensor methods and neural networks in electronic structure calculations

Mathias Dus, Geneviève Dusson, Virginie Ehrlacher, Clément Guillot, Joel Pascal Soffo Wambo

Published Jan 1, 2025• 0 citations
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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 Schrödinger 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 Schrödinger 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.

DOI
10.1051/proc/202581055

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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 Schrödinger 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 Schrödinger 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.
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 Schrödinger 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 Schrödinger 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.
The solution of Schrödinger equation generates the quantisation properties of the system, which can completely describe the quantum behaviour of microscopic particles in a physical system. However, solving the Schrödinger equation is a non-deterministic polynomial time (NP)-hard problem. Numerical methods for solving the Schrödinger equation require careful selection of basis configurations, and the complete basis set has high accuracy but its complexity increases exponentially in the number of particles. Therefore, it is crucial to find an effective numerical method. To solve this problem, this paper present a deep learning architecture, VMCNet, using the powerful computational efficiency of neural networks to improve the speed of numerical computation. Moreover, this paper proposes a suitable wavefunction ansatz, witch determines the accuracy of neural networks, that achieves more accurate energy solutions of the many-electron Schrödinger equation compared to the conventional single-determinant variational molecular Monte Carlo. We introduce non-local and local blocks to represent quantum mechanical interactions, which contain more physical information about the particle system. The experimental results show that VMCNet can cover more than 93% of the correlation energy of the main group elements (Be–Ne) for the monoatomic system and more than 90% of the correlation energy for the diatomic system.
Deep learning techniques have opened a new venue for electronic structure theory in recent years. In contrast to traditional methods, deep neural networks provide much more expressive and flexible wave function Ans\"atze, resulting in better accuracy and timescale behavior. In order to study larger systems while retaining sufficient accuracy, we integrate a powerful neural-network-based model (FermiNet) with the effective core potential method, which helps to reduce the complexity of the problem by replacing inner core electrons with additional semilocal potential terms in the Hamiltonian. In this work, we calculate the ground-state energy of 3d3d3d transition metal atoms and their monoxides, which is quite challenging for the original FermiNet work, and the results are consistent with both experimental data and other state-of-the-art computational methods. Our work is an important step for a broader application of deep learning in the electronic structure calculation of molecules and materials.
One of the main challenges of quantum many-body physics is that the dimensionality of the Hilbert space grows exponentially with the system size, which makes it extremely difficult to solve the Schrodinger equations of the system. But typically, many physical systems have a simplified internal structure which makes the parameters needed to characterize their ground states exponentially smaller. This makes many numerical methods possible in capture the physics of the system. Among these modern numerical techniques, neural networks, which show great power in approximating functions and extracting features of the big data, are now attracting many interests. Neural network representation of quantum many-body states shows great potential in solving some traditionally difficult quantum problems involving large number of freedoms. In this work, we briefly review the progress of using the artificial neural network to build quantum many-body ansatz states. We take Boltzmann machine representation as a prototypical example to illustrate various aspects of neural network representation of quantum many-body states. We first briefly review the classical neural networks, then we illustrate how to use neural networks to represent quantum states and density operators. Some physical properties of the neural network states, including entanglement features, representational power, and the relation with tensor network states, are discussed. For applications, we briefly review the progress of many-body calculating based on neural network states, neural network state approach to tomography, and also the classical simulation of quantum computing based on Boltzmann machine states. At the end of the work, some outlooks and open problems are given.
Understanding strongly correlated quantum many-body systems requires dealing with a largeconfiguration space comprised of a set of all possible measurable configurations. The total number of configurations grows exponentially with the size of the system, which makes exact numerical calculations impossible for even moderately sized systems. Since the advent of high-temperature superconductivity and the subsequent interest in strongly correlated phases of matter, progress in the field has been marked by ingenuity of the field to overcome these computational limitations. The first techniques to emerge were inspired by quantum chemistry and based on exact diagonalization[1], which is limited to small clusters, and different variants of quantum Monte Carlo[2], that suffers from the sign problem in fermionic and frustrated systems. Circa 1992, the density matrix renormalization group (DMRG) [3, 4, 5, 6, 7] became one of the major breakthroughs of the past few decades. Despite the success of DMRG for one-dimensional(1D) and quasi-one-dimensional geometries, extensions to actual two-dimensional systems remain challenging and applications are constrained to long cylinders and strips. One promising alternative is to combine one the oldest methods, variational wave functions (VWFs), with neural networks, which have been used to recently with great success in machine learning. The focus of this dissertation is on developing novel variational Monte Carlo algorithms to make neural network variational wave functions applicable to many problems in quantum many-body physics. These algorithms draw from existing variational wave-function, tensor network, and machine learning methods. We calculate dynamical spectral functions using the vector correction and Chebyshev expansion method. We do finite temperature calculations with minimally entangled thermal states. Finally, we do ground state calculations for the frustrated J1−J2 Heisenberg model. We benchmark these results using restricted Boltzmann machines as the neural network for the the variational wave-functions.--Author's abstract