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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Maximum-entropy principle and neural networks that learn to construct approximate wave functions

Luis Anibal Diambra, Angel Ricardo Plastino

Published Jan 1, 1996• 6 citations
Publisher

Abstract

By recourse to a maximum-entropy-based method for the training of perceptrons, we show that an appropriately prepared network is able to construct approximate ground state wave functions of good quality with the sole knowledge of a few expectation values, even if nothing is known concerning the interaction potential. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physreve.53.1021

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By recourse to a maximum-entropy-based method for the training of perceptrons, we show that an appropriately prepared network is able to construct approximate ground state wave functions of good quality with the sole knowledge of a few expectation values, even if nothing is known concerning the interaction potential. \textcopyright{} 1996 The American Physical Society.
Quantum theory has been remarkably successful in providing an understanding of physical systems at foundational scales. Solving the Schrödinger equation provides full knowledge of all dynamical quantities of the physical system. However closed form solutions to this equation are only available for a few systems and approximation methods are typically used to find solutions. In this paper we address the problem of solving the time independent Schrödinger equation for the ground state solution of physical systems. We propose using end-to-end deep learning approach in a variational optimization scheme for approximating the ground state energies and wave functions of these systems. A neural network realizes a universal trial wave function and is trained in an unsupervised learning framework by optimizing the expectation value of the Hamiltonian of a physical system. The proposed approach is evaluated on physical systems consisting of a particle in a box with and without a perturbation. We demonstrate that our approach obtains approximations of ground state energies and wave functions that are highly accurate, which makes it a potentially plausible candidate for solving more complex physical systems for which analytical solutions are beyond reach.
Recently developed neural network-based wave function methods are capable of achieving state-of-the-art results for finding the ground state in real space. In this work, a neural network-based method is used to compute excited states. We train our network via variational principle, along a further penalty term that imposes the orthogonality with lower-energy eigenfunctions. As a demonstration of the effectiveness of this approach, results from numerical calculations for one-dimensional and two-dimensional harmonic oscillators are presented.
We study neural-network approximation of density functionals in a Banach-space setting. Since the exact continuum functionals are discontinuous in the natural density topology, we apply ppp-Moreau-Yosida regularization. We construct finite-rank ReLU networks that preserve convexity and nonnegativity, lie below the regularized functional, and approximate it uniformly on compact sets. For energies obtained by minimization over compact density sets, we account exactly for the energy shift caused by regularization and bound the remaining error by the localization on these compact sets and network approximation errors. We apply this framework to canonical and grand-canonical density-functional theory at fixed particle number and fixed mean particle number, respectively. We obtain uniform energy convergence on norm-compact classes of external potentials, including molecular Coulomb potentials. We illustrate our results through numerical experiments on a simple model system.
We present a machine-learning approach to a long-standing issue in quantum many-body physics, namely, analytic continuation. This notorious ill-conditioned problem of obtaining spectral function from an imaginary time Green's function has been a focus of new method developments for past decades. Here we demonstrate the usefulness of modern machine-learning techniques including convolutional neural networks and the variants of a stochastic gradient descent optimizer. The machine-learning continuation kernel is successfully realized without any ``domain knowledge,'' which means that any physical ``prior'' is not utilized in the kernel construction and the neural networks ``learn'' the knowledge solely from ``training.'' The outstanding performance is achieved for both insulating and metallic band structure. Our machine-learning-based approach not only provides the more accurate spectrum than the conventional methods in terms of peak positions and heights, but is also more robust against the noise which is the required key feature for any continuation technique to be successful. Furthermore, its computation speed is 104–105{10}^{4}\text{--}{10}^{5}104–105 times faster than the maximum entropy method.