Year: 2018
Type: dissertation
Abstract: Over the last few years the field of Machine Learning had great success in employing Deep Learning techniques to efficiently represent highly complex multivariate functions.Thereby, guidelines were found that help to design Neural Networks specifically to tackle a certain task.A great issue in quantum many-body physics is finding efficient representations of quantum wave-functions.Thus, we investigate Artificial Neural Networks as variational approach to find the ground-state wavefunction of the anti-ferromagnetic Heisenberg Hamiltonian.Goal of our investigation is to derive guidelines for how to build a suitable network for this particular task, similar to the guidelines known from Machine Learning.We restricted our investigation to small system sizes, that are solvable by Exact Diagonalization, but we aimed for a network that can yield computationally exact ground-state energies.In our tests we observed that increasing the depth of neural networks increases the expressiveness of a network.Networks that are composed from activation functions with a specific symmetry are more successful to represent functions that agree with this symmetry.Increasing the depth makes networks less biased towards this intrinsic symmetry.Though to have computationally exact energies, activation functions are required to be unbiased towards symmetry.Other aspects of interest are training success and required number of parameters.We found that training success can significantly be increased with activation functions that have at no point a vanishing gradient.Regrading number of parameters, increasing depth to a moderate number (slightly more than two) allows to drastically reduce network parameters.In conclusion we found that one must respect the symmetry of the wave-function when trying to approximate it.Further redundant neurons and functions with strong gradient allow to increase training success, while depth can help reducing the necessary number of parameters (to accomplish a certain precision in energy).Ideally the derived guidelines help constructing networks to tackle larger systems and different Hamiltonians. Eidesstattliche ErklärungIch erkläre hiermit an Eides statt durch meine eigenhändige Unterschrift, dass ich die vorliegende Arbeit selbständig verfasst und keine anderen als die angegebenen Quellen und Hilfsmittel verwendet habe.Alle Stellen, die wörtlich oder inhaltlich den angegebenen Quellen entnommen wurden, sind als solche kenntlich gemacht.Die vorliegende Arbeit wurde bisher in gleicher oder ähnlicher Form noch nicht als Magister-/Master-/Diplomarbeit/Dissertation eigereicht.
Author: Rigo, Jonas
Institution: Universität Innsbruck
Language: English
Topic: Quantum many-body systems
Subfield: Atomic and Molecular Physics, and Optics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7017510460