Learning Potentials of Quantum Systems using Deep Neural Networks

Abstract

Machine Learning has wide applications in a broad range of subjects, including physics. Recent works have shown that neural networks can learn classical Hamiltonian mechanics. The results of these works motivate the following question: Can we endow neural networks with inductive biases coming from quantum mechanics and provide insights for quantum phenomena? In this work, we try answering these questions by investigating possible approximations for reconstructing the Hamiltonian of a quantum system given one of its wave-functions. Instead of handcrafting the Hamiltonian and a solution of the Schrödinger equation, we design neural networks that aim to learn it directly from our observations. We show that our method, termed Quantum Potential Neural Networks (QPNN), can learn potentials in an unsupervised manner with remarkable accuracy for a wide range of quantum systems, such as the quantum harmonic oscillator, particle in a box perturbed by an external potential, hydrogen atom, Pöschl-Teller potential, and a solitary wave system. Furthermore, in the case of a particle perturbed by an external force, we also learn the perturbed wave function in a joint end-to-end manner. * Equal contribution Preprint. Under review.

Predicting quantum potentials by deep neural network and metropolis sampling

“…In Ref. [23], the authors propose to use a deep neural network named as quantum potential neural network (QPNN) to predict the unknown potential V (r). In detail, the QPNN maps the coordinates to the values of the potential, denoted as U θ (r) with θ the variational parameters of the QPNN.…”

“…The authors of Ref. [23] proposed to use E (r), considering that the square root of the probability density is much easier to access in experiments.…”

“…Considering the quantum lattice models, for example, it has been proposed to predict the coupling constants from the measurements of the target states [21] or the local reduced density matrices [22]. Sehanobish et al consider the Schrödinger equation and propose the quantum potential NN (QPNN) to predict the potential term provided the eigen wave-function [23]. These works indicates the feasibility of using ML to investigate quantum phenomena by reformulating the quantum mechanical systems as the solutions of certain inverse problems.…”

Observing how deep neural networks understand physics through the energy spectrum of one-dimensional quantum mechanics

“…In this paper, the subject of this study is the Schrödinger equation in quantum mechanics, and even if we focus only on the surrounding area, we can see many recent developments related to NNs. Partial differential equations, including the Schrödinger equation, have been solved using NNs [8], potentials have been estimated inversely from wave functions [9,10], and soliton solutions to the nonlinear Schrödinger equation have been investigated using NNs [11]. A new family of NNs inspired by the Schrödinger equation has also been proposed [12].…”

Deep Neural Network Based Differential Equation Solver for HIV Enzyme Kinetics

“…Liu et al [5] applied a NN approach to solving the Laplace Equation. In quantum mechanics, Sehanobish et al [10] used NNs to compute solutions for the potential energy function from Schrodinger's Equation. In classical mechanics, Mattheakis et al [6] solved Hamilton's Equations for positions and momenta, besting the fidelity of numerical solution phase space diagrams for both periodic and chaotic dynamical systems.…”