TY - CONF

T1 - SELF-CONSISTENT EVOLUTION: NEW NEURAL NETWORK APPROACH TO BOUND STATE CALCULATIONS

AU - Руднев, Владимир Александрович

N1 - Conference code: LXXIV

PY - 2024/7/1

Y1 - 2024/7/1

N2 - An application of neural networks for solving quantum mechanical problems has beensuggested in [1,2]. Many improvements, including an adaptation of deep neural network techniques[3], have been proposed since. Development of a new computational technology which could liftthe curse of dimensionality, however, has not yet been completed, although some steps in thisdirection have already been made [4,5].We propose a new approach to training neural networks for approximation of quantumHamiltonian invariant subspaces corresponding to bound states. The approach is based on trainingan artificial neural network to solve the Schr ̈odinger equation in imaginary time with initialconditions that put the solution into an invariant subspace.The advantage of the proposed approach is a simpler objective function which leads to betterperformance.Theoretical results are illustrated with numerical examples.1. I.E. Lagaris, A. Likas, and D.I. Fotiadis, Artificial Neural Networks for Solving Ordinary andPartial Differential Equations // IEEE TRANSACTIONS ON NEURAL NETWORKS 1998, V. 9, N. 5,P.9872. I.E. Lagaris, A. Likas, and D.I. Fotiadis, Artificial neural networks in quantum mechanics // Comp.Phys. Comm. 1997, V.104, P.1-143. Sirignano, J., Spiliopoulos, K., DGM: A deep learning algorithm for solving partial differentialequations// arXiv preprint arXiv:1708.074694. Hong Li, Qilong Zhai, Jeff Z. Y. Chen, Neural-network-based multistate solver for a staticSchr ̈odinger equation // Phys.Rev. A 2021,V. 103, P. 0324055. V.A. Roudnev, M.M. Stepanova, Deep learning approach to high dimensional problems of quantummechanics // Proceedings of Science 2022, V.429, P. 13

AB - An application of neural networks for solving quantum mechanical problems has beensuggested in [1,2]. Many improvements, including an adaptation of deep neural network techniques[3], have been proposed since. Development of a new computational technology which could liftthe curse of dimensionality, however, has not yet been completed, although some steps in thisdirection have already been made [4,5].We propose a new approach to training neural networks for approximation of quantumHamiltonian invariant subspaces corresponding to bound states. The approach is based on trainingan artificial neural network to solve the Schr ̈odinger equation in imaginary time with initialconditions that put the solution into an invariant subspace.The advantage of the proposed approach is a simpler objective function which leads to betterperformance.Theoretical results are illustrated with numerical examples.1. I.E. Lagaris, A. Likas, and D.I. Fotiadis, Artificial Neural Networks for Solving Ordinary andPartial Differential Equations // IEEE TRANSACTIONS ON NEURAL NETWORKS 1998, V. 9, N. 5,P.9872. I.E. Lagaris, A. Likas, and D.I. Fotiadis, Artificial neural networks in quantum mechanics // Comp.Phys. Comm. 1997, V.104, P.1-143. Sirignano, J., Spiliopoulos, K., DGM: A deep learning algorithm for solving partial differentialequations// arXiv preprint arXiv:1708.074694. Hong Li, Qilong Zhai, Jeff Z. Y. Chen, Neural-network-based multistate solver for a staticSchr ̈odinger equation // Phys.Rev. A 2021,V. 103, P. 0324055. V.A. Roudnev, M.M. Stepanova, Deep learning approach to high dimensional problems of quantummechanics // Proceedings of Science 2022, V.429, P. 13

KW - Nural networks

KW - Bound states

KW - Quantum mechanics

UR - https://drive.google.com/file/d/1BuXs9FVMraOODIb-0lF1F2Vf79UoyYCf/view

M3 - Abstract

T2 - LXXIV International Conference “Nucleus-2024: Fundamental problems and applications”, Dubna, July 1–5, 2024:

Y2 - 1 July 2024 through 5 July 2024

ER -