Neural network approach for the calculation of potential coefficients in quantum mechanics

Abstract A numerical method based on artificial neural networks is used to solve the inverse Schrodinger equation for a multi-parameter class of potentials. First, the finite element method was used to solve repeatedly the direct problem for different parametrizations of the chosen potential function. Then, using the attainable eigenvalues as a training set of the direct radial basis neural network a map of new eigenvalues was obtained. This relationship was later inverted and refined by training an inverse radial basis neural network, allowing the calculation of the unknown parameters and therefore estimating the potential function. Three numerical examples are presented in order to prove the effectiveness of the method. The results show that the method proposed has the advantage to use less computational resources without a significant accuracy loss.

[1]  Sebastián Ossandón,et al.  On the neural network calculation of the Lamé coefficients through eigenvalues of the elasticity operator , 2016 .

[2]  B. M. Fulk.  MATH , 1992 .

[3]  Y. Shirvany,et al.  Numerical solution of the nonlinear Schrodinger equation by feedforward neural networks , 2008 .

[4]  M. Fortin,et al.  Mixed Finite Element Methods and Applications , 2013 .

[5]  Dimitris G. Papageorgiou,et al.  An Efficient Chebyshev-Lanczos Method for Obtaining Eigensolutions of the Schrödinger Equation on a Grid , 1996 .

[6]  Michel Fortin,et al.  Mixed and Hybrid Finite Element Methods , 2011, Springer Series in Computational Mathematics.

[7]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[8]  James J. Carroll,et al.  Approximation of nonlinear systems with radial basis function neural networks , 2001, IEEE Trans. Neural Networks.

[9]  D. Fotiadis,et al.  Artificial neural network methods in quantum mechanics , 1997, quant-ph/9705029.

[10]  Andrew G. Glen,et al.  APPL , 2001 .

[11]  Tom E. Simos,et al.  A finite-difference method for the numerical solution of the Schro¨dinger equation , 1997 .

[12]  Daniele Boffi,et al.  Finite element approximation of eigenvalue problems , 2010, Acta Numerica.

[13]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[14]  Carlos M. Reyes,et al.  Neural network solution for an inverse problem associated with the Dirichlet eigenvalues of the anisotropic Laplace operator , 2016, Comput. Math. Appl..

[15]  Arif Masud,et al.  Stabilized Finite Element Methods for the Schrdinger Wave Equation , 2009 .

[16]  B. Mercier,et al.  Eigenvalue approximation by mixed and hybrid methods , 1981 .

[17]  Thad G. Walker,et al.  Matrix Numerov method for solving Schrödinger's equation , 2012 .

[18]  H. Ishikawa,et al.  An accurate method for numerical calculations in quantum mechanics , 2002 .

[19]  Tomaso A. Poggio,et al.  A Sparse Representation for Function Approximation , 1998, Neural Computation.