Artificial neural network methods in quantum mechanics

Abstract

In a previous article [1] we have shown how one can employ Artificial Neural Networks (ANNs) in order to solve non-homogeneous ordinary and partial differential equations. In the present work we consider the solution of eigenvalue problems for differential and integrodifferential operators, using ANNs. We start by considering the Schrödinger equation for the Morse potential that has an analytically known solution, to test the accuracy of the method. We then proceed with the Schrödinger and the Dirac equations for a muonic atom, as well as with a non-local Schrödinger integrodifferential equation that models the n + α system in the framework of the resonating group method. In two dimensions we consider the well studied [2] Henon-Heiles Hamiltonian and in three dimensions the model problem of three coupled anharmonic oscillators. The method in all of the treated cases proved to be highly accurate, robust and efficient. Hence it is a promising tool for tackling problems of higher complexity and dimensionality.

Genetic Programming Approaches for Solving Elliptic Partial Differential Equations

“…This makes finding a solution that satisfies the boundary conditions inefficient at best and almost intractable at worst. This problem motivated us to employ the formulations developed in [10]- [12] to ensure that the boundary conditions are implicitly satisfied. In a GP context, such approaches for implicitly satisfying the boundary conditions can be interpreted as chromosome repair strategies, which are similar in spirit to those commonly employed in the evolutionary optimization literature for tackling constrained numerical optimization problems; see, for example, Michalewicz [33].…”

“…To ensure that the boundary conditions are satisfied by , we need to enforce at the boundary nodes in the set , that is (10) Substituting (9) in (10) and assuming the boundary operator to be linear, we obtain a system of linear equations for the undetermined coefficients, , as given below (11) where…”

“…These approximations are then aggregated to form the final expression for the field variables (see [10] for a description of this formulation in the context of increasing the accuracy of neural network solutions of linear PDEs).…”

Piecewise Neural Networks for Function Approximation, Cast in a Form Suitable for Parallel Computation

“…Other non-polynomial splines have been developed in the past, for instance we mention the "Tension Splines" that are based on the exponential function [2]. Neural Networks are well known for their universal approximation capabilities [3], [4] and have been employed for interpolation, approximation and modeling tasks in many cases, ranging from pattern recognition [5], signal processing, control and the solution of ordinary and partial differential equations [6], [7], [8].…”

Artificial neural networks with an infinite number of nodes

“…Usual types are the so called feed-forward "Multi-Layered Perceptrons (MLP)" and the "Radial Basis Function (RBF) networks". The usefulness of ANNs is indisputable and there is a vast literature describing applications in different fields, such as pattern recognition and data fitting [3,4], the solution of ordinary and partial differential equations [5,6,7], stock price prediction [8], etc. ANNs can learn from existing data the underlying law (i.e.…”