Artificial Neural Network Methods in Quantum Mechanics

In a previous article 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 Schr\"odinger equation for the Morse potential that has an analytically known solution, to test the accuracy of the method. We then proceed with the Schr\"odinger and the Dirac equations for a muonic atom, as well as with a non-local Schr\"odinger integrodifferential equation that models the $n+\alpha$ system in the framework of the resonating group method. In two dimensions we consider the well studied 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.Comment: Latex file, 29pages, 11 psfigs, submitted in CP

Neural Network Methods for Boundary Value Problems Defined in Arbitrarily Shaped Domains

Partial differential equations (PDEs) with Dirichlet boundary conditions defined on boundaries with simple geometry have been succesfuly treated using sigmoidal multilayer perceptrons in previous works. This article deals with the case of complex boundary geometry, where the boundary is determined by a number of points that belong to it and are closely located, so as to offer a reasonable representation. Two networks are employed: a multilayer perceptron and a radial basis function network. The later is used to account for the satisfaction of the boundary conditions. The method has been successfuly tested on two-dimensional and three-dimensional PDEs and has yielded accurate solutions

Quadratic momentum dependence in the nucleon-nucleon interaction

We investigate different choices for the quadratic momentum dependence required in nucleon-nucleon potentials to fit phase shifts in high partial-waves. In the Argonne v18 potential L**2 and (L.S)**2 operators are used to represent this dependence. The v18 potential is simple to use in many-body calculations since it has no quadratic momentum-dependent terms in S-waves. However, p**2 rather than L**2 dependence occurs naturally in meson-exchange models of nuclear forces. We construct an alternate version of the Argonne potential, designated Argonne v18pq, in which the L**2 and (L.S)**2 operators are replaced by p**2 and Qij operators, respectively. The quadratic momentum-dependent terms are smaller in the v18pq than in the v18 interaction. Results for the ground state binding energies of 3H, 3He, and 4He, obtained with the variational Monte Carlo method, are presented for both the models with and without three-nucleon interactions. We find that the nuclear wave functions obtained with the v18pq are slightly larger than those with v18 at interparticle distances < 1 fm. The two models provide essentially the same binding in the light nuclei, although the v18pq gains less attraction when a fixed three-nucleon potential is added.Comment: v.2 important corrections in tables and minor revisions in text; reference for web-posted subroutine adde

Nuclear matter hole spectral function in the Bethe-Brueckner-Goldstone approach

The hole spectral function is calculated in nuclear matter to assess the relevance of nucleon-nucleon short range correlations. The calculation is carried out within the Brueckner scheme of many-body theory by using several nucleon-nucleon realistic interactions. Results are compared with other approaches based on variational methods and transport theory. Discrepancies appear in the high energy region, which is sensitive to short range correlations, and are due to the different many-body treatment more than to the specific N-N interaction used. Another conclusion is that the momentum dependence of the G-matrix should be taken into account in any self consistent approach.Comment: 7 pages, 5 figure

Short-range Correlations in a CBF description of closed-shell nuclei

The Correlated Basis Function theory (CBF) provides a theoretical framework to treat on the same ground mean-field and short-range correlations. We present, in this report, some recent results obtained using the CBF to describe the ground state properties of finite nuclear systems. Furthermore we show some results for the excited state obtained with a simplified model based on the CBF theory.Comment: 10 latex pages plus 6 uuencoded figure

Variational Monte Carlo Calculations of 3^3H and 4^4He with a relativistic Hamiltonian - II

In relativistic Hamiltonians the two-nucleon interaction is expressed as a sum of $\tilde{v}_{ij}$, the interaction in the ${\bf P}_{ij}=0$ rest frame, and the ``boost interaction'' $\delta v({\bf P}_{ij})$ which depends upon the total momentum ${\bf P}_{ij}$ and vanishes in the rest frame. The $\delta v$ can be regarded as a sum of four terms: $\delta v_{RE}$, $\delta v_{LC}$, $\delta v_{TP}$ and $\delta v_{QM}$; the first three originate from the relativistic energy-momentum relation, Lorentz contraction and Thomas precession, while the last is purely quantum. The contributions of $\delta v_{RE}$ and $\delta v_{LC}$ have been previously calculated with the variational Monte Carlo method for $^3$H and $^4$He. In this brief note we report the results of similar calculations for the contributions of $\delta v_{TP}$ and $\delta v_{QM}$. These are found to be rather small.Comment: 7 pages, P-94-09-07

Fast Neural Network Predictions from Constrained Aerodynamics Datasets

Incorporating computational fluid dynamics in the design process of jets, spacecraft, or gas turbine engines is often challenged by the required computational resources and simulation time, which depend on the chosen physics-based computational models and grid resolutions. An ongoing problem in the field is how to simulate these systems faster but with sufficient accuracy. While many approaches involve simplified models of the underlying physics, others are model-free and make predictions based only on existing simulation data. We present a novel model-free approach in which we reformulate the simulation problem to effectively increase the size of constrained pre-computed datasets and introduce a novel neural network architecture (called a cluster network) with an inductive bias well-suited to highly nonlinear computational fluid dynamics solutions. Compared to the state-of-the-art in model-based approximations, we show that our approach is nearly as accurate, an order of magnitude faster, and easier to apply. Furthermore, we show that our method outperforms other model-free approaches

Phaseshift equivalent NN potentials and the deuteron

Different modern phase shift equivalent NN potentials are tested by evaluating the partial wave decomposition of the kinetic and potential energy of the deuteron. Significant differences are found, which are traced back to the matrix elements of the potentials at medium and large momenta. The influence of the localisation of the one-pion-exchange contribution to these potentials is analyzed in detail.Comment: 11 pages, LaTeX, 4 figures include

Spin-Isospin Structure and Pion Condensation in Nucleon Matter

We report variational calculations of symmetric nuclear matter and pure neutron matter, using the new Argonne v18 two-nucleon and Urbana IX three-nucleon interactions. At the equilibrium density of 0.16 fm^-3 the two-nucleon densities in symmetric nuclear matter are found to exhibit a short-range spin-isospin structure similar to that found in light nuclei. We also find that both symmetric nuclear matter and pure neutron matter undergo transitions to phases with pion condensation at densities of 0.32 fm^-3 and 0.2 fm^-3, respectively. Neither transtion occurs with the Urbana v14 two-nucleon interaction, while only the transition in neutron matter occurs with the Argonne v14 two-nucleon interaction. The three-nucleon interaction is required for the transition to occur in symmetric nuclear matter, whereas the the transition in pure neutron matter occurs even in its absence. The behavior of the isovector spin-longitudinal response and the pion excess in the vicinity of the transition, and the model dependence of the transition are discussed.Comment: 44 pages RevTeX, 15 postscript figures. Minor modifications to original postin