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

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

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

Microscopic calculations of the enhancement factor in the electric dipole sum rule

Correlated basis function perturbation theory with state-dependent correlations is used to calculate the nuclear photoabsorp- tion enhancement factor K in the electric dipole sum rule for some realistic models of nuclear matter. The contribution due to 2p-2h admixtures in the ground state wave function turns out to be only a few percent of the unperturbed value. The values obtained for K are about 1.8 at experimental equilibrium density and increase almost linearly with density. We also give estimates of K for finite nuclei, obtained within the local density approximation framework. The surface effects give a contribution which is - 20% of the volume term. state of the non-relativistic hamiltonian having V as nuclear potential and D z = ~Ei= 1 ,A rizZi is the z component of the electric dipole operator, with riz being the third component of the isospin opertor for the ith nucleon. The theoretical estimates (2) of K do not depend very much on the realistic interaction adopted, and are more than a factor of two larger than the experimental value (3), Kex p = 0.76 + 0.10, obtained from the integrated photo- nuclear cross sections up to the rr-meson production threshold. It is important to know how much of this discre- pancy is due to effects not explicitly taken into account in the Bethe-Levinger sum rule, like tail corrections of the integrated cross section, higher multipoles and dipole retardation effects, and how much is due to the in- adequacy of the variational wave function used in the calculation. In this letter we present the results obtained for K when the variational ground state is corrected by adding 2p2h correlated basis functions (CBF) components to it. The 2p2h admixtures are calculated by using second order CBF perturbation theory (4--6). The CBF states are normalized but not orthogonal, and are given by (koi) = F( (bi)/(cb i (F+Fltbi )1/2, (2) where I(I)i) are Fermi gas states and F = S H

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

Artificial Neural Networks for Solving Ordinary and Partial Differential Equations

We present a method to solve initial and boundary value problems using artificial neural networks. A trial solution of the differential equation is written as a sum of two parts. The first part satisfies the boundary (or initial) conditions and contains no adjustable parameters. The second part is constructed so as not to affect the boundary conditions. This part involves a feedforward neural network, containing adjustable parameters (the weights). Hence by construction the boundary conditions are satisfied and the network is trained to satisfy the differential equation. The applicability of this approach ranges from single ODE's, to systems of coupled ODE's and also to PDE's. In this article we illustrate the method by solving a variety of model problems and present comparisons with finite elements for several cases of partial differential equations.Comment: LAtex file, 26 pages, 21 figs, submitted to IEEE TN

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

Exploring datasets to solve partial differential equations with TensorFlow

The version of record is available online at: http://dx.doi.org/10.1007/978-3-030-57802-2_42This paper proposes a way of approximating the solution of partial differential equations (PDE) using Deep Neural Networks (DNN) based on Keras and TensorFlow, that is capable of running on a conventional laptop, which is relatively fast for different network architectures. We analyze the performance of our method using a well known PDE, the heat equation with Dirichlet boundary conditions for a non-derivable non-continuous initial function. We have tried the use of different families of functions as training datasets as well as different time spreadings aiming at the best possible performance. The code is easily modifiable and can be adapted to solve PDE problems in more complex scenarios by changing the activation functions of the different layers.This work has been partially supported by the Spanish Ministry of Science, Innovation and Universities, Gobierno de España, under Contracts No. PGC2018-093854-BI00, and ICMAT Severo Ochoa SEV-2015-0554, and from the People Programme (Marie Curie Actions) of the European Union’s Horizon 2020 Research and Innovation Program under Grant No. 734557.Peer ReviewedPostprint (published version

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