Isaac E. Lagaris

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 initial/boundary conditions and contains no adjustable parameters. The second part is constructed so as not to affect the initial/boundary conditions. This part involves a feedforward neural network containing adjustable parameters (the weights). Hence by construction the initial/boundary conditions are satisfied and the network is trained to satisfy the differential equation. The applicability of this approach ranges from single ordinary differential equations (ODEs), to systems of coupled ODEs and also to partial differential equations (PDEs). In this article, we illustrate the method by solving a variety of model problems and present comparisons with solutions obtained using the Galekrkin finite element method for several cases of partial differential equations. With the advent of neuroprocessors and digital signal processors the method becomes particularly interesting due to the expected essential gains in the execution speed.

A spatially constrained mixture model for image segmentation

Gaussian mixture models (GMMs) constitute a well-known type of probabilistic neural networks. One of their many successful applications is in image segmentation, where spatially constrained mixture models have been trained using the expectation-maximization (EM) framework. In this letter, we elaborate on this method and propose a new methodology for the M-step of the EM algorithm that is based on a novel constrained optimization formulation. Numerical experiments using simulated images illustrate the superior performance of our method in terms of the attained maximum value of the objective function and segmentation accuracy compared to previous implementations of this approach.

PHENOMENOLOGICAL TWO NUCLEON INTERACTION OPERATOR

Abstract We report a phenomenological two-nucleon interaction operator obtained by fitting the nucleon-nucleon phase shifts up to 425 MeV in S, P, D and F waves, and the deuteron properties. The operator has the standard eight potentials associated with the two-body operators 1, σ 1 · σ 2 , τ 1 · τ 2 , σ 1 · σ 2 τ 1 · τ 2 , S 12 , S 12 τ 2 , S 12 τ 1 · τ 2 , L · S and L · S τ 1 · τ 2 ; and six phenomenological potentials associated with operators L 2 , L 2 σ 1 · σ 2 , L 2 τ 1 · τ 2 , L 2 σ 1 · σ 2 τ 1 · τ 2 , ( L · S ) 2 and ( L · S ) 2 τ 1 · τ 2 . The six “quadratic L” terms are relatively weak, and are chosen in order to make many-body calculations with this operator simpler.

Neural-network methods for boundary value problems with irregular boundaries

Partial differential equations (PDEs) with boundary conditions (Dirichlet or Neumann) defined on boundaries with simple geometry have been successfully 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 exact satisfaction of the boundary conditions. The method has been successfully tested on two-dimensional and three-dimensional PDEs and has yielded accurate results.

Variational calculations of realistic models of nuclear matter

Abstract We report variational calculations of nuclear matter with a semi-realistic Reid v12 model, and a realistic v14 model of the two-nucleon interaction operator. The v14 model fits the available nucleon-nucleon scattering data up to 425 MeV lab energy, and has relatively weak L2 and ( L · S ) 2 interactions in addition to the standard central, tensor and ( L · S ). The L2 and ( L · S ) 2 interactions are treated semiperturbatively; their contribution reduces the overbinding of nuclear matter. However, the equilibrium kF = 1.7 fm−1 and E0 = −17.5 MeV obtained with the v14 model are both higher than their empirical values kF = 1.33 fm− and E0 = −16 MeV. We assume that the difference between the calculated and empirical E(ρ) is entirely due to three-nucleon interactions (TNI). The TNI contributions are phenomenologically added to the nuclear matter energy, and their parameters are adjusted to obtain the correct equilibrium energy, density and compressibility. The required TNI contributions appear to be of reasonable magnitude.

Mixture model analysis of DNA microarray images

In this paper, we propose a new methodology for analysis of microarray images. First, a new gridding algorithm is proposed for determining the individual spots and their borders. Then, a Gaussian mixture model (GMM) approach is presented for the analysis of the individual spot images. The main advantages of the proposed methodology are modeling flexibility and adaptability to the data, which are well-known strengths of GMM. The maximum likelihood and maximum a posteriori approaches are used to estimate the GMM parameters via the expectation maximization algorithm. The proposed approach has the ability to detect and compensate for artifacts that might occur in microarray images. This is accomplished by a model-based criterion that selects the number of the mixture components. We present numerical experiments with artificial and real data where we compare the proposed approach with previous ones and existing software tools for microarray image analysis and demonstrate its advantages.

Solving differential equations with genetic programming

A novel method for solving ordinary and partial differential equations, based on grammatical evolution is presented. The method forms generations of trial solutions expressed in an analytical closed form. Several examples are worked out and in most cases the exact solution is recovered. When the solution cannot be expressed in a closed analytical form then our method produces an approximation with a controlled level of accuracy. We report results on several problems to illustrate the potential of this approach.

Artificial neural network methods in quantum mechanics

Abstract 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 Schrodinger equation for the Morse potential that has an analytically known solution, to test the accuracy of the method. We then proceed with the Schrodinger and the Dirac equations for a muonic atom, as well as with a nonlocal Schrodinger integrodifferential equation that models the n + α 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.

Merlin-3.0 A multidimensional optimization environment

We present an optimization environment for multidimensional continuous functions. Robust and powerful algorithms are used that guarantee its effectiveness. The environment offers programmability and ease of use by providing a specialized operating system and a control language that can be used to create successful optimization strategies. We report on several applications where this software has been successfully used.

VARIATIONAL CALCULATIONS OF ASYMMETRIC NUCLEAR MATTER

Abstract We report on variational calculations of the energy E ( ρ , β ) of asymmetric nuclear matter having ϱ = ϱ n + ϱ p = 0.05 to 0.35 fm −3 , and β = ( ϱ n − ϱ p / g 9 = 0 to 1. The nuclear h used in this work consists of a realistic two-nucleon interaction, called v 14 , that fits the available nucleon-nucleon scattering data up to 425 MeV, and a phenomenological three nucleon interaction adjusted to reproduce the empirical properties of symmetric nuclear matter. The variational many-body theory of symmetric nuclear matter is extended to treat matter with neutron excess. Numerical and analytic studies of the β-dependence of various contributions to the nuclear matter energy show that at ϱ −3 the β 4 terms are very small, and that the interaction energy EI(ρ, β) defined as E ( ρ , β ) − T F ( ρ , β ), where T F is the Fermi-gas energy, is well approximated by EI 0 (ϱ) + β 2 EI 2 (ρ). The calculated symmetry energy at equilibrium density is 30 MeV and it increases from 15 to 38 MeV as ϱ increases from 0.05 to 0.35 fm −3 .