International Journal of Computational Methods

Artificial neural networks (NNs) with various architectures are widely used for practical problems, including multilayer perceptron (MLP), the Convolutional Neural Networks (CNNs), Recurrent Neural Networks (RNNs), among others. The TrumpetNets and TubeNets were also recently proposed for creating two-way deepnets for both forward and inverse problems. All these studies have demonstrated the power of NNs, but the theoretical foundation needs a more thorough study. This paper first conducts an intensive study on affine transformations and its role as a prediction function. This reveals a data-to-parameter (or $x-w$) encoding mechanism and its uniqueness with proof. Then, effects of the architecture and the roles of nonlinear activation functions wrapping the affine transformations are investigated. The effects of stacking of affine transformations and chaining of the wrapped affine transformations in an NN are examined, leading finally to a novel Universal Prediction Theory (UPT).

In this paper, polynomial-based projection type and modified projection-type methods for approximating the solution of Hammerstein integral equations with a kernel of Green’s function type are proposed. The projection is either an orthogonal projection or an interpolatory projection using Legendre polynomial basis. The orders of convergence of these methods and those of superconvergence of the iterated modified projection-type methods are analyzed. A numerical example is given to illustrate the theoretical estimates.

A key difficulty of the conventional unstaggered central schemes for the shallow water equations (SWEs) is the well-balanced property that may be missed when the computational domain contains wet-dry fronts. To avoid the numerical difficulty caused by the nonconservative product, we construct a linear piecewise continuous bottom topography. We propose a new discretization of the source term on the staggered cells, and a novel “backward” step based on the water surface elevation. The core of this paper is that, we construct a map between the water surface elevation and the cell average of the free surface on the staggered cells to discretize the source term for maintaining the stationary solutions. The positivity-preserving property is obtained by using the “draining” time-step technique. A number of classical problems of the SWEs can be solved reasonably.

T lymphocytes have a primary role in both health and disease. Extracellular and intracellular signals determine whether a T-cell activates different cells, divides, or begins apoptosis. The reaction–diffusion process of Ca²⁺ ions is critical for the initiation, sustenance, and termination of the immunological function of T cell. A nonlinear spatio-temporal dynamics of Ca²⁺ in T cells is modeled incorporating parameters Sarco/endoplasmic reticulum Ca²⁺-ATPase (SERCA) pump, Ryanodine receptor, source amplitude, and buffers. A numerical meshless approach using multiquadric radial basis functions (MQRBF), differential quadrature, and Runge–Kutta method is developed for the solution. The results obtained here give better insights of calcium dynamics in T cells.

In molecular solvation, the size-modified electrostatic free energy is investigated. With the uniform ionic and solvent molecular sizes, the generalized Poisson–Boltzmann (PB) equation is considered. The first variation of the size-modified electrostatic free energy with respect to the location variation of the interface is derived. The concept of shape derivative is used to define such variations. The explicit formula of the dielectric boundary force is derived.

To meet the needs of industrial production, an improved evolution structural optimization (ESO) method with high efficiency is proposed. The optimized design variables with intermediate density were designed using the windowed evolution structural optimization (WESO) method to increase the stability of the algorithm. The efficient calculation method of the element node sensitivity was established, which realizes the establishment of level set functions, smooth topological design of structures and the updating of design variables. The stability of the proposed algorithm was verified by the Zhou–Rozvany problem, two- and three-dimensional (3D) numerical results. The effectiveness and efficiency of the proposed algorithm was further verified by numerical comparison with other topology optimization frameworks. Lastly, the improved windowed ESO method was applied to the initial configuration design of the double-deck bridge structure, which not only provides guidance for its initial design but also demonstrates the applicability of the method in complex structural systems.