ANNs-Based Method for Solving Partial Differential Equations : A Survey

Conventionally, partial differential equations (PDE) problems are solved numerically through discretization process by using finite difference approximations. The algebraic systems generated by this process are then finalized by using an iterative method. Recently, scientists invented a short cut approach, without discretization process, to solve the PDE problems, namely by using machine learning (ML). This is potential to make scientific machine learning as a new sub-field of research. Thus, given the interest in developing ML for solving PDEs, it makes an abundance of an easy-to-use methods that allows researchers to quickly set up and solve problems. In this review paper, we discussed at least three methods for solving high dimensional of PDEs, namely PyDEns, NeuroDiffEq, and Nangs, which are all based on artificial neural networks (ANNs). ANN is one of the methods under ML which proven to be a universal estimator function. Comparison of numerical results presented in solving the classical PDEs such as heat, wave, and Poisson equations, to look at the accuracy and efficiency of the methods. The results showed that the NeuroDiffEq and Nangs algorithms performed better to solve higher dimensional of PDEs than the PyDEns.

The Riemann-Lebesgue Integral of Interval-Valued Multifunctions

We study Riemann-Lebesgue integrability for interval-valued multifunctions relative to an interval-valued set multifunction. Some classic properties of the RL integral, such as monotonicity, order continuity, bounded variation, convergence are obtained. An application of interval-valued multifunctions to image processing is given for the purpose of illustration; an example is given in case of fractal image coding for image compression, and for edge detection algorithm. In these contexts, the image modelization as an interval valued multifunction is crucial since allows to take into account the presence of quantization errors (such as the so-called round-off error) in the discretization process of a real world analogue visual signal into a digital discrete one.

Dynamics of Some Discretized Fractional-Order Differential Equations

This chapter deals with fractional-order differential equations and their discretization. First of all, a discretization process for discretizing ordinary differential equations with piecewise constant arguments is presented. Secondly, a discretization method is proposed for discretizing fractional-order differential equations. Stability of fixed points of the discretized equations are investigated. Numerical simulations are carried out to show the dynamic behavior of the resulting difference equations such as bifurcation and chaos.

Solving Partial Integro-Differential Equations with Weakly Singular Kernel

Equations with a combination of integrals and derivatives are known as integro-differential equations. They are a combination of science and engineering. Many models are implemented with the help of integro-differential equations. Various techniques are available to solve integro-differential equations. In the present study, the Radial Basis Function and Adomain Decomposition Method-based numerical algorithms are used to solve a linear partial integro-differential equation with weakly singular kernel, which arises from viscoelasticity. In the discretization process, singular integrals were compared with the product trapezoidal method. Implementation of various radial basis functions was carried out. The proposed system was found to be useful and to provide reproducible results.

A Concept of the Non-Stationary Filtering Network with Reduced Transient Response

This paper presents a concept of the non-stationary filtering network with reduced transient response consisting of the first-order digital elements with time-varying parameters. The digital filter section is based on the analog system. In order to design the filtering network, the analog prototype was subjected to the discretization process. The time constant and the gain factor were then temporarily varied in time in order to suppress the transient response of the designed filtering structure. The optimization method, based on the Particle Swarm Optimization (PSO) algorithm which is aimed at reducing the settling time by a proper parameter manipulation, is presented. Simulation results proving the usefulness of the proposed concept are also shown and discussed.

Short Transient Parameter-Varying IIR Filter Based on Analog Oscillatory System

This paper presents a concept for digital infinite impulse response (IIR) lowpass filter with reduced transient response. The proposed digital filtering structure is based on an analog oscillatory system. In order to design the considered digital filter, the analog prototype is subjected to a discretization process and, then, the parameters describing the dynamical properties of the oscillatory system are temporarily varied in time, so as to suppress the transient response of the designed filter. An optimization method, aimed at reducing the settling time by proper parameter manipulation, is presented. Simulation results, along with a real-life application proving the usefulness of the proposed concept, are also shown and discussed.

Calculation Formulas and Simulation Algorithms for Entropy of Function of LR Fuzzy Intervals

Entropy has continuously arisen as one of the pivotal issues in optimization, mainly in portfolios, as an indicator of risk measurement. Aiming to simplify operations and extending applications of entropy in the field of LR fuzzy interval theory, this paper first proposes calculation formulas for the entropy of function via the inverse credibility distribution to directly calculate the entropy of linear function or simple nonlinear function of LR fuzzy intervals. Subsequently, to deal with the entropy of complicated nonlinear function, two novel simulation algorithms are separately designed by combining the uniform discretization process and the numerical integration process with the proposed calculation formulas. Compared to the existing simulation algorithms, the numerical results show that the advantage of the algorithms is well displayed in terms of stability, accuracy, and speed. On the whole, the simplified calculation formulas and the effective simulation algorithms proposed in this paper provide a powerful tool for the LR fuzzy interval theory, especially in entropy optimization.

A class of strongly stable approximation for unbounded operators

We derive new sufficient conditions to solve the spectral pollution problem by using the generalized spectrum method. This problem arises in the spectral approximation when the approximate matrix may possess eigenvalues which are unrelated to any spectral properties of the original unbounded operator. We develop the theoretical background of the generalized spectrum method as well as illustrate its effectiveness with the spectral pollution. As a numerical application, we will treat the Schr¨odinger’s operator where the discretization process based upon the Kantorovich’s projection.