A comparative investigation of neural networks in solving differential equations

Methods for solving differential equations based on neural networks have been widely proposed in recent years. However, limited open literature to date has reported the choice of loss functions and the hyperparameters of the network and how it influences the quality of numerical solutions. In the present work we intend to address this issue. Precisely we will focus on possible choices of loss functions and compare their efficiency in solving differential equations through a series of numerical experiments. In particular, a comparative investigation is performed between the natural neural networks and Ritz neural networks, with and without penalty for the boundary conditions. The sensitivity on the accuracy of the neural networks with respect to the size of training set, the number of nodes, and the penalty parameter is also studied. In order to better understand the training behavior of the proposed neural networks, we further investigate the approximation properties of the neural networks in function fitting. A particular attention is paid to approximating Müntz polynomials by neural networks.

Müntz Spectral Methods for the Time-Fractional Diffusion Equation

In this paper, we propose and analyze a fractional spectral method for the time-fractional diffusion equation (TFDE). The main novelty of the method is approximating the solution by using a new class of generalized fractional Jacobi polynomials (GFJPs), also known as Müntz polynomials. We construct two efficient schemes using GFJPs for TFDE: one is based on the Galerkin formulation and the other on the Petrov–Galerkin formulation. Our theoretical or numerical investigation shows that both schemes are exponentially convergent for general right-hand side functions, even though the exact solution has very limited regularity (less than {H^{1}}). More precisely, an error estimate for the Galerkin-based approach is derived to demonstrate its spectral accuracy, which is then confirmed by numerical experiments. The spectral accuracy of the Petrov–Galerkin-based approach is only verified by numerical tests without theoretical justification. Implementation details are provided for both schemes, together with a series of numerical examples to show the efficiency of the proposed methods.

A collocation method for solving the fractional calculus of variation problems

In this paper we use a family of Muntz polynomials and a computational technique based on the collocation method to solve the calculus variation problem. This approach is utilizedto reduce the solution of linear and nonlinear fractional order dierential equations to the solution of a system of algebraic equations. Thus we can obtain a good approximation evenby using a smaller of collocation points.

A time-optimal boundary controllability problem for the heat equation in a ball

The aim of this paper is to study a boundary time-optimal control problem for the heat equation in a two-dimensional ball. The main ingredient is the extension of a result concerning Müntz polynomials due to Borwein and Erdélyi that allows us to prove an observability inequality for the dynamical system's truncation to a finite number of modes. This result, combined with a well-known Lebeau–Robbiano argument used to show the null-controllability of parabolic type equations, enables us to deduce the existence, uniqueness and bang-bang properties for the boundary time-optimal control.