Solving initial-boundary value problems for systems of partial differential equations using neural networks and optimization techniques

A Dual-Dimer method for training physics-constrained neural networks with minimax architecture

“…It was shown that neural networks such as multi-layer perceptron (MLP) and radial basis function (RBF) neural networks can solve ODEs and PDEs with higher accuracy and lower memory requirement than traditional numerical methods (Shirvany, Hayati, & Moradian, 2009). The prior knowledge of initial and boundary conditions can be incorporated in the trial solutions to improve the training efficiency of neural networks (I. E. Lagaris, Likas, & Fotiadis, 1998;Shekari Beidokhti & Malek, 2009). However, it may be difficult to find trial solutions for boundary value problems which are defined on irregular boundaries.…”

A discretization-free deep neural network-based approach for advection-dispersion-reaction mechanisms

“…Especially for advection-dominated cases, the numerical solutions can develop oscillations or numerical dispersion that cause increased computational costs [14,15]. Even if there are several analytical methods for accomplishing this, they are restricted to particular unique types to reduce errors [16][17][18][19]. Additionally, several numerical techniques such as the boundary element method (BEM), the finite volume method (FVM), the finite difference method (FDM), and the finite element method (FEM) are designed to address constraints associated with the model structure [20][21][22].…”

Unsupervised neural networks for solving Troesch's problem

“…It is well known that such solvers, based on artificial neural network (ANN) mathematical modeling hybridized with global and local search methodologies, have a universal capability to solve a variety of problems associated with linear and non-linear differential equations. [20][21][22][23][24] For example, these solvers provide viable solutions for nonlinear Van der Pol oscillators, [25,26] nonlinear singular systems associated with the Emden-Fowler type equations, [27] the moveable singularity problem of Painlevé equation I, [28] nonlinear Schrodinger equations, [29] fluid flow problems, [30] nonlinear Bratu type equations arising in the fuel ignition model of combustion theory [31,32] and a transformed form of Troesch's boundary value problem. [33] A good source of reference in this regard is a survey article, [34] which describes the history, recent trend and future prospects of research in this field.…”