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Solving partial differential equations using neural networks is mostly a proof of concept approach. In the case of direct function approximation, a single neural network is constructed to be the solution of a particular boundary value problem. Independent variables are fed into the input layer, and a single output is considered as the solution's value. The network is substituted into the equation, and the residual is then minimized with respect to the weights of the network using a gradient-based method. Our previous work showed that by minimizing all derivatives of the residual up to the third order one can obtain a machine precise solution for 2D boundary value problem using very sparse grids. The goal of this paper is to use this grid complexity advantage in order to obtain a solution faster than finite differences. However, the number of all possible high-order derivatives (and therefore the training time) increases with the number of dimensions and it was unclear whether this goal can be achieved. Here, we demonstrate that this increase can be compensated by using random directional derivatives instead. In 2D case neural networks are slower than finite differences, but for each additional dimension the complexity increases approximately 4 times for neural networks and 125 times for finite differences. This allows neural networks to catch up in 3D case for memory complexity and in 5D case for time complexity. For the first time a machine precise solution was obtained with neural network faster than with finite differences method. [ABSTRACT FROM AUTHOR]

A nonlinear Poisson–Boltzmann equation with inhomogeneous Robin type boundary conditions at the interface between two materials is investigated. The model describes the electrostatic potential generated by a vector of ion concentrations in a periodic multiphase medium with dilute solid particles. The key issue stems from interfacial jumps, which necessitate discontinuous solutions to the problem. Based on variational techniques, we derive the homogenisation of the discontinuous problem and establish a rigorous residual error estimate up to the first-order correction. [ABSTRACT FROM PUBLISHER]

A steady-state Poisson-Nernst-Planck system is investigated, which is conformed into a nonlinear Poisson equation by means of the Boltzmann statistics. It describes the electrostatic potential generated by multiple concentrations of ions in a heterogeneous (porous) medium with diluted (solid) particles. The nonlinear elliptic problem is singularly perturbed with the Debye length as a small parameter related to the electric double layer near the solid particle boundary. For star-shaped solid particles, we prove rigorously that the solution of the problem in spatial dimensions 1d, 2d and 3d is uniformly and super-asymptotically approximated by a constant reference state. Copyright © 2015 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Abstract: In this note, we study a Modica type gradient estimate for smooth solutions to general non-linear Poisson equation where is a complete Riemannian manifold with bounded geometry and non-negative Ricci curvature and f is the derivative of the non-negative smooth function on R. Then we use this gradient estimate to conclude a Liouville theorem. [Copyright &y& Elsevier]

This paper applies radial point interpolation collocation method (RPICM) for solving nonlinear Poisson equations arising in computational chemistry and physics. Thin plate spline (TPS) Radial basis functions are used in the work. A series of test examples are numerically analysed using the present method, including 2D Liouville equation, Bratu problem and Poisson-Boltzmann equation, in order to test the accuracy and efficiency of the proposed schemes. Several aspects have been numerically investigated, namely the enforcement of additional polynomial terms; and the application of the Hermite-type interpolation which makes use of the normal gradient on Neumann boundary for the solution of PDEs with Neumann boundary conditions. Particular emphasis was on an efficient scheme, namely Hermite-type interpolation for dealing with Neumann boundary conditions. The numerical results demonstrate that a good accuracy can be obtained. The h-convergence rates are also studied for RPICM with coarse and fine discretization models. [ABSTRACT FROM AUTHOR]

Abstract: Numerical schemes for approximating the inverse of a given matrix, using an ordinary differential equation (ODE) model, were recently developed. We extend that approach to solve nonlinear minimization problems, within the framework of a preconditioned gradient method. The main idea is to develop an automatic and implicit scheme to approximate directly the preconditioned search direction at every iteration, without an a priori knowledge of the Hessian of the objective function, and involving only a reduced and controlled amount of storage and computational cost. The new scheme allows us to obtain asymptotically the Newton''s direction by improving the accuracy in the ODE solver associated with the implicit scheme. We will present extensive and encouraging numerical results on some well-known test problems, on the problem of computing the square root of a given symmetric and positive definite matrix, and also on a nonlinear Poisson type equation. [Copyright &y& Elsevier]

We study a singular perturbation problem for the nonlinear Poisson equation and we classify its solutions. This classification is based on their monotonicity properties, which mainly depend on the stationary points of the so‐called Sagdeev potential. For each class of solution, we give necessary and sufficient conditions for the resolution of the problem and provide a boundary layer analysis. [ABSTRACT FROM AUTHOR]