Abstract
Seeking efficient solutions to nonlinear boundary value problems is a crucial challenge in the mathematical modelling of many physical phenomena. A well-known example of this is solving the Biharmonic equation relating to numerous problems in fluid and solid mechanics. One must note that, in general, it is challenging to solve such boundary value problems due to the higher-order partial derivatives in the differential operators. An artificial neural network is thought to be an intelligent system that learns by example. Therefore, a well-posed mathematical problem can be solved using such a system. This paper describes a mesh free method based on a suitably crafted deep neural network architecture to solve a class of well-posed nonlinear boundary value problems. We show how a suitable deep neural network architecture can be constructed and trained to satisfy the associated differential operators and the boundary conditions of the nonlinear problem. To show the accuracy of our method, we have tested the solutions arising from our method against known solutions of selected boundary value problems, e.g., comparison of the solution of Biharmonic equation arising from our convolutional neural network subject to the chosen boundary conditions with the corresponding analytical/numerical solutions. Furthermore, we demonstrate the accuracy, efficiency, and applicability of our method by solving the well known thin plate problem and the Navier-Stokes equation.
Highlights
Artificial neural networks (ANNs) can be utilised to create suitable tools to solve large-scale computational problems [2, 37]
The challenge we address in this paper is that we propose a general ANN architecture for solving a broad range of nonlinear boundary value problems
The work we describe here is novel in that we demonstrate that an ANN-based nonlinear machine learning model is used to solve general nonlinear boundary value problems
Summary
Artificial neural networks (ANNs) can be utilised to create suitable tools to solve large-scale computational problems [2, 37]. ANNs have shown great promise in solving many challenging problems in applied mathematics and computing In this sense, several ways to solve partial differential equations (PDEs) using feedforward neural networks by substituting approximate solutions into the corresponding differential operator [22, 38] have been proposed. The challenge we address in this paper is that we propose a general ANN architecture for solving a broad range of nonlinear boundary value problems. For this purpose, we utilise a deep neural network that can learn patterns from the data [27] using trial solutions as input during the learning phase.
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