A high resolution Physics-informed neural networks for high-dimensional convection–diffusion–reaction equations

Abstract

In practical problems, some partial differential equations defined in high-dimensional domains or complex surfaces are difficult to calculate by traditional methods. In this paper, a novel data-driven deep learning algorithm is proposed to solve high-dimensional convection–diffusion–reaction equations. The main idea of the method is to use the neural network which combines the physical characteristics of the equation to get high accuracy numerical solution. The proposed method not only avoids the high cost of mesh generation, but also effectively reduces the numerical oscillation caused by the domination of the convection. In addition, two types of loss functions are designed to force physical properties, such as the positivity or maximum principle of the solution. Various numerical examples are performed to demonstrate the validity and accuracy of the proposed method.