A Physically Informed Neural Network for Solving the Convective Diffusion Equation in Natural Dispersed Media

Abstract

This paper examines the application of physically informed neural networks to modeling pollutant migration processes in natural dispersed media. A one-dimensional convective diffusion equation describing substance transport under the influence of advective and diffusion mechanisms was used as the basic mathematical model. A neural network architecture based on a multilayer perceptron is proposed, enabling the approximation of the solution in a continuous spatiotemporal domain without the need to construct a computational grid. The model was trained by minimizing the loss function, which includes the residual of the differential equation, as well as deviations from the initial and boundary conditions. Collocation points generated within the computational domain were used for training. Computational experiments demonstrated that the developed model correctly reproduces the fundamental physical laws of the transport process, including the shift of the maximum concentration due to advection and its smoothing due to diffusion. It is demonstrated that the use of physically informed neural networks ensures a smooth, stable, and physically consistent solution even with a limited amount of initial information. The advantages of the method are noted, which include the absence of the need for training data and the ability to work in areas of complex geometry.