A deep neural network approach on solving the linear transport model under diffusive scaling

Abstract: In this work, we propose a learning method for solving the linear transport equation under the diffusive scaling. Due to the multiscale nature of our model equation, the model is challenging to solve by using conventional methods. We employ the physical informed neural network (PINN) framework, a mesh-free learning method that can numerically solve partial differential equations. Compared to conventional methods (such as finite difference or finite element type), our proposed learning method is able to obtain the solution at any given points in the chosen domain accurately and efficiently, which enables us to better understand the physics underlying the model. In our framework, the solution is approximated by a neural network that satisfies both the governing equation and other constraints. The network is then trained with a combination of different loss terms. Using the approximation theory and energy estimates for kinetic models, we prove theoretically that the total loss vanishes as the neural network converges, upon which the neural network approximated solution converges pointwisely to the analytic solution of the linear transport model. Numerical experiments for two benchmark examples are conducted to verify the effectiveness and accuracy of our proposed method.