  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.
