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In this work, we propose a learning method for solving the linear transportequation under the diffusive scaling. Due to the multiscale nature of our modelequation, the model is challenging to solve by using conventional methods. Weemploy the physical informed neural network (PINN) framework, a mesh-freelearning method that can numerically solve partial differential equations.Compared to conventional methods (such as finite difference or finite elementtype), our proposed learning method is able to obtain the solution at any givenpoints in the chosen domain accurately and efficiently, which enables us tobetter understand the physics underlying the model. In our framework, thesolution is approximated by a neural network that satisfies both the governingequation and other constraints. The network is then trained with a combinationof different loss terms. Using the approximation theory and energy estimatesfor kinetic models, we prove theoretically that the total loss vanishes as theneural network converges, upon which the neural network approximated solutionconverges pointwisely to the analytic solution of the linear transport model.Numerical experiments for two benchmark examples are conducted to verify theeffectiveness and accuracy of our proposed method.

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