RNN-Attention Based Deep Learning for Solving Inverse Boundary Problems in Nonlinear Marshak Waves

. Radiative transfer, described by the radiative transfer equation (RTE), is one of the dominant energy exchange processes in the inertial confinement fusion (ICF) experiments. The Marshak wave problem is an important benchmark for time-dependent RTE. In this work, we present a neural network architecture termed RNN-attention deep learning (RADL) as a surrogate model to solve the inverse boundary problem of the nonlinear Marshak wave in a data-driven fashion. We train the surrogate model by numerical simulation data of the forward problem, and then solve the inverse problem by minimizing the distance between the target solution and the surrogate predicted solution concerning the boundary condition. This minimization is made efficient because the surrogate model by-passes the expensive numerical solution, and the model is differentiable so the gradient-based optimization algorithms are adopted. The effectiveness of our approach is demonstrated by solving the inverse boundary problems of the Marshak wave benchmark in two case studies: where the transport process is modeled by RTE and where it is modeled by its nonlinear diffusion approximation (DA). Last but not least, the importance of using both the RNN and the factor-attention blocks in the RADL model is illustrated, and the data efficiency of our model is investigated in this work.