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.

Inertial confinement fusion (ICF) refers to the fusion energy released when a small amount of hot nuclear fuel is ignited by high-power substances to make it reach the ignition conditions under inertial confinement. The ICF implosion is characterized by the equations of radiation hydrodynamics, which are mainly composed of equations describing fluid motion, electron heat conduction, ion heat conduction, photon transport, nuclear reaction and charged particle transport. 

In practical applications, the focus of research in inertial confinement fusion (ICF) is on determining the appropriate boundary conditions that will achieve the desired temperature control of both material and radiation during the fusion process. In mathematics, this is a typical inverse problem, and traditional numerical methods require multiple solutions of differential equations, which is a highly costly task. Moreover, due to the physical conditions of ICF, acquiring the raw data can be extremely expensive. 

This article proposes a method of using RADL neural network as a surrogate model to obtain approximate solutions to inverse problems, which efficiently solves the problems of high computational cost and difficult data acquisition in traditional numerical methods.