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. 

We present a new combined mean field control game (MFCG) problem which can be interpreted as a competitive game between collaborating groups and its solution as a Nash equilibrium between groups. Players coordinate their strategies within each group. An example is a modification of the classical trader’s problem. Groups of traders maximize their wealth. They face cost for their transactions, for their own terminal positions, and for the average holding within their group. The asset price is impacted by the trades of all agents. We propose a three-timescale reinforcement learning algorithm to approximate the solution of such MFCG problems. We test the algorithm on benchmark linear-quadratic specifications for which we provide analytic solutions.

We present a new combined mean field control game (MFCG) problem which can be interpreted as a competitive game between collaborating groups and its solution as a Nash equilibrium between groups. Players coordinate their strategies within each group. An example is a modification of the classical trader's problem. Groups of traders maximize their wealth. They face cost for their transactions, for their own terminal positions, and for the average holding within their group. The asset price is impacted by the trades of all agents. We propose a three-timescale reinforcement learning algorithm to approximate the solution of such MFCG problems. We test the algorithm on benchmark linear-quadratic specifications for which we provide analytic solutions.

Loss functions with non-isolated minima have emerged in several machine-learning problems, creating a gap between theoretical predictions and observations in practice. In this paper, we formulate a new type of local convexity condition that is suitable to describe the behavior of loss functions near non-isolated minima. We show that such a condition is general enough to encompass many existing conditions. In addition, we study the local convergence of the stochastic gradient descent (SGD) method under this mild condition by adopting the notion of stochastic stability. In the convergence analysis, we establish concentration inequalities for the iterates in SGD, which can be used to interpret the empirical observation from some practical training results.

A huge success in modern machine learning applications is attributed to the feasibility of non-convex optimization via effective optimization techniques such as the stochastic gradient descent (SGD). But at the theoretical level, it is still not clear why the SGD works for the task of non-convex optimization, especially when non-isolated local minima are present. In this paper, we present a local convergence analysis of the SGD in non-convex optimization with non-isolated minima. We propose new non-convex conditions that characterize a large class of non-convex loss functions with non-isolated minima. Under those conditions, we prove the concentration inequalities of the SGD that characterized the convergence behavior. Some numerical results confirm the existence of a deep neural network satisfying the weakest non-convex condition among the proposed ones.