Physics informed neural networks with variable Eddington factor iteration for linear radiative transfer equations

Abstract

In this paper, a Physics Informed Neural Networks (PINNs) method based on Variable Eddington Factor (VEF) acceleration iteration is proposed to address the time-dependent linear radiative transfer equations (LRTEs), which exhibit the characteristics of multi-scale and high dimensionality. Firstly, the factors relating to the failure of the vanilla PINNs in solving LRTEs within the diffusion regime are analyzed by the Neural Tangent Kernel (NTK) theory. Subsequently, the VEF-PINNs method is established, where PINNs are employed to handle the radiative transfer equations and the analytic VEF equations that are used to accelerate the iteration process. It is demonstrated that as the Knudsen number ε approaches 0, the VEF-PINNs method converges to the iteration of diffusion limit equations, thereby ensuring the proposed method maintains the asymptotic preserving property. A theoretical analysis about the approximation errors of the iterative solution of the VEF-PINNs method is given. To evaluate the performance of the method, comparisons are made with the vanilla PINNs and Asymptotic Preserving Neural Networks (APNNs) based on micro-macro decomposition. The results reveal that the proposed VEF-PINNs can effectively solve LRTEs in various opacity regimes and can enhance the solving efficiency to a certain extent.