Solving multiscale steady radiative transfer equation using neural networks with uniform stability

This paper concerns solving the steady radiative transfer equation with diffusive scaling, using physics informed neural networks (PINNs). The idea of PINNs is to minimize a least-square loss function, that consists residual from governing equation, mismatch boundary conditions, and other physical constraints such as conservation. It advantageous being flexible easy execute, brings potential for high dimensional problems. Nevertheless, due presence small scales, vanilla can be extremely unstable multiscale equations. In this paper, we propose new formulation based on macro-micro decomposition. We prove that, function uniformly stable respect Knudsen number in sense $$L^2$$ -error network solution controlled by loss. When condition an-isotropic, layer emerges diffusion limit therefore an additional difficulty training network. To resolve issue, include corrector carries over sharp transition part leaves rest approximated. effectiveness methodology demonstrated extensive numerical examples.