Error estimates of asymptotic-preserving neural networks in approximating stochastic linearized Boltzmann equation

In this paper, we construct an asymptotic-preserving neural network (APNN) for the linearized Boltzmann equation with uncertain parameters. Utilizing the micro-macro decomposition, we design the loss function based on the stochastic-Galerkin system conducted from the micro-macro equations. Rigorous analysis is provided to show the capability of the neural network in approximating solutions near the global Maxwellian. By employing hypocoercivity techniques, we demonstrate two key results: (i) the existence of APNN leading to arbitrarily small loss function, and (ii) the convergence of the APNN's approximated solution as the loss tends to zero, with the error exhibiting an exponential decay in time.