Deep learning-based moment closure for multi-phase computation of semiclassical limit of the Schr\"odinger equation

We present a deep learning approach for computing multi-phase solutions to the semiclassical limit of the Schr\"odinger equation. Traditional methods require deriving a multi-phase ansatz to close the moment system of the Liouville equation, a process that is often computationally intensive and impractical. Our method offers an efficient alternative by introducing a novel two-stage neural network framework to close the $2N\times 2N$ moment system, where $N$ represents the number of phases in the solution ansatz. In the first stage, we train neural networks to learn the mapping between higher-order moments and lower-order moments (along with their derivatives). The second stage incorporates physics-informed neural networks (PINNs), where we substitute the learned higher-order moments to systematically close the system. We provide theoretical guarantees for the convergence of both the loss functions and the neural network approximations. Numerical experiments demonstrate the effectiveness of our method for one- and two-dimensional problems with various phase numbers $N$ in the multi-phase solutions. The results confirm the accuracy and computational efficiency of the proposed approach compared to conventional techniques.