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Abstract

In this paper, we integrate neural networks and Gaussian wave packets to numerically solve the Schr\"odinger equation with a smooth potential near the semi-classical limit. Our focus is not only on accurately obtaining solutions when the non-dimensional Planck's constant, $\varepsilon$, is small, but also on constructing an operator that maps initial values to solutions for the Schr\"odinger equation with multiscale properties. Using Gaussian wave packets framework, we first reformulate the Schr\"odinger equation as a system of ordinary differential equations. For a single initial condition, we solve the resulting system using PINNs or MscaleDNNs. Numerical simulations indicate that MscaleDNNs outperform PINNs, improving accuracy by one to two orders of magnitude. When dealing with a set of initial conditions, we adopt an operator-learning approach, such as physics-informed DeepONets. Numerical examples validate the effectiveness of physics-informed DeepONets with Gaussian wave packets in accurately mapping initial conditions to solutions.