标题： 基于深度学习的多相计算半经典极限的矩闭方法 显示英文标题

标题： Deep learning-based moment closure for multi-phase computation of semiclassical limit of the Schrödinger equation

摘要： 我们提出了一种深度学习方法，用于计算薛定谔方程半经典极限的多相解。 传统方法需要推导一个多相假设来闭合Liouville方程的矩系统，这一过程通常计算量大且不切实际。 我们的方法通过引入一种新颖的两阶段神经网络框架来闭合$2N\times 2N$矩系统，其中$N$表示解假设中的相数。 在第一阶段，我们训练神经网络以学习高阶矩与低阶矩（以及它们的导数）之间的映射关系。 第二阶段结合了物理信息神经网络（PINNs），我们将学习到的高阶矩代入以系统地闭合该系统。 我们提供了对损失函数和神经网络近似收敛性的理论保证。 数值实验表明，我们的方法在具有各种相数$N$的一维和二维问题中是有效的。 结果证明了所提出方法相对于传统技术的准确性和计算效率。 显示英文摘要

摘要： 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.