Time-dependent wave equations represent an important class of partial differential equations (PDE) for describing wave propagation phenomena, which are often formulated over unbounded domains. Given a compactly supported initial condition, classical numerical methods reduce such problems to bounded domains using artificial boundary condition (ABC). In this work, we present a machine-learning method to solve this equation as an alternative to ABCs. Specifically, the mapping from the initial conditions to the PDE solution is represented by a neural network, trained using wave packets that are parameterized by their band width and wave numbers. The accuracy is tested for both the second-order wave equation and the Schrodinger equation, including the nonlinear Schrodinger. We examine the accuracy from both interpolations and extrapolations. For initial conditions lying in the training set, the learned map has good interpolation accuracy, due to the approximation property of deep neural networks. The learned map also exhibits some good extrapolation accuracy. Therefore, the proposed method provides an interesting alternative for finite-time simulation of wave propagation.
翻译:取决于时间的波方程式代表了描述波波传播现象(PDE)的一个重要的局部差异方程式类别(PDE),波传播现象往往在不受约束的域域上形成。根据一个紧密支持的初步条件,古典数字方法将这类问题用人工边界条件(ABC)降低到封闭域。在这项工作中,我们提出了一个机器学习方法来解决这个方程式,作为ABC的替代。具体地说,从初始条件到PDE溶液的映射由神经网络代表,经过培训的波包使用其波宽和波数参数的波包进行参数测量。第二波方程式和Schr ⁇ odinger方程式的精确度都经过测试,包括非线性Schrodinger的精确度。我们研究了来自内插和外推的精确度。对于训练组的初始条件,由于深神经网络的近似特性,所学地图具有良好的内插精确度。因此,所学的地图还显示了一些好的外推精确性。因此,拟议的方法为波波的定时模拟模拟提供了一种有趣的替代方法。