We propose a deep learning approach for wave propagation in media with multiscale wave speed, using a second-order linear wave equation model. We use neural networks to enhance the accuracy of a given inaccurate coarse solver, which under-resolves a class of multiscale wave media and wave fields of interest. Our approach involves generating training data by the given computationally efficient coarse solver and another sufficiently accurate solver, applied to a class of wave media (described by their wave speed profiles) and initial wave fields. We find that the trained neural networks can approximate the nonlinear dependence of the propagation on the wave speed as long as the causality is appropriately sampled in training data. We combine the neural-network-enhanced coarse solver with the parareal algorithm and demonstrate that the coupled approach improves the stability of parareal algorithms for wave propagation and improves the accuracy of the enhanced coarse solvers.
翻译:我们提出一种深层次的学习方法,用于在多波速的媒体中以多波速传播波浪,使用二阶线性波等方程式。我们使用神经网络来提高某一不准确的粗粗求解器的准确性,该求解器在解答一个多波级媒体和引起兴趣的波流领域时不足。我们的方法包括由特定的计算效率高的粗粗求解器和另一个足够准确的求解器生成培训数据,该方法应用于一个波级媒体(由波速剖面描述)和最初的波场。我们发现,经过训练的神经网络只要在培训数据中适当采样了因果关系,就可以接近波浪速传播的非线性依赖性。我们把神经网络增强的粗粗求解器与准真实的算法结合起来,并表明这种结合方法可以提高波传播的准实际算法的稳定性,并提高强化粗粗求解器的准确性。