SUMMARY Frequency-domain wavefield solutions corresponding to the anisotropic acoustic wave equation can be used to describe the anisotropic nature of the Earth. To solve a frequency-domain wave equation, we often need to invert the impedance matrix. This results in a dramatic increase in computational cost as the model size increases. It is even a bigger challenge for anisotropic media, where the impedance matrix is far more complex. In addition, the conventional finite-difference method produces numerical dispersion artefacts in solving acoustic wave equations for anisotropic media. To address these issues, we use the emerging paradigm of physics-informed neural networks (PINNs) to obtain wavefield solutions for an acoustic wave equation for transversely isotropic (TI) media with a vertical axis of symmetry (VTI). PINNs utilize the concept of automatic differentiation to calculate their partial derivatives, which are free of numerical dispersion artefacts. Thus, we use the wave equation as a loss function to train a neural network to provide functional solutions to the acoustic VTI form of the wave equation. Instead of predicting the pressure wavefields directly, we solve for the scattered pressure wavefields to avoid dealing with the point-source singularity. We use the spatial coordinates as input data to the network, which outputs the real and imaginary parts of the scattered wavefields and auxiliary function. After training a deep neural network, we can evaluate the wavefield at any point in space almost instantly using this trained neural network without calculating the impedance matrix inverse. We demonstrate these features on a simple 2-D anomaly model and a 2-D layered model. Additional tests on a modified 3-D Overthrust model and a 2-D model with irregular topography further validate the effectiveness of the proposed method.

中文翻译：

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使用物理信息神经网络求解频域声学 VTI 波动方程

发明内容对应于各向异性声波方程的频域波场解可用于描述地球的各向异性性质。为了求解频域波动方程，我们经常需要对阻抗矩阵求逆。随着模型大小的增加，这会导致计算成本急剧增加。对于阻抗矩阵要复杂得多的各向异性介质来说，这甚至是一个更大的挑战。此外，传统的有限差分法在求解各向异性介质的声波方程时会产生数值色散伪影。为了解决这些问题，我们使用新兴的物理信息神经网络 (PINN) 范式来获得具有垂直对称轴 (VTI) 的横向各向同性 (TI) 介质的声波方程的波场解。PINN 利用自动微分的概念来计算它们的偏导数，这些偏导数没有数值色散伪影。因此，我们使用波动方程作为损失函数来训练神经网络，为波动方程的声学 VTI 形式提供函数解。我们不是直接预测压力波场，而是求解散射压力波场以避免处理点源奇点。我们使用空间坐标作为网络的输入数据，输出散射波场的实部和虚部以及辅助函数。在训练一个深度神经网络之后，我们可以使用这个训练有素的神经网络几乎立即评估空间中任何点的波场，而无需计算阻抗矩阵逆。我们在一个简单的二维异常模型和一个二维分层模型上演示了这些特征。对修改后的 3-D Overthrust 模型和具有不规则地形的 2-D 模型的附加测试进一步验证了所提出方法的有效性。