利用物理信息神经网络求解声波VTI波动方程
Solving the acoustic VTI wave equation using physics-informed neural networks
摘要：
对应于各向异性声波方程的频域波场解可以用来描述地球的各向异性性质。为了求解频域波动方程，我们经常需要求阻抗矩阵的逆。这导致随着模型大小的增加，计算成本急剧增加。对于阻抗矩阵复杂得多的各向异性介质，这甚至是一个更大的挑战。为了解决这个问题，我们使用物理信息神经网络（PINNs）的新兴范式，以获得波场解的声波方程的横向各向同性（TI）介质的垂直对称轴（VTI）。PINN利用自动微分的概念来计算其偏导数。因此，我们使用波动方程作为损失函数来训练神经网络，以提供函数解来形成声波VTI波动方程。我们不直接预测压力波场，而是求解散射压力波场，以避免处理点源奇异性。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。在训练了深度神经网络（NN）之后，我们可以使用这个训练好的NN立即评估空间中任何点的波场。我们展示了一个简单的异常模型和分层模型的这些功能。在一个改进的三维逆掩断层模型和一个不规则地形模型上的试验也表明了该方法的有效性。
英文摘要：
Frequency-domain wavefield solutions corresponding to the anisotropic acoustic wave equations 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. To address this issue, 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 its partial derivatives. Thus, we use the wave equation as a loss function to train a neural network to provide functional solutions to form of the acoustic VTI 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 (NN), we can evaluate the wavefield at any point in space instantly using this trained NN. We demonstrate these features on a simple anomaly model and a layered model. Additional tests on a modified 3D Overthrust model and a model with irregular topography also show the effectiveness of the proposed method.
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