来自机器学习函数的波场解决方案
Wavefield solutions from machine learned functions
摘要：
当我们试图用记录的地震资料来解释地球时，求解波动方程式是我们面临的最基本的问题之一。亥姆霍兹方程提供了与时域相比每个频率在维度上减小的波场解，这对于许多应用（如全波形反演（FWI））是有用的。然而，我们获得这种波场解的能力往往取决于模型的大小和波动方程式的复杂性。因此，我们在这里使用一个最近推出的框架，基于神经网络预测功能的解决方案，通过设置底层的物理方程作为一个损失函数，以优化神经网络参数。对于由模型空间中的位置给出的输入，网络学习预测该位置处的波场值，以及使用称为自动微分的概念来拟合其偏导数，在我们的情况下，拟合亥姆霍兹方程的形式。我们特别寻求散射波场的解决方案，考虑一个简单的均匀背景模型，允许背景波场的解析解。为神经网络（NN）提供来自模型空间的合理数量的随机点将最终训练完全连接的深度NN以预测散射波场函数。网络的大小主要取决于所需波场的复杂性，这种复杂性随着频率的增加和模型复杂性的增加而增加。然而，较小的网络可以提供更平滑的波场，这可能对反演应用有用。两个盒子形的散射体模型与源在中间，以及，Marmousi模型与表面上的源的初步测试证明了这种应用的NN的潜力。在3D模型上的附加测试证明了该方法的潜在通用性。
英文摘要：
Solving the wave equation is one of the most (if not the most) fundamental problems we face as we try to illuminate the Earth using recorded seismic data. The Helmholtz equation provides wavefield solutions that are dimensionally reduced, per frequency, compared to the time domain, which is useful for many applications, like full waveform inversion (FWI). However, our ability to attain such wavefield solutions depends often on the size of the model and the complexity of the wave equation. Thus, we use here a recently introduced framework based on neural networks to predict functional solutions through setting the underlying physical equation as a loss function to optimize the neural network parameters. For an input given by a location in the model space, the network learns to predict the wavefield value at that location, and its partial derivatives using a concept referred to as automatic differentiation, to fit, in our case, a form of the Helmholtz equation. We specifically seek the solution of the scattered wavefield considering a simple homogeneous background model that allows for analytical solutions of the background wavefield. Providing the neural network (NN) a reasonable number of random points from the model space will ultimately train a fully connected deep NN to predict the scattered wavefield function. The size of the network depends mainly on the complexity of the desired wavefield, with such complexity increasing with increasing frequency and increasing model complexity. However, smaller networks can provide smoother wavefields that might be useful for inversion applications. Preliminary tests on a two-box-shaped scatterer model with a source in the middle, as well as, the Marmousi model with a source on the surface demonstrate the potential of the NN for this application. Additional tests on a 3D model demonstrate the potential versatility of the approach.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
