Similar to many fields of sciences, recent deep learning advances have been applied extensively in geosciences for both small- and large-scale problems. However, the necessity of using large training data and the ‘black box’ nature of learning have limited them in practice and difficult to interpret. Furthermore, including the governing equations and physical facts in such methods is also another challenge, which entails either ignoring the physics or simplifying them using unrealistic data. To address such issues, physics informed machine learning methods have been developed which can integrate the governing physics law into the learning process. In this work, a 1-dimensional (1D) time-dependent seismic wave equation is considered and solved using two methods, namely Gaussian process (GP) and physics informed neural networks. We show that these meshless methods are trained by smaller amount of data and can predict the solution of the equation with even high accuracy. They are also capable of inverting any parameter involved in the governing equation such as wave velocity in our case. Results show that the GP can predict the solution of the seismic wave equation with a lower level of error, while our developed neural network is more accurate for velocity (P- and S-wave) and density inversion.

与许多科学领域类似，最近的深度学习进展已在地球科学中广泛应用于小型和大型问题。但是，使用大量训练数据的必要性和学习的“黑匣子”性质限制了它们的实际应用，并且难以解释。此外，在这种方法中包括控制方程和物理事实也是另一个挑战，这需要要么忽略物理学，要么使用不切实际的数据来简化它们。为了解决这些问题，已经开发了基于物理的机器学习方法，该方法可以将控制物理定律整合到学习过程中。在这项工作中，考虑并使用高斯过程（GP）和物理信息神经网络这两种方法来求解一维（1D）时变地震波方程。我们表明，这些无网格方法通过较少的数据量进行训练，甚至可以以较高的精度预测方程的解。它们还能够反转控制方程中涉及的任何参数，例如波速。结果表明，GP可以以较低的误差水平预测地震波方程的解，而我们开发的神经网络对于速度（P波和S波）和密度反演更准确。