Physics informed machine learning: Seismic wave equation

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.