D-SWE: Data-driven discovery of a seismic wave equation

We propose a data-driven method for the wave equation D-SWE, which includes machine learning and genetic algorithm (GA), capable of discovering a wave equation from the observed spatial-temporal wavefields. D-SWE first trains a neural network (NN) in a supervised fashion to establish the mapping between the spatial-temporal locations (x, y, z, t) and observation displacement wavefield function u(x, y, z, t). The trained NN serves to generate meta-data and provide the time and spatial derivatives of the wavefield (e.g., utt and uxx) by automatic differentiation. Then, the overcomplete library, including candidate function terms, is produced by using GA. Finally, to choose the proper candidate terms and determine the accurate form of a seismic wave equation, we use a physics-informed information criterion to evaluate precision and parsimony of potential equations and determine the best structure of a wave equation, and we further train a physics-informed neural network to identify the corresponding coefficients of each functional term. Examples in discovering the 2D acoustic wave equation are used to validate the feasibility and effectiveness of the proposed method. Results demonstrate that D-SWE enables the identification of the general form of the acoustic wave equation, also, is robust to noisy and sparse wavefield data.