Robust data driven discovery of a seismic wave equation

Despite the fact that our physical observations can often be described by derived physical laws, such as the wave equation, in many cases, we observe data that do not match the laws or have not been described physically yet. Therefore recently, a branch of machine learning has been devoted to the discovery of physical laws from data. We test this approach for discovering the wave equation from the observed spatial-temporal wavefields. The algorithm first pre-trains a neural network (NN) in a supervised fashion to establish the mapping between the spatial-temporal locations (x, y, z, t) and the observation displacement wavefield function u(x, y, z, t). The trained NN serves to generate metadata and provide the time and spatial derivatives of the wavefield (e.g. utt and uxx) by automatic differentiation. Then, a preliminary library of potential terms for the wave equation is optimized from an overcomplete library by using a genetic algorithm. We, then, use a physics-informed information criterion to evaluate the precision and parsimony of potential equations in the preliminary library and determine the best structure of the wave equation. Finally, we train the 'physics-informed' neural network to identify the corresponding coefficients of each functional term. Examples in discovering the 2-D acoustic wave equation validate the feasibility and effectiveness of our implementation. We also verify the robustness of this method by testing it on noisy and sparsely acquired wavefield data.