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Frequency-domain simulation of seismic waves plays an important role inseismic inversion, but it remains challenging in large models. The recentlyproposed physics-informed neural network (PINN), as an effective deep learningmethod, has achieved successful applications in solving a wide range of partialdifferential equations (PDEs), and there is still room for improvement on thisfront. For example, PINN can lead to inaccurate solutions when PDE coefficientsare non-smooth and describe structurally-complex media. In this paper, we solvethe acoustic and visco-acoustic scattered-field wave equation in the frequencydomain with PINN instead of the wave equation to remove source singularity. Wefirst illustrate that non-smooth velocity models lead to inaccurate wavefieldswhen no boundary conditions are implemented in the loss function. Then, we addthe perfectly matched layer (PML) conditions in the loss function of PINN anddesign a quadratic neural network to overcome the detrimental effects ofnon-smooth models in PINN. We show that PML and quadratic neurons improve theresults as well as attenuation and discuss the reason for this improvement. Wealso illustrate that a network trained during a wavefield simulation can beused to pre-train the neural network of another wavefield simulation afterPDE-coefficient alteration and improve the convergence speed accordingly. Thispre-training strategy should find application in iterative full waveforminversion (FWI) and time-lag target-oriented imaging when the modelperturbation between two consecutive iterations or two consecutive experimentscan be small.

Wave simulation in non-smooth media by PINN with quadratic neural network and PML condition