Multi-frequency wavefield modeling of acoustic VTI wave equation using physics informed neural networks

Incorporating anisotropy is crucial for accurately modeling seismic wave propagation. However, numerical solutions are susceptible to dispersion artifacts, and they often require considerable computational resources. Moreover, their accuracy is dependent on the size of discretization, which is a function of the operating frequency. Physics informed neural networks (PINNs) have demonstrated the potential to tackle long-standing challenges in seismic modeling and inversion, addressing the associated computational bottleneck and numerical dispersion artifacts. Despite progress, PINNs exhibit spectral bias, resulting in a stronger capability to learn low-frequency features over high-frequency ones. This paper proposes the use of a simple fully-connected PINN model, and evaluates its potential to interpolate and extrapolate scattered wavefields that correspond to the acoustic VTI wave equation across multiple frequencies. The issue of spectral bias is tackled by incorporating the Kronecker neural network architecture with composite activation function formed using the inverse tangent (atan), exponential linear unit (elu), locally adaptive sine (l-sin), and locally adaptive cosine (l-cos) activation functions. This allows the construction of an effectively wider neural network with a minimal increase in the number of trainable parameters. The proposed scheme keeps the network size fixed for multiple frequencies and does not require repeated training at each frequency. Numerical results demonstrate the efficacy of the proposed approach in fast and accurate, anisotropic multi-frequency wavefield modeling.