Modeling multisource multifrequency acoustic wavefields by a multiscale Fourier feature physics-informed neural network with adaptive activation functions

Recently, the physics-informed neural network (PINN) was adopted to solve partial differential equation (PDE)-based forward and inverse problems. Compared to numerical differentiation, a PINN calculates derivatives by mesh-free automatic differentiation without dispersion artifacts. The Fourier feature PINN was applied to solve the frequency-domain acoustic wave equation to model multifrequency scattered wavefields. Although solving for scattered wavefields avoids the source singularity problem, it has drawbacks (e.g., requiring an analytic formula for computing the background wavefield, which only exists for the wave equation for simple models). We evaluated an approach for modeling multisource multifrequency acoustic wavefields using a multiscale Fourier feature mapping (MFFM) PINN with adaptive activations, directly solving for full wavefields instead of scattered wavefields and naturally avoiding the drawbacks of solving the scattered wave equation. For the MFFM, we explored the determination of the maximum and number of Fourier scales. Our inputs to the MFFM were only the spatial coordinates of the subsurface model; this result is lower than that of previous work (improving the efficiency of the PINN while maintaining its accuracy). Because the activation function is extremely important for a PINN, we use an existing technique and adapt it to a new architecture and develop an adaptive amplitude-scaled and phase-shifted sine activation function, which performs the best among the studied activation functions. Experiments show that the MFFM, adaptive activation, an appropriate learning rate, a linearly shrinking NN, and transfer learning greatly improve the convergence rate, accuracy, and efficiency of the PINN for simulating multisource multifrequency wavefields, laying the foundation for applying a PINN to wave equation-based inversion and imaging. We shared our codes, data, and results via a public repository.