A modified physics-informed neural network with positional encoding

Recently developed physics-informed neural network (PINN) for solving for the scattered wavefield in the Helmholtz equation showed large potential in seismic modeling because of its flexibility, low memory requirement, and no limitations on the shape of the solution space. However, the predicted solutions were somewhat smooth and the convergence of the training was slow. Thus, we propose a modified PINN using sinusoidal activation functions and positional encoding, aiming to accelerate the convergence and fit better. We transform the scalar input coordinate parameters using positional encoding into high-dimensional embedded vectors and train a fully-connected neural network to predict the real and imaginary parts of the scattered waveifeld. Numerical results show that, compared to the commonly used PINN, the proposed modified PINN using positional encoding exhibits notable superiority in terms of convergence and accuracy.