Enhanced full-waveform inversion through Fourier MLP and cosine-similarity optimization: A neural tangent kernel and multiscale loss approach

Full-waveform inversion (FWI) is a powerful tool for high-resolution subsurface imaging, but its practical use is limited by cycle skipping, strong sensitivity to the initial velocity model, and high computational cost. We present a Fourier-enhanced dynamics FWI (FED-FWI) framework that combines three components within a recurrent-neural-network formulation of the forward problem. First, Fourier feature mapping is applied to the coordinate inputs of the multilayer perceptron that parameterizes the velocity model, which enriches the neural tangent kernel spectrum and improves the representation of sharp velocity contrasts. Second, a composite misfit couples an optimal-transport term, which is robust to phase shifts, with a wavelet-domain term that enforces multiscale consistency; the two terms are adaptively balanced by exponential-moving-average weighting according to their smoothed relative magnitudes, and the resulting coefficient is not required to vary monotonically with iteration. Third, an optimizer that modulates the update magnitude according to the cosine similarity between successive gradients suppresses oscillatory updates and stabilizes convergence. The framework is implemented with automatic differentiation, so the adjoint equations need not be derived manually. Starting from a linear initial model that spans 1200 to 4800 m/s, with no water layer assumed to be known, FED-FWI recovers the Marmousi II model with a structural similarity index of 0.804 and a mean squared error of 0.075, compared with 0.626 and 0.212 for optimal-transport FWI and 0.695 and 0.134 for Sobolev-norm FWI; it also attains the shortest convergence time of the methods compared while requiring less than 200 MiB of additional graphics processing unit memory. Ablation tests indicate that each component contributes to this improvement, and two salt-model experiments show that the principal salt geometry is recovered under strong velocity contrasts. The results indicate that FED-FWI improves inversion quality when initialized with the linear velocity model considered in this study.