Accurate subsurface velocity models are essential for seismic imaging, yet conventional full waveform inversion (FWI) often suffers from cycle skipping, noise sensitivity, and reliance on good initial models. We develop a physics-guided Fourier representation neural network (PGFRNN) for unsupervised acoustic FWI and simultaneous-source FWI (SSFWI), which embeds Fourier-transformed seismic data into a latent space and iteratively updates the velocity model using a softplus-approximated log-cosh (SALC) loss and a physics-guided optimizer. Numerical tests on the Overthrust model demonstrate that PGFRNN outperforms conventional L2- and SALC-loss-based FWI methods, achieving higher inversion accuracy and robustness to noise and challenging initial models.