A variational neural network for the reconstruction of compressed sensing 4D flow MRI is presented. Nine iterations of an iterative reconstruction are unfolded in a neural network which was trained using eight retrospectively undersampled datasets. A phase-invariant network architecture was designed with two types of filter operations, one with equal real and imaginary component and the other operating on image magnitude only. The method is shown to outperform spatial regularization in the Wavelet domain. A retrospectively undersampled patient scan demonstrates that the network can reconstruct pathologies based on healthy training samples. Reconstruction of prospectively undersampled 4D flow MRI shows good agreement of peak velocities and peak flow.