In this contribution, the Helmholtz decomposition of a velocity field into vortical and compressible structures is implemented using a finite element framework and physics-informed neural networks. These two implementations of Helmholtz’s decomposition are compared for a verification example and a 2D mixing layer flow. The work shows how neural networks can leverage physical knowledge to perform the inverse task of post-processing a compressible flow field into subparts. Furthermore, different input variables, network setups, network parameters, network types, and formulation of the objective function for the optimizer are investigated and compared to each other. The physics-informed neural network formulation results on the verification example outline promising directions for further applications to post-process compressible flow fields.