Proton radiography is a technique extensively used to resolve magnetic field structures in high-energy-density plasmas, revealing a whole variety of interesting phenomena such as magnetic reconnection and collisionless shocks found in astrophysical systems. Existing methods of analyzing proton radiographs give mostly qualitative results or specific quantitative parameters, such as magnetic field strength, and recent work showed that the line-integrated transverse magnetic field can be reconstructed in specific regimes where many simplifying assumptions were needed. Using artificial neural networks, we demonstrate for the first time 3D reconstruction of magnetic fields in the nonlinear regime, an improvement over existing methods, which reconstruct only in 2D and in the linear regime. A proof of concept is presented here, with mean reconstruction errors of less than 5% even after introducing noise. We demonstrate that over the long term, this approach is more computationally efficient compared to other techniques. We also highlight the need for proton tomography because (i) certain field structures cannot be reconstructed from a single radiograph and (ii) errors can be further reduced when reconstruction is performed on radiographs generated by proton beams fired in different directions.