Subspace based reconstructions can be used for quantitative imaging. We extend the recently added neural network framework in BART to support non-Cartesian trajectories and linear subspace constraint reconstructions. The network is trained to reconstruct coefficient maps from single-shot radial FLASH inversion recovery acquisitions. T1 maps are estimated using pixel-wise fitting to the signal model. The reconstruction quality of the coefficient and T1 maps are improved compared to an $$$\ell_1$$$-Wavelet regularized reconstruction. The subspace constraint network can be used for any linear subspace constraint reconstruction.