We propose a dictionary-matching-free pipeline for multi-parametric quantitative MRI image computing. Our approach has two stages based on compressed sensing reconstruction and deep learned quantitative inference. The reconstruction phase is convex and incorporates efficient spatiotemporal regularisations within an accelerated iterative shrinkage algorithm. This minimises the under-sampling (aliasing) artefacts from aggressively short scan times. The learned quantitative inference phase is purely trained on physical simulations (Bloch equations) that are flexible for producing rich training samples. We propose a deep and compact encoder-decoder network with residual blocks in order to embed Bloch manifold projections through multi-scale piecewise affine approximations, and to replace the non-scalable dictionary-matching baseline. Tested on a number of datasets we demonstrate effectiveness of the proposed scheme for recovering accurate and consistent quantitative information from novel and aggressively subsampled 2D/3D quantitative MRI acquisition protocols.