A multivariate regression model of affine and diffeomorphic transformation sequences—FineMorphs—is presented. Leveraging concepts from shape analysis, model states are optimally "reshaped" by diffeomorphisms generated by smooth vector fields during learning. Affine transformations and vector fields are optimized within an optimal control setting, and the model can naturally reduce (or increase) dimensionality and adapt to large data sets via sub-optimal vector fields. An existence proof of solution and necessary conditions for optimality for the model are derived. Experimental results on real data sets from the UCI repository are presented, with favorable results in comparison with state-of-the-art in the literature, neural ordinary differential equation models, and densely-connected neural networks in TensorFlow.