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Abstract An unsupervised machine learning method is introduced to align medical images in the context of the large deformation elasticity coupled with growth and remodeling biophysics. The technique, which stems from the principle of minimum potential energy in solid mechanics, consists of two steps: Firstly, in the predictor step, the geometric registration is achieved by minimizing a loss function composed of a dissimilarity measure and a regularizing term. Secondly, the physics of the problem, including the equilibrium equations along with growth mechanics, are enforced in a corrector step by minimizing the potential energy corresponding to a Dirichlet problem, where the predictor solution defines the boundary condition and is maintained by distance functions. The features of the new solution procedure, as well as the nature of the registration problem, are highlighted by considering several examples. In particular, registration problems containing large non-uniform deformations caused by extension, shearing, and bending of multiply-connected regions are used as benchmarks. In addition, we analyzed a benchmark biological example (registration for brain data) to showcase that the new deep learning method competes with available methods in the literature. We then applied the method to various datasets. First, we analyze the regrowth of the zebrafish embryonic fin from confocal imaging data. Next, we evaluate the quality of the solution procedure for two examples related to the brain. For one, we apply the new method for 3D image registration of longitudinal magnetic resonance images of the brain to assess cerebral atrophy, where a first-order ODE describes the volume loss mechanism. For the other, we explore cortical expansion during early fetal brain development by coupling the elastic deformation with morphogenetic growth dynamics. The method and examples show the ability of our framework to attain high-quality registration and, concurrently, solve large deformation elasticity balance equations and growth and remodeling dynamics.