<p>In cardiac modelling, it is essential to accurately capture anatomical variations of the heart over timescales ranging from seconds to months, such as strain estimation and tracking the progression of cardiac dysfunction. In this study, we introduce physics-enhanced and geometry-enhanced variants of large deformation diffeomorphic metric mapping (LDDMM) framework to map a reference configuration of the heart to another state, without explicitly solving the underlying mathematical models. Two scenarios are considered: (1) passive filling of the heart within a single heartbeat, and (2) the growth and remodelling process following myocardial infarction. The first scenario has traditionally been solved using nonlinear finite element methods based on known physical laws, whereas the second remains an open research question. The physics-enhanced LDDMM incorporates a predefined strain energy function of the myocardium and the embedded myofibre structure, while the geometry-enhanced LD-DMM introduces three additional geometric metrics of the deformed configuration. Our results demonstrate that both frameworks can accurately recover biomechanical mappings during diastolic filling. Notably, the deformation tensor estimated via geometry-enhanced LDDMM is locally homogeneous, potentially reflecting more biologically realistic remodelling patterns. The estimated growth tensor accounts for at least 70% of the structural changes observed post-myocardial infarction when analyzed using the kinematic growth theory. These enhanced variant LDDMM frameworks thus offer an alternative approach for mapping cardiac dynamics—whether driven by short-term mechanical loading or long-term disease processes such as growth and remodelling—particularly in contexts where fully-established or tractable mechanistic models are unavailable.</p>