Reliable motion estimation and strain analysis using 3D+ time echocardiography (4DE) for localization and characterization of myocardial injury is valuable for early detection and targeted interventions. However, motion estimation is difficult due to the low-SNR that stems from the inherent image properties of 4DE, and intelligent regularization is critical for producing reliable motion estimates. In this work, we incorporated the notion of domain adaptation into a supervised neural network regularization framework. We first propose a semi-supervised Multi-Layered Perceptron (MLP) network with biomechanical constraints for learning a latent representation that is shown to have more physiologically plausible displacements. We extended this framework to include a supervised loss term on synthetic data and showed the effects of biomechanical constraints on the network's ability for domain adaptation. We validated the semi-supervised regularization method on in vivo data with implanted sonomicrometers. Finally, we showed the ability of our semi-supervised learning regularization approach to identify infarct regions using estimated regional strain maps with good agreement to manually traced infarct regions from postmortem excised hearts.