Comprehensive patient-specific computational models continue to be developed for cardiac simulations of health and disease. However, due to the complex multiphysics and multiscale nature of cardiac biomechanical function, traditional finite element methods remain prohibitively slow for clinical applications. To meet the requirements of speed as well as accuracy, we have developed and utilized a novel neural network finite element (NNFE) approach for soft tissue simulations that can produce simulation results within clinically relevant timeframes. The NNFE approach is a physics-based approach for rapid simulations that uses the neural network (NN) to represent the nodal displacements, and finite elements to map the displacement output from the NN on the problem domain, as well as to enforce boundary conditions and perform numerical integrations. In other words, this approach does not rely on data generated from physical experiments or simulations for training, rather, the NNFE model is trained to learn the governing PDE. In this work, we present a feasibility study using an extension of the NNFE approach towards complete organ level cardiac simulations to predict the P-V loop responses of the left ventricle, accounting for active contraction and transmural fiber distributions.The NNFE model predicted the displacements and corresponding PV loop, with mean nodal error between the NNFE solution and the FE solution was 2.32x10-2 mm (Fig. 2). The trained NNFE model could accurately produce the twisting experienced by the left ventricle under active contraction. The trained NNFE model took 2-3 seconds whereas FEniCS took 10-20 min. Once trained, the NNFE approach produced results for any condition within the training range without a need for retraining. Consequentially, the NNFE approach is well-suited for training in advance, and when presented with the patient-specific data, rapid simulation results can be produced with the trained model.