Cardiac simulations are promising tools in the electrophysiological study of cardiovascular diseases related to cardiac rhythm disturbance such as ventricular fibrillation. Realistic 3D simulations of the heart tissue require applying differential equation solvers (DES) over a massive amount of data, which pose as a major barrier to make it feasible for clinical applications despite efforts on parallelization and the use of adaptive techniques. The traditional adaptive stencil techniques that work well for regular cubic mesh structures and homogeneous tissue are not suitable for the complex geometry and inhomogeneity of the heart tissue. We address this challenge with a two-level autonomous computational approach. We treat each node of the simulation mesh as an autonomous element at the lower level of autonomy. Each node in our 3D mesh structure retrieves data from the surrounding nodes, acts as an asynchronous DES, and distributes information. At the upper level of autonomy, we introduce a closed-loop autonomic self-tuning system composed of a machine learning (ML) and an optimization module. The system receives error-related information from the nodes, learns the rules based on the ML module, and tunes parameters of the time step adaptivity functions based on the optimization module. This new Finite Element Method based approach is scalable and enables an efficient asynchronous adaptive technique, which is well suited for parallelizing the computations effectively on a single Nvidia K20x GPU. We show that the proposed approach reduces the execution time by a factor that ranges between 30 and 120 (that depends on the geometry and phase of the simulation) while maintaining 99.9% accuracy with respect to the baseline GPU stencil implementation without adaptivity.