  Reducing the computational time required by high-fidelity, full order models
(FOMs) for the solution of problems in cardiac mechanics is crucial to allow
the translation of patient-specific simulations into clinical practice. While
FOMs, such as those based on the finite element method, provide valuable
information of the cardiac mechanical function, up to hundreds of thousands
degrees of freedom may be needed to obtain accurate numerical results. As a
matter of fact, simulating even just a few heartbeats can require hours to days
of CPU time even on powerful supercomputers. In addition, cardiac models depend
on a set of input parameters that we could let vary in order to explore
multiple virtual scenarios. To compute reliable solutions at a greatly reduced
computational cost, we rely on a reduced basis method empowered with a new
deep-learning based operator approximation, which we refer to as Deep-HyROMnet
technique. Our strategy combines a projection-based POD-Galerkin method with
deep neural networks for the approximation of (reduced) nonlinear operators,
overcoming the typical computational bottleneck associated with standard
hyper-reduction techniques. This method is shown to provide reliable
approximations to cardiac mechanics problems outperforming classical
projection-based ROMs in terms of computational speed-up of orders of
magnitude, and enhancing forward uncertainty quantification analysis otherwise
unaffordable.
