  Non-invasive assessment of the electrical activation pattern can
significantly contribute to the diagnosis and treatment of cardiac arrhythmias,
due to faster and safer diagnosis, improved surgical planning and easier
follow-up. One promising path is to measure the mechanical contraction via
echocardiography and utilize this as an indirect way of measuring the original
activation pattern. To solve this demanding inversion task, we make use of
physics-informed neural networks, an upcoming methodology to solve forward and
inverse physical problems governed by partial differential equations. In this
study, synthetic data sets were created, consisting of 2D excitation waves
coupled to an isotropic and linearly deforming elastic medium. We show that for
both focal and spiral patterns, the underlying excitation waves can be
reconstructed accurately. We test the robustness of the method against Gaussian
noise, reduced spatial resolution and projected tri-planar data. In situations
where the data quality is heavily reduced, we show how to improve the
reconstruction by additional regularization on the wave speed. Results on the
optimization of hyperparameters are also discussed. Our findings suggest that
physics-informed neural networks hold the potential to solve sparse and noisy
bio-mechanical inversion problems and may offer a pathway to non-invasive
assessment of certain cardiac arrhythmias.
