Electrocardiography is a valuable tool to aid in medical understanding and treatment of heart-related ailments, specifically atrial fibrillation (AF) and other irregular cardiac behavior. Although signs of AF will manifest in conventional electrocardiogram (ECG) recordings, interpretation and localization of AF sources require significant clinical expertise. In this vein, electrocardiographic imaging has emerged as an important medical imaging modality that provides reconstructions of the heart's electrical activity from non-invasive multi-lead body-surface ECG and anatomical x-ray computed tomography images. In this paper, we present a nonlinear inversion model for computing this mapping to improve upon the reconstruction performance of current methods. While contemporary techniques typically determine an inverse solution by discretizing and inverting an underdetermined linear system of partial differential equations governing the relationship between voltage potentials of the heart and torso, the presented technique re-casts this problem as a task in function approximation and provides a direct parameterization of the inverse operator using a polynomial neural network. That is, the outlined nonlinear inversion technique is a generalization of contemporary reconstruction techniques which allows geometrical and material parameterizations of the forward-model to be optimized using real experimental data collected from patients suffering from AF, as to better represent the inverse operator with respect to reconstruction metrics applicable to electrophysiology. The accuracy of our model is evaluated against a dataset of real-patient recordings to demonstrate its validity, and mathematical analysis is provided to support the polynomial expansion used in our inversion model.