A unified framework for geometry-independent operator learning in cardiac electrophysiology simulations

Learning neural operators on heterogeneous and irregular geometries remains a fundamental challenge, as existing approaches typically rely on structured discretisations or explicit mappings to a shared reference domain. We propose a unified framework for geometry-independent operator learning that reformulates the learning problem in an intrinsic coordinate space defined on the underlying manifold. By expressing both inputs and outputs in this shared coordinate domain, the framework decouples operator learning from mesh discretisation and geometric variability, while preserving meaningful spatial organisation and enabling faithful reconstruction on the original geometry. We demonstrate the framework on cardiac electrophysiology, a particularly challenging setting due to extreme anatomical variability across heart geometries. Leveraging a GPU-accelerated simulation pipeline, we generate large-scale datasets of high-fidelity electrophysiology simulations across diverse patient-specific anatomies and train customised neural operators to predict full-field local activation time maps. The proposed approach outperforms established neural operators on both atrial and ventricular geometries. Beyond cardiac electrophysiology, we further show that the same representation enables operator learning in cardiac biomechanics, a distinct problem involving volumetric deformation, highlighting the generality of the proposed framework. Together, these results establish intrinsic coordinate representations as a principled and extensible pathway for neural operator learning on complex physical systems characterised by heterogeneous geometry.