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

arXiv: https://arxiv.org/abs/2512.01702 · HTML: https://mlsanity.com/paper/2512.01702/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 22 of its 73 references that are in the map.
- First posted on arXiv 1 Dec 2025.

## Builds on (most cited first, 22 of 22)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
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- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
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- [CvT: Introducing Convolutions to Vision Transformers](https://mlsanity.com/paper/2103.15808/) (2021)
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- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [GNOT: A General Neural Operator Transformer for Operator Learning](https://mlsanity.com/paper/2302.14376/) (2023)
- [Transolver: A Fast Transformer Solver for PDEs on General Geometries](https://mlsanity.com/paper/2402.02366/) (2024)
- [Geometry-Informed Neural Operator for Large-Scale 3D PDEs](https://mlsanity.com/paper/2309.00583/) (2023)
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- [Transformers as Neural Operators for Solutions of Differential Equations with Finite Regularity](https://mlsanity.com/paper/2405.19166/) (2024)
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- [Graph Fourier Neural Kernels (G-FuNK): Learning Solutions of Nonlinear Diffusive Parametric PDEs on Multiple Domains](https://mlsanity.com/paper/2410.04655/) (2024)
- [Diffeomorphic Latent Neural Operators for Data-Efficient Learning of Solutions to Partial Differential Equations](https://mlsanity.com/paper/2411.18014/) (2024)
- [Cardiac Digital Twins at Scale from MRI: Open Tools and Representative Models from ~55000 UK Biobank Participants](https://mlsanity.com/paper/2505.21019/) (2025)
- [Evaluation of an Open-Source Pipeline to Create Patient-Specific Left Atrial Models: A Reproducibility Study](https://mlsanity.com/paper/2301.06998/) (2023)

## Abstract (arXiv)

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.