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.

💡 Research Summary

This paper addresses a fundamental obstacle in scientific machine learning: how to learn neural operators that are robust to heterogeneous, irregular geometries without relying on structured discretizations or explicit mappings to a reference domain. The authors propose a unified framework that reformulates the operator learning problem in an intrinsic coordinate space defined directly on the underlying manifold. By projecting both inputs (e.g., tissue conductivities, anisotropy ratios, global anatomical measures) and outputs (local activation time, LAT, maps) onto a shared intrinsic coordinate system—Universal Atrial Coordinates (UAC) for atria and Universal Ventricular Coordinates (UVC) for ventricles—the learning task becomes geometry‑independent. This decouples the operator from mesh topology, resolution, and patient‑specific shape while preserving physiologically meaningful spatial organization.

To validate the approach, the authors built a massive GPU‑accelerated finite‑element electrophysiology pipeline. They simulated high‑fidelity LAT maps for 147 patient‑specific left atria (100 in Cohort A, 47 in Cohort B). For each heart they generated data from seven pacing sites and sampled 300 combinations of longitudinal conductivity (σₗ) and anisotropy ratio (σₗ/σₜ) using Latin Hypercube Sampling, yielding 308,700 simulations—one of the largest publicly available cardiac EP datasets. All simulated fields were projected onto the standardized UAC domain, providing a consistent 2‑D representation across all anatomies.

The authors performed extensive analyses. Dimensionality reduction (PCA + UMAP) revealed clear clustering by pacing site and a systematic domain shift between Cohort A and B, despite identical physiological parameters, highlighting the impact of anatomical variability. Feature‑sensitivity studies showed that global surface area and the intrinsic coordinates themselves are critical for accurate LAT prediction, whereas the current atlas‑based fiber orientation did not improve performance and was therefore omitted in subsequent experiments. Spatial regularization was investigated by augmenting the H¹ loss with total variation (TV) and Laplacian penalties. The combination of both regularizers consistently yielded the lowest mean absolute error (MAE) and highest structural similarity index measure (SSIM), indicating that enforcing first‑ and second‑order smoothness jointly preserves physiologically realistic activation patterns.

Benchmarking against state‑of‑the‑art neural operators (DeepONet, Fourier Neural Operator, Wavelet Neural Operator) and convolutional architectures (U‑Net, ResNet) demonstrated the superiority of the proposed model. While DeepONet with a Vision‑Transformer backbone achieved respectable results, it was outperformed by the new architecture in both numerical accuracy (MAE ≈ 5.2 ms) and structural fidelity (SSIM ≈ 0.976). FNO and WNO struggled to capture the heterogeneous, noisy activation fields on complex manifolds. Convolutional models performed well locally but failed to capture long‑range pacing effects. Error maps showed that residual errors concentrate along the UAC seam, with low errors elsewhere, confirming high spatial fidelity across most of the atrial surface.

Generalization was further examined in two ways. First, a unified model trained jointly on all seven pacing sites achieved a 16.5 % reduction in MAE and a 1.67 % increase in SSIM compared with seven separate single‑site models, indicating that the network learns shared electrophysiological structure rather than site‑specific memorization. Second, cross‑domain experiments (training on Cohort A, testing on Cohort B and vice‑versa) demonstrated that the intrinsic coordinate representation mitigates domain shift, preserving performance despite systematic anatomical differences.

Finally, the authors illustrated the framework’s generality by applying the same intrinsic coordinate representation to a cardiac biomechanics problem involving volumetric deformation. The successful transfer underscores that any physical system admitting a smooth intrinsic chart can benefit from this geometry‑independent operator learning paradigm.

In summary, the paper delivers a comprehensive solution to geometry‑independent neural operator learning: (1) intrinsic coordinate mapping that normalizes heterogeneous anatomy, (2) joint learning of physical parameters and field outputs in this space, (3) large‑scale GPU‑driven data generation, and (4) carefully designed regularization that preserves spatial structure. The resulting model not only surpasses existing neural operators in accuracy and robustness but also achieves real‑time inference (≈0.12 ms per sample on a modern GPU), opening the door to rapid, patient‑specific cardiac electrophysiology predictions and broader applications in computational science.