On the Hybrid Approach for Cardiac Electrophysiology Modeling Utilizing Spectral Neural Operators
DOI:
https://doi.org/10.32603/2071-2340-2026-1-40-56Keywords:
bidomain model, cardiac electrophysiology, neural operators, spectral methodsAbstract
Purpose. To investigate the applicability of a hybrid approach combining spectral neural operators with classical numerical methods for accelerated cardiac electrophysiology simulation in the bidomain formulation.
Materials and methods. The bidomain model is considered on rectangular anisotropic 3D domains with the ten Tusscher-Panfilov ionic model. A hybrid scheme is investigated: an autoregressive Fourier neural operator (AR-FNO) approximates the nonlinear parabolic evolution of the transmembrane potential, while the elliptic coupling equation is solved by the conjugate gradient method. Gradient-aware training is employed to improve wavefront reproduction accuracy.
Results. On test 3D anisotropic slabs with a 2 ms time step, a conduction velocity error of 3–6 % relative to the reference finite element solution was obtained. An ablation study of individual method components was performed. Limitations were identified: error accumulation during prolonged autoregressive rollout and accuracy dependence on time step size.
Conclusion. The principal feasibility of applying hybrid neural network architectures for computational electrophysiology problems on model domains is demonstrated. The applicability boundaries and directions for further research are identified.
References
N. A. Trayanova, "Whole-heart modeling: applications to cardiac electrophysiology and electromechanics," Circ. Res., vol. 108, no. 1, pp. 113-128, 2011; doi:10.1161/CIRCRESAHA.110.223610 DOI: https://doi.org/10.1161/CIRCRESAHA.110.223610
S. A. Niederer, J. Lumens, and N. A. Trayanova, "Computational models in cardiology," Nat. Rev. Cardiol., vol. 16, no. 2, pp. 100-111, 2019; doi:10.1038/s41569-018-0104-y DOI: https://doi.org/10.1038/s41569-018-0104-y
C. S. Henriquez, "Simulating the electrical behavior of cardiac tissue using the bidomain model," Crit. Rev. Biomed. Eng., vol. 21, no. 1, pp. 1-77, 1993.
E. Y. Shchetinin and A. V. Pestryakova, "On modeling electrocardiograms with correlated leads using deep learning methods," Soft Meas. Comput., vol. 10, no. 95, pp. 74-87, 2025 (in Russian); doi:10.36871/2618-9976.2025.10.006 DOI: https://doi.org/10.36871/2618-9976.2025.10.006
E. Y. Shchetinin, D. V. Moiseev, and A. V. Pestryakova, "Automatic detection of cardiac arrhythmias based on a hybrid model with generative augmentation," Soft Meas. Comput., vol. 11, no. 96, pp. 157-169, 2025 (in Russian); doi:10.36871/2618-9976.2025.11.013 DOI: https://doi.org/10.36871/2618-9976.2025.11.013
B. J. Roth, "Bidomain modeling of electrical and mechanical properties of cardiac tissue," Biophys. Rev., vol. 2, no. 4, p. 041301, 2021; doi:10.1063/5.0059358 DOI: https://doi.org/10.1063/5.0059358
G. Plank et al., "The openCARP simulation environment for cardiac electrophysiology," Comput. Methods Programs Biomed., vol. 208, p. 106223, 2021; doi:10.1016/j.cmpb.2021.106223 DOI: https://doi.org/10.1016/j.cmpb.2021.106223
Z. Li et al., "Fourier neural operator for parametric partial differential equations," in Proc. 9th Int. Conf. Learn. Represent. (ICLR), Appleton, WI, USA, 2021.
N. Kovachki et al., "Neural operator: learning maps between function spaces," J. Mach. Learn. Res., vol. 24, no. 89, pp. 1-97, 2023.
N. Rahaman et al., "On the spectral bias of neural networks," in Proc. 36th Int. Conf. Mach. Learn. (ICML). — PMLR, 2019, vol. 97, pp. 5301-5310.
F. Sahli Costabal et al., "Physics-informed neural networks for cardiac activation mapping," Front. Phys., vol. 8, p. 42, 2020; doi:10.3389/fphy.2020.00042 DOI: https://doi.org/10.3389/fphy.2020.00042
G. E. Karniadakis et al., "Physics-informed machine learning," Nat. Rev. Phys., vol. 3, no. 6, pp. 422-440, 2021; doi:10.1038/s42254-021-00314-5 DOI: https://doi.org/10.1038/s42254-021-00314-5
S. Wang, Y. Teng, and P. Perdikaris, "Understanding and mitigating gradient flow pathologies in physics-informed neural networks," SIAM J. Sci. Comput., vol. 43, no. 5, pp. A3055-A3081, 2021; doi:10.1137/20M1318043 DOI: https://doi.org/10.1137/20M1318043
K. H. W. J. ten Tusscher and A. V. Panfilov, "Alternans and spiral breakup in a human ventricular tissue model," Am. J. Physiol. Heart Circ. Physiol., vol. 291, no. 3, pp. H1088-H1100, 2006; doi:10.1152/ajpheart.00109.2006 DOI: https://doi.org/10.1152/ajpheart.00109.2006
Y. Saad, Iterative methods for sparse linear systems, 2nd ed. Philadelphia, PA, USA: SIAM, 2003; doi:10.1137/1.9780898718003 DOI: https://doi.org/10.1137/1.9780898718003
W. M. Czarnecki et al., "Sobolev training for neural networks," in Proc. 31st Conf. Adv. Neural Inf. Process. Syst. 30 (NeurIPS), Long Beach, CA, USA, 2017.
Z. Li, D. Z. Huang, B. Liu, and A. Anandkumar, "Fourier neural operator with learned deformations for PDEs on general geometries," J. Mach. Learn. Res., vol. 24, no. 388, pp. 1-26, 2023.
T. Pfaff, M. Fortunato, A. Sanchez-Gonzalez, and P. Battaglia, "Learning mesh-based simulation with graph networks," in Proc. 9th Int. Conf. Learn. Represent. (ICLR), Appleton, WI, USA, 2021.
Downloads
Published
Issue
Section
License
Copyright (c) 2026 Eugene Shchetinin
This work is licensed under a Creative Commons Attribution 4.0 International License.
