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Learning High-dimensional Ionic Model Dynamics Using Fourier Neural Operators

T0 review · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A single Fourier Neural Operator can learn the full state dynamics of stiff ionic models up to 41 variables with roughly 2% relative L2 test error.

arxiv 2505.14039 v1 pith:FU7MZFBM submitted 2025-05-20 cs.LG cs.NAmath.NAstat.ML

Luca Pellegrini , Massimiliano Ghiotto , Edoardo Centofanti , Luca Franco Pavarino This is my paper
classification cs.LGcs.NAmath.NAstat.ML
keywords dynamicsneuralfouriermodelmodelsdynamicalionicnumber
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper studies whether Fourier Neural Operators (FNOs), a neural network architecture popular in scientific machine learning, can learn to simulate ionic models: systems of stiff ordinary differential equations that describe how cell membranes generate electrical signals such as action potentials. The authors train a single FNO to map a simple applied current (a rectangular pulse with two parameters, amplitude and duration) to the full time evolution of every state variable in the model. They test three models of increasing size: the 2-variable FitzHugh-Nagumo model, the 4-variable Hodgkin-Huxley model, and the 41-variable O'Hara-Rudy model of a human heart cell.

The results show held-out test errors of about 0.9% for FitzHugh-Nagumo and 2.2 to 2.7% for the Hodgkin-Huxley and O'Hara-Rudy models, with errors spread relatively evenly across variables. The authors also compare two hyperparameter tuning strategies: unconstrained and constrained to about 500,000 parameters, finding similar accuracy but faster convergence for the unconstrained network. They interpret the results as evidence that FNOs scale to high-dimensional stiff systems.

The main caveats are that the input space is only two-dimensional, the ground truth relies on an unspecified stiff ODE solver, and no code or data is currently available. The comparison to other operator learning methods is also missing, so the paper establishes capability more than superiority.

Extended reading notes

Core claim

The central claim is that FNO parameters and relative errors are bounded independently of the state dimension, and that a single FNO can learn the full solution operator of stiff ionic models including the 41-variable O'Hara-Rudy model with about 2% relative L2 test error. If correct, FNOs are a viable surrogate for high-dimensional ionic dynamics.

Load-bearing premise

The applied current input is restricted to a two-parameter family of piecewise constant functions (amplitude i and stimulus duration Tstim), defined in Section 2. The paper's success on this low-dimensional input family is taken as evidence that the learned operator generalizes to arbitrary applied currents; if FNOs fail on richer input waveforms, the central capability claim is overstated.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper makes no theoretical derivation and introduces no scientific free parameters; the network weights and hyperparameters are fitted in the standard ML sense. The central claim rests on the correctness of the reference ODE solver, the faithful implementation of the published ionic models, and the extrapolation of scaling behavior from three test models.

assumptions (4)
  • domain assumption System (1) admits a unique solution for the chosen parameter ranges.
    Stated in Section 2 without proof; the FNO is trained on solutions to these ODEs, so existence and uniqueness are needed.
  • domain assumption The O'Hara-Rudy model equations and parameters from [24] are correctly implemented in the data generation.
    The paper refers to the original paper for the full equations and does not reproduce them; correctness of the surrogate depends on the correctness of the reference implementation.
  • domain assumption The reference stiff ODE solver produces accurate ground truth.
    The dataset is generated with a Runge-Kutta method for stiff ODEs [28]; no tolerances are reported, so the reference solutions are trusted as exact.
  • standard math FNO universality and mesh-independence properties from cited works hold for the present setting.
    The architecture design follows the FNO literature; the paper relies on these theoretical properties to justify the approach.

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Cite this review

Pith. "Pith review of Learning High-dimensional Ionic Model Dynamics Using Fourier Neural Operators." pith.science (2026). https://pith.science/paper/FU7MZFBM

@misc{pith2026250514039,
  author       = {Pith},
  title        = {Pith review of: Learning High-dimensional Ionic Model Dynamics Using Fourier Neural Operators},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FU7MZFBM}},
  note         = {Machine review of arXiv:2505.14039}
}
read the original abstract

Ionic models, described by systems of stiff ordinary differential equations, are fundamental tools for simulating the complex dynamics of excitable cells in both Computational Neuroscience and Cardiology. Approximating these models using Artificial Neural Networks poses significant challenges due to their inherent stiffness, multiscale nonlinearities, and the wide range of dynamical behaviors they exhibit, including multiple equilibrium points, limit cycles, and intricate interactions. While in previous studies the dynamics of the transmembrane potential has been predicted in low dimensionality settings, in the present study we extend these results by investigating whether Fourier Neural Operators can effectively learn the evolution of all the state variables within these dynamical systems in higher dimensions. We demonstrate the effectiveness of this approach by accurately learning the dynamics of three well-established ionic models with increasing dimensionality: the two-variable FitzHugh-Nagumo model, the four-variable Hodgkin-Huxley model, and the forty-one-variable O'Hara-Rudy model. To ensure the selection of near-optimal configurations for the Fourier Neural Operator, we conducted automatic hyperparameter tuning under two scenarios: an unconstrained setting, where the number of trainable parameters is not limited, and a constrained case with a fixed number of trainable parameters. Both constrained and unconstrained architectures achieve comparable results in terms of accuracy across all the models considered. However, the unconstrained architecture required approximately half the number of training epochs to achieve similar error levels, as evidenced by the loss function values recorded during training. These results underline the capabilities of Fourier Neural Operators to accurately capture complex multiscale dynamics, even in high-dimensional dynamical systems.

Figures

Figures reproduced from arXiv: 2505.14039 by the authors.

Figure 1
Figure 1. Visual representation of a Fourier Neural Operator. [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FHN model: FNO training relative L 2 (blue) loss and relative test L 1 (green), L 2 (orange), and H1 (red) loss functions. (a) Unconstrained FNO model, (b) constrained FNO model. (a) Unconstrained FNO bar plot. (b) Constrained FNO bar plot. (c) Box plot for unconstrained and constrained FNO [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. FHN model: FNO performance comparison. Figure [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (9 more)
Figure 5
Figure 5. Figure 5: FHN model: examples of FNO performance. Each column represents a single example from a [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: HH model: FNO training relative L 2 (blue) loss and test relative L 1 (green), L 2 (orange), and H1 (red) loss functions. (a) Unconstrained FNO model, (b) constrained FNO model. (a) Unconstrained FNO bar plot. (b) Constrained FNO bar plot. 13 [PITH_FULL_IMAGE:figures/…
Figure 7
Figure 7. Figure 7: HH model: FNO performance comparison. Fig. [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]
Figure 8
Figure 8. Figure 8: HH model: examples of FNO performance. Each column represents a single example from a [PITH_FULL_IMAGE:figures/full_fig_p014_8.png]
Figure 9
Figure 9. Figure 9: ORd model: FNO training relative L 2 (blue) loss and test relative L 1 (green), L 2 (orange), and H1 (red) loss functions. (a) Unconstrained FNO model, (b) constrained FNO model. (a) Unconstrained FNO bar plot. (b) Constrained FNO bar plot. (c) Box plot for the unconst…
Figure 10
Figure 10. Figure 10: ORd model: FNO performance comparison. Fig. [PITH_FULL_IMAGE:figures/full_fig_p016_10.png]
Figure 11
Figure 11. Figure 11: ORd model: examples of FNO performance. Each figure contains five plots, arranged in order [PITH_FULL_IMAGE:figures/full_fig_p017_11.png]
Figure 12
Figure 12. Figure 12: Dimensionality dependence of FNO parameters (from the summary Tables [PITH_FULL_IMAGE:figures/full_fig_p018_12.png]
Figure 13
Figure 13. Figure 13: Component-wise box plot for the unconstrained FNO relative errors on the test set. [PITH_FULL_IMAGE:figures/full_fig_p019_13.png]

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