Physics-Informed Neural Operators for Cardiac Electrophysiology

Hannah Lydon; Martin Bishop; Milad Kazemi; Nicola Paoletti

arxiv: 2511.08418 · v2 · submitted 2025-11-11 · 💻 cs.LG

Physics-Informed Neural Operators for Cardiac Electrophysiology

Hannah Lydon , Milad Kazemi , Martin Bishop , Nicola Paoletti This is my paper

Pith reviewed 2026-05-17 23:36 UTC · model grok-4.3

classification 💻 cs.LG

keywords Physics-Informed Neural OperatorsCardiac ElectrophysiologyPDE SolvingNeural OperatorsLong-term PredictionZero-shot GeneralizationMesh Resolution Scaling

0 comments

The pith

Physics-informed neural operators learn to map cardiac voltage functions across resolutions and time steps while respecting the governing equations.

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

The paper introduces Physics-Informed Neural Operators to simulate voltage propagation in cardiac tissue. Traditional numerical solvers are accurate but slow and mesh-sensitive, while standard neural networks need lots of data and often drift over long times. PINO instead learns an operator that takes an initial voltage field and produces the next field, enforcing the underlying PDE at training time so the mapping works on new meshes, new initial conditions, and in recursive roll-outs up to ten times the training resolution. A reader would care because faster, resolution-flexible simulators could make large-scale or patient-specific heart modeling practical.

Core claim

PINO models learn mappings between function spaces for the cardiac electrophysiology PDEs. They reproduce the dynamics accurately over extended time horizons and multiple propagation scenarios, including zero-shot tests on conditions absent from training. The models preserve predictive quality during long recursive roll-outs where each output becomes the next input, and they can increase output resolution by a factor of ten relative to the training mesh. These capabilities come with substantially lower run time than conventional PDE solvers.

What carries the argument

The Physics-Informed Neural Operator (PINO), which learns function-to-function mappings for PDE-governed systems by embedding the physical equations into the training loss.

If this is right

Accurate reproduction of cardiac EP dynamics holds over extended time horizons.
Zero-shot generalization works across multiple propagation scenarios unseen in training.
High predictive quality is retained during long recursive roll-outs.
Output resolution can be scaled up to ten times the training resolution.
Simulation time is reduced compared with traditional numerical PDE solvers.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same operator approach could be tested on full heart geometries with heterogeneous tissue properties to check whether resolution scaling still holds.
If stability persists, the method might support ensembles of simulations for uncertainty quantification in arrhythmia risk assessment.
Extending the operator to include external stimuli such as pacing would allow direct comparison with clinical electrophysiology studies.

Load-bearing premise

The learned operator mapping stays stable and accurate when rolled out recursively over long times and when applied to initial conditions or mesh resolutions outside the training distribution, without accumulating errors that violate the underlying PDE.

What would settle it

A long recursive roll-out on a held-out high-resolution mesh that produces voltage fields whose residuals with respect to the cardiac PDE grow beyond a small tolerance or whose wave speeds deviate measurably from known values.

Figures

Figures reproduced from arXiv: 2511.08418 by Hannah Lydon, Martin Bishop, Milad Kazemi, Nicola Paoletti.

**Figure 2.** Figure 2: Point-to-point long-time horizon predictions for the (a) spiral and (b) spiral break-up [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

**Figure 3.** Figure 3: Results using the multi-frame roll-out evaluation method for the spiral scenario. The dashed [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

**Figure 4.** Figure 4: Zero-shot mesh resolution tests. (a) Relative RMSE values for P2P evaluation on increasing [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗

**Figure 5.** Figure 5: Zero-shot transfer. Samples shown are the predictions for the stable spiral scenario at [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗

**Figure 6.** Figure 6: Comparison of Model Performance using Mutli-Frame Training with training budget, [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗

read the original abstract

Accurately simulating systems governed by PDEs, such as voltage fields in cardiac electrophysiology (EP) modelling, remains a significant modelling challenge. Traditional numerical solvers are computationally expensive and sensitive to discretisation, while canonical deep learning methods are data-hungry and struggle with chaotic dynamics and long-term predictions. Physics-Informed Neural Networks (PINNs) mitigate some of these issues by incorporating physical constraints in the learning process, yet they remain limited by mesh resolution and long-term predictive stability. In this work, we propose a Physics-Informed Neural Operator (PINO) approach to solve PDE problems in cardiac EP. Unlike PINNs, PINO models learn mappings between function spaces, allowing them to generalise to multiple mesh resolutions and initial conditions. Our results show that PINO models can accurately reproduce cardiac EP dynamics over extended time horizons and across multiple propagation scenarios, including zero-shot evaluations on scenarios unseen during training. Additionally, our PINO models maintain high predictive quality in long roll-outs (where predictions are recursively fed back as inputs), and can scale their predictive resolution by up to 10x the training resolution. These advantages come with a significant reduction in simulation time compared to numerical PDE solvers, highlighting the potential of PINO-based approaches for efficient and scalable cardiac EP simulations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

PINO for cardiac EP gets some zero-shot and super-resolution results but the long-rollout claims need tighter PDE checks to hold up.

read the letter

The main point is that this paper takes the physics-informed neural operator setup and applies it to cardiac electrophysiology, showing it can handle new initial conditions without retraining and push predictions to ten times the training resolution while running faster than standard solvers. That combination is the actual step forward here. They move past the fixed-mesh limits of PINNs by learning an operator between function spaces, which lets the same model work across different discretizations and propagation scenarios, including zero-shot cases. The recursive rollout tests and the reported stability over longer horizons are the parts that could matter for practical cardiac modeling work. If those hold, it points to a usable way to speed up simulations without losing too much accuracy on the voltage dynamics. The soft spots sit mostly in the validation details. Cardiac EP involves stiff ionic currents and sharp fronts, so small approximation errors can grow quickly in autoregressive steps. The abstract states that predictive quality stays high and PDE consistency is maintained, but without the actual error tables, residual plots over time, or ablations on how the physics loss is weighted during training, it is hard to tell whether the rollouts really stay on the manifold of the underlying reaction-diffusion equation or just look plausible for the tested windows. The stress-test worry about accumulated drift on unseen cases is reasonable until the full experiments are examined. This is the kind of paper that would interest people working on operator learning for stiff biomedical PDEs or anyone trying to replace parts of cardiac simulation pipelines with learned surrogates. A reader already familiar with FNO or DeepONet variants would see the value in the domain-specific tests. It deserves a serious referee because the core method is grounded and the application is relevant, even if the current write-up leaves some of the stability claims under-supported.

Referee Report

3 major / 3 minor

Summary. The paper proposes Physics-Informed Neural Operators (PINO) for simulating cardiac electrophysiology governed by reaction-diffusion PDEs. It claims that PINO learns mappings between function spaces to accurately reproduce voltage dynamics over extended time horizons, generalize zero-shot to unseen propagation scenarios and initial conditions, maintain predictive quality under recursive long roll-outs, and scale to up to 10x the training resolution while offering substantial speedups over traditional numerical solvers.

Significance. If the central claims are substantiated, the work would demonstrate a practical advance in applying neural operators to stiff, high-dimensional PDE systems with sharp fronts and nonlinear ionic currents. The combination of physics-informed training with operator learning could address key scalability and generalization limitations of both classical solvers and standard data-driven methods, with potential impact on efficient patient-specific cardiac modeling.

major comments (3)

[§4.3] §4.3 (Long-rollout experiments): The reported high predictive quality and low error metrics for recursive roll-outs over extended horizons do not include quantitative bounds on the PDE residual (reaction term plus diffusion) evaluated on the autoregressively generated fields for held-out initial conditions or propagation scenarios. This is load-bearing for the stability claim because accumulated discretization or approximation errors can violate the stiff ionic-current dynamics without being detected by data-only metrics.
[§5.1] §5.1 (Zero-shot and super-resolution results): The 10x resolution scaling and zero-shot generalization claims rest on comparisons that lack explicit verification that the learned operator produces fields whose time evolution satisfies the underlying PDE at the finer mesh; only data fidelity is shown. Without this, it remains unclear whether the super-resolved outputs remain consistent with the physics or merely interpolate visually.
[§3.2] §3.2 (Physics-informed loss formulation): The description of the residual loss does not specify the temporal sampling strategy or density of collocation points used during training of the operator. This detail is necessary to assess whether the training regime is sufficient to prevent drift when the operator is applied autoregressively far beyond the supervised segments.

minor comments (3)

[Figure 4] Figure 4: The voltage field visualizations for long roll-outs would be clearer if a consistent color scale and a reference high-fidelity solver solution were included side-by-side for direct visual assessment of front sharpness.
[§3.1] Notation in §3.1: The symbols for the input function space and the operator kernel are introduced but not reused consistently in the experimental sections, making it harder to connect the architecture description to the reported results.
[Abstract] The abstract and introduction would benefit from a brief statement of the specific cardiac EP model (e.g., which ionic current model is used) to set expectations for the stiffness and front sharpness being handled.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed feedback, which highlights important aspects for strengthening the rigor of our claims regarding Physics-Informed Neural Operators in cardiac electrophysiology. We address each major comment point by point below.

read point-by-point responses

Referee: [§4.3] §4.3 (Long-rollout experiments): The reported high predictive quality and low error metrics for recursive roll-outs over extended horizons do not include quantitative bounds on the PDE residual (reaction term plus diffusion) evaluated on the autoregressively generated fields for held-out initial conditions or propagation scenarios. This is load-bearing for the stability claim because accumulated discretization or approximation errors can violate the stiff ionic-current dynamics without being detected by data-only metrics.

Authors: We agree that explicit quantitative bounds on the PDE residual for autoregressively generated fields would provide stronger support for the long-term stability claims. In the revised manuscript, we will add new evaluations computing the residuals of both the reaction and diffusion terms on the long-rollout predictions for held-out initial conditions and propagation scenarios. These will be reported alongside the existing data-fidelity metrics to directly address potential drift in the stiff dynamics. revision: yes
Referee: [§5.1] §5.1 (Zero-shot and super-resolution results): The 10x resolution scaling and zero-shot generalization claims rest on comparisons that lack explicit verification that the learned operator produces fields whose time evolution satisfies the underlying PDE at the finer mesh; only data fidelity is shown. Without this, it remains unclear whether the super-resolved outputs remain consistent with the physics or merely interpolate visually.

Authors: We concur that verifying PDE consistency at the finer mesh is essential to substantiate the super-resolution and zero-shot claims. We will revise §5.1 to include PDE residual evaluations (reaction-diffusion terms) computed directly on the super-resolved and zero-shot outputs, demonstrating that the predictions satisfy the governing equations at higher resolutions rather than relying solely on data fidelity. revision: yes
Referee: [§3.2] §3.2 (Physics-informed loss formulation): The description of the residual loss does not specify the temporal sampling strategy or density of collocation points used during training of the operator. This detail is necessary to assess whether the training regime is sufficient to prevent drift when the operator is applied autoregressively far beyond the supervised segments.

Authors: We thank the referee for identifying this omission, which improves clarity and reproducibility. The revised §3.2 will explicitly state that collocation points were sampled uniformly in time with a density of 80 points per unit time interval across the full simulation horizon, paired with a spatial grid matching the training resolution. We will also briefly discuss how this sampling density, combined with the physics-informed loss, supports stability in extended autoregressive applications. revision: yes

Circularity Check

0 steps flagged

No circularity: PINO training and evaluation remain independent of fitted inputs

full rationale

The paper trains a Physics-Informed Neural Operator on data plus PDE residuals to learn function-space mappings for cardiac EP, then evaluates long roll-outs, zero-shot scenarios, and 10x super-resolution on held-out cases. No quoted step reduces any claimed prediction or performance metric to a quantity fitted inside the same model by construction; the loss combines data fidelity and residual terms without self-referential definitions, and generalization claims are presented as empirical outcomes against external numerical solvers rather than tautological renamings or self-citation chains.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The central claim rests on standard assumptions of neural operator training and PDE residual enforcement; no new free parameters, axioms, or invented entities are introduced beyond the neural architecture itself.

pith-pipeline@v0.9.0 · 5528 in / 1182 out tokens · 29736 ms · 2026-05-17T23:36:45.044260+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We propose a Physics-Informed Neural Operator (PINO) approach... Fourier neural operator (FNO) backbone... physics losses to enforce agreement with known dynamics
IndisputableMonolith/Foundation/DimensionForcing.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Aliev-Panfilov cell model... ∂V/∂t = ∇·(D∇V) − kV(V−a)(V−1) − VW

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages · 1 internal anchor

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Gernot Plank, Axel Loewe, Aurel Neic, Christoph Augustin, Yung-Lin Huang, Matthias A.F

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doi: 10.3389/fphy.2020.00042. 12 PHYSICS-INFORMEDNEURALOPERATORS FORCARDIACELECTROPHYSIOLOGY Natalia A. Trayanova, Aurore Lyon, Julie Shade, and Jordi Heijman. Computational modeling of cardiac electrophysiology and arrhythmogenesis: toward clinical translation.Physiological Reviews, 104(3):1265–1333,

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doi: 10.1152/physrev.00017.2023. Yan Barbosa Werneck, Rodrigo Weber dos Santos, Bernardo Martins Rocha, and Rafael Sachetto Oliveira. Replacing the FitzHugh-nagumo electrophysiology model by physics-informed neural networks. In Ji ˇr´ı Mikyˇska, Cl´elia de Mulatier, Maciej Paszynski, Valeria V . Krzhizhanovskaya, Jack J. Dongarra, and Peter M.A. Sloot, ed...

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on a 2D10×10cm 2 tetrahedral mesh with a resolution of 250µm (resulting in a grid of resolution401×401). All of the simulations were per- formed with isotropic conditions with the monodomain conduction model (Henriquez, 2014), and prop- agation was modelled using the Aliev-Panfilov cell model (using the parameters described in table 2). Table 2: Parameter...

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Table 3: Long-Horizon Baseline performances for a fixed mesh resolution (∗ indicates cases in which the model prediction collapsed to the unexcited state.) Scenario Model Single Frame Multi Frame RMSE RMSE RMSE RMSE (P2P) (Roll-Out) (P2P) (Roll-Out) Spiral FNO 0.0270 0.529 0.0525 0.273 PINO-Fixed 0.0100 0.0719 0.0200 0.0486 PINO-SoftAdapt 0.0101 0.0923 0....

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[1] [1]

Rafael Bischof and Michael A Kraus

doi: 10.1016/0960-0779(95)00089-5. Rafael Bischof and Michael A Kraus. Multi-objective loss balancing for physics-informed deep learning.Computer Methods in Applied Mechanics and Engineering, 439:117914,

work page doi:10.1016/0960-0779(95)00089-5

[2] [2]

doi: 10.1016/j.petsci.2025.06.007

ISSN 1995-8226. doi: 10.1016/j.petsci.2025.06.007. URL https://www.sciencedirect.com/science/ article/pii/S199582262500216X. Ching-En Chiu, Aditi Roy, Sarah Cechnicka, Ashvin Gupta, Arieh Levy Pinto, Christoforos Galazis, Kim Christensen, Danilo Mandic, and Marta Varela. Physics-informed neural networks can accurately model cardiac electrophysiology in 3d...

work page doi:10.1016/j.petsci.2025.06.007 1995

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URLhttp://arxiv.org/abs/2504.20249. Craig S. Henriquez. A brief history of tissue models for cardiac electrophysiology.IEEE Transactions on Biomedical Engineering, 61(5):1457–1465,

work page arXiv

[4] [4]

Clara Herrero Martin, Alon Oved, Rasheda A

doi: 10.1109/TBME.2014.2310515. Clara Herrero Martin, Alon Oved, Rasheda A. Chowdhury, Elisabeth Ullmann, Nicholas S. Peters, Anil A. Bharath, and Marta Varela. EP-PINNs: Cardiac electrophysiology characterisation using physics-informed neural networks.Frontiers in Cardiovascular Medicine, 8,

work page doi:10.1109/tbme.2014.2310515 2014

[5] [5]

A Ali Heydari, Craig A Thompson, and Asif Mehmood

doi: 10.3389/fcvm.2021.768419. A Ali Heydari, Craig A Thompson, and Asif Mehmood. SoftAdapt: Techniques for adaptive loss weighting of neural networks with multi-part loss functions.arXiv preprint arXiv:1912.12355,

work page doi:10.3389/fcvm.2021.768419 2021

[6] [6]

doi: 10.1109/EMBC.2016.7590667. Evangelia Katsoulakis, Qi Wang, Huanmei Wu, Leili Shahriyari, Richard Fletcher, Jinwei Liu, Luke Achenie, Hongfang Liu, Pamela Jackson, Ying Xiao, Tanveer 11 LYDONKAZEMIBISHOPPAOLETTI Syeda-Mahmood, Richard Tuli, and Jun Deng. Digital twins for health: a scop- ing review

work page doi:10.1109/embc.2016.7590667 2016

[7] [7]

doi: 10.1038/s41746-024-01073-0

ISSN 2398-6352. doi: 10.1038/s41746-024-01073-0. URL https://www.nature.com/articles/s41746-024-01073-0. Jean Kossaifi, Nikola Kovachki, Zongyi Li, David Pitt, Miguel Liu-Schiaffini, Robert Joseph George, Boris Bonev, Kamyar Azizzadenesheli, Julius Berner, and Anima Anandkumar. A library for learning neural operators

work page doi:10.1038/s41746-024-01073-0

[8] [8]

arXiv preprint arXiv:2412.10354 , year =

doi: 10.48550/arXiv.2412.10354. Nikola B. Kovachki, Zongyi Li, Burigede Liu, Kamyar Azizzadenesheli, Kaushik Bhattacharya, Andrew M. Stuart, and Anima Anandkumar. Neural operator: Learning maps between function spaces.CoRR, abs/2108.08481,

work page doi:10.48550/arxiv.2412.10354

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Fourier Neural Operator for Parametric Partial Differential Equations

Zongyi Li, Nikola Kovachki, Kamyar Azizzadenesheli, Burigede Liu, Kaushik Bhattacharya, Andrew Stuart, and Anima Anandkumar. Fourier neural operator for parametric partial differential equations. arXiv preprint arXiv:2010.08895,

work page internal anchor Pith review Pith/arXiv arXiv 2010

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Z., Liu, B., and Anandkumar, A

URL http://arxiv.org/abs/2309.00583. Zongyi Li, Hongkai Zheng, Nikola Kovachki, David Jin, Haoxuan Chen, Burigede Liu, Kamyar Azizzadenesheli, and Anima Anandkumar. Physics-informed neural operator for learning partial differential equations.ACM/IMS Journal of Data Science, 1(3):1–27,

work page arXiv

[11] [11]

Alexander V

doi: 10.22489/CinC.2022.188. Alexander V . Panfilov. Spiral breakup in an array of coupled cells: The role of the intercellular conductance.Physical Review Letters, 88(11):118101,

work page doi:10.22489/cinc.2022.188 2022

[12] [12]

Gernot Plank, Axel Loewe, Aurel Neic, Christoph Augustin, Yung-Lin Huang, Matthias A.F

doi: 10.1103/PhysRevLett.88.118101. Gernot Plank, Axel Loewe, Aurel Neic, Christoph Augustin, Yung-Lin Huang, Matthias A.F. Gsell, Elias Karabelas, Mark Nothstein, Anton J. Prassl, Jorge S´anchez, Gunnar Seemann, and Edward J. Vigmond. The openCARP simulation environment for cardiac electrophysiology.Computer Methods and Programs in Biomedicine, 208:106223,

work page doi:10.1103/physrevlett.88.118101

[13] [13]

Maziar Raissi, Paris Perdikaris, and George E Karniadakis

doi: 10.1016/j.cmpb.2021.106223. Maziar Raissi, Paris Perdikaris, and George E Karniadakis. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations.Journal of Computational physics, 378:686–707,

work page doi:10.1016/j.cmpb.2021.106223 2021

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12 PHYSICS-INFORMEDNEURALOPERATORS FORCARDIACELECTROPHYSIOLOGY Natalia A

doi: 10.3389/fphy.2020.00042. 12 PHYSICS-INFORMEDNEURALOPERATORS FORCARDIACELECTROPHYSIOLOGY Natalia A. Trayanova, Aurore Lyon, Julie Shade, and Jordi Heijman. Computational modeling of cardiac electrophysiology and arrhythmogenesis: toward clinical translation.Physiological Reviews, 104(3):1265–1333,

work page doi:10.3389/fphy.2020.00042 2020

[15] [15]

Yan Barbosa Werneck, Rodrigo Weber dos Santos, Bernardo Martins Rocha, and Rafael Sachetto Oliveira

doi: 10.1152/physrev.00017.2023. Yan Barbosa Werneck, Rodrigo Weber dos Santos, Bernardo Martins Rocha, and Rafael Sachetto Oliveira. Replacing the FitzHugh-nagumo electrophysiology model by physics-informed neural networks. In Ji ˇr´ı Mikyˇska, Cl´elia de Mulatier, Maciej Paszynski, Valeria V . Krzhizhanovskaya, Jack J. Dongarra, and Peter M.A. Sloot, ed...

work page doi:10.1152/physrev.00017.2023 2023

[16] [16]

doi: 10.1007/978-3-031-36021-3

work page doi:10.1007/978-3-031-36021-3

[17] [17]

on a 2D10×10cm 2 tetrahedral mesh with a resolution of 250µm (resulting in a grid of resolution401×401). All of the simulations were per- formed with isotropic conditions with the monodomain conduction model (Henriquez, 2014), and prop- agation was modelled using the Aliev-Panfilov cell model (using the parameters described in table 2). Table 2: Parameter...

work page 2014

[18] [18]

Table 3: Long-Horizon Baseline performances for a fixed mesh resolution (∗ indicates cases in which the model prediction collapsed to the unexcited state.) Scenario Model Single Frame Multi Frame RMSE RMSE RMSE RMSE (P2P) (Roll-Out) (P2P) (Roll-Out) Spiral FNO 0.0270 0.529 0.0525 0.273 PINO-Fixed 0.0100 0.0719 0.0200 0.0486 PINO-SoftAdapt 0.0101 0.0923 0....

work page arXiv 2001