ODE-Constrained Generative Modeling of Cardiac Dynamics for 12-Lead ECG Synthesis
Authors: Yakir Yehuda
Kira Radinsky
Venue: Accepted by TMLR ()

Abstract: Generating realistic training data for supervised learning remains a significant challenge in artificial intelligence. This is particularly true in the synthesis of electrocardiograms (ECGs), where the objective is to develop a synthetic 12-lead ECG model. The primary challenge in this task lies in accurately modeling the intricate biological and physiological interactions among different ECG leads. Although mathematical process models have shed light on these dynamics, effectively incorporating this understanding into generative models is not straightforward. We introduce an innovative method that employs ordinary differential equations (ODEs) to enhance the fidelity of 12-lead ECG data generation. This approach integrates cardiac dynamics directly into the generative optimization process via a novel Euler Loss, producing biologically plausible data that respects real-world variability and inter-lead constraints. Empirical analysis on the G12EC and PTB-XL datasets demonstrates that augmenting training data with MultiODE-GAN yields consistent, statistically significant improvements in specificity across multiple cardiac abnormalities. This highlights the value of enforcing physiological coherence in synthetic medical data.

Published in Transactions on Machine Learning Research (02/2026)
ODE- Constraidd Generative Modeling of Cardiac Dynamics for 12-Lead ECG Synthesis
Yakir Yehuda
y j.uukr@ss..eccnion.cc.
Department of Computer Science Technson Israel Institute of Technology
Kira Radinsky
kirar Bcs. technion.ac..
Department Computer Science Technion Israel Institute of Technology
Reviewed on OpenReview: https: o/penreveeeeeeeeeee net/ forum? id= 4N566uusPu i
Abstract
Generating realistic training data for supervised learning remains significant challenge in artificial intelligence, particularly in domains where large, expert-labeled datasets are scarce or costly to obtain. This is especially true for electrocardiograms (ECGs), where privacy constraints, class imbalance, and the need for physician annotation limit the availability of labeled 12-lcad rocordings, motivating the development of high-fidelity synthetic ECG data. The primary challenge in this task lies in accurately modeling the intricate biological and physiological interactions among different ECG leads. Although mathematical process models have shed light on these dynamics, effectively incorporating this understanding into generative models is not straightforward. We introduce an innovative method that em- ploys ordinary differential equations (ODEs) to enhance the fidelity of 12-lead ECG data generation. This approach integrates cardiac dynamics directly into the generative opti- mization process via a novel Euler Loss, producing biologically plausible data that respects real-world variability and inter-lead constraints. Empirical analysis on the G12EC and PTB-XI. datasets demonstrates that augmenting training data with MultiODE-GAN yields consistent, statistically significant improvements in specificity across multiple cardiac abnor- malities. This highlights the value of enforcing physiological coherence in synthetic medical data.
1 Introduction
The generation of synthetic 12-lead electrocardiogram (ECG) data has emerged significant area of research, driven by the need for large and diverse datasets to train machine learning models for various medical applications (de Melo et al. 2022). ECG data, which records the electrical activity of the heart, is critical for diagnosing and monitoring cardiac conditions. While real ECG datasets are available, they are often biased toward common conditions, exhibit severe class imbalance for rare disorders, and are constrained by privacy and data security concerns (Voigt & Bussche 2017). Moreover, acquiring annotated multi-lead ECG data scale requires physician review, making large-scale expert labeling costly and further limiting the availability of rare cardiac disorder cases. Generating synthetic data offers promising solution to address privacy concerns associated with the distribution of sensitive health information (Giuffrè & Shung, 2023). In this context, ensuring that synthctic signals are physiologically realistic, not just visually plausible, is essential for their utility in clinical and machine learning contexts, especially for tasks such as anomaly detection and disease classification. Recent advances suggest that embedding domain-specific mathematical models into deep generative models can lead to more faithful data reprosentations. In this work, we leverage such mathematical models by incorporating ordinary differential equations (ODEs) that describe cardiac dynamics directly into the generative process.

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ODE-based models are well-established in the sciences for simulating complex systems, enabling manccdd manipulation and interpretation of variables over time. Realistic heart models, for example, require extensive knowledge of cardiac mechanics and anatomy to simulate phenomena such as pressure tressure-volume volume relationships and various pathologics. In contrast, our work focuses on the ECG Dynamical Model (EDM), which cap- tures electrophysiological dynamics via a system of ODEs McSharry et al. (2003). Embedding this model within our generative framework enables continuous simulation and synthesis of ECG signals with enhanced physiological fidelity.
To this end, we introduce MultiODE-GAN, a novel Generative Adversarial Network (GAN) framework specifically designed for generating synthetic 12-lead ECG data. This framework is meticulously tailored to replicate cardiac cycles from ECG signals across all 12 leads. Our GAN not only employs an adversarial loss to encourage realistic outputs, but also introduces an Euler Loss that quantifies the discrepancy between the temporal dynamics of generated heartbeats and those predicted by the EDM. This loss enforces two levels of physiological consistency: (1) individual leads follow the ODE-preseribed dynamics, and (2) inter-lead dependencies (e-g., Einthoven's law), which is essential for synthesizing coherent 12-lead ECG signals and capturing both temporal and spatial dependencies. Since the ODEs cannot generally be solved analytically, we leverage the Euler method to numerically evaluate this loss. We validate MultiODE-GAN on gold- standard 12-lead ECG datasets and show that incorporating ODE-based based constraints into the generative process significantly enhances the physiological realism of synthetic signals. To assess clinical utility, we further evaluate MultiODE-GAN in collaboration with the Interventional Cardiology Unit for early detection of left ventricular systolic dysfunction (LVSD) using real patient data. Additionally, classifiers trained synthetic FCGs generated by MultiODE-GAN show improved performance in downst heartbeat classification tasks, demonstrating its value for anomaly detection and carly screening applications. In retrospective clinical study (Appendix B}, the method achieves specificity improvements comparable to those of board-certifed cardiologists for early LVSD detection, highlighting its potential to support clinical decision-making.
Our contributions are threefold: (1) We propose MuliODE-GAN, a novel ODE-constrained GAN that incorporates physiological heart model and explicit inter-lead dependencies for synthetic 12-lead ECG generation. (2) We provide extensive empirical evidence, showing that MultiODE-GAN consistently improves downstream 12- ECG classification performance across multiple abnormalities and benchmarks. (3) We release our implementation open source to support reproducibility and further research in medical AT.
2 Related Work
The availability of annotated data often represents major bottleneck in the development of deep learning
models (de Melo et al., 2022). Unlike biological systems, deep lcarning methods still lack capabilities for richer representations and sophisticated learning, which synthetic data and simulators can help to address
(Cichy & Kaiser) 2019).. Synthetic data plays an essential role in training deep neural networks across
domains such as detection, segmentation, and classification (Tremblay et al. 2018 Ros et al. 2016 de Mclo et al. 2022).
In the medical and machine learning communities, the generation of synthetic ECG data has gained signifi- cant interest due to the critical need for large, high-quality datasets. Various generative models have been proposed, each with unique approaches to simulate realistic ECG signals.
A cornerstone of recent advancements is the use of physiological mathematical models, such as partial dif- ferential equations, to represent. biomedical phenomena including electrical activation in the heart (Boulakia) et al. 2010) and neuronal activity and disorders (Fenwick et al. 1971 Hallez et al. 2007). A notable ad- vancement in ECG generation is the dynamical models based on ordinary differential equations (ODEs). McSharry et al. (2003) proposed set of coupled ODEs to simulate the heart's electrophyssologicall activity providing a physiologically grounded approach to ECG generation.
The advent of deep learning has introducod innovative methods for generating synthetic ECG data, among which Generative Adversarial Networks (GANs) (Goodfellow et al.) (2014) stand out. GANs, through ad- versarial training of generator and discriminator, have been effectively employed to create high-fidelity
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synthetic data (Fai et al.
2018 Karras et al., 2017 Variants such DCGAN (Radford et al. 2016
In our research, we utilize WaveGAN
(Dnahue et al. 2018), GAN variant designed for waveform data, making it particularly well-suited for modeling the temporal dynamics of ECG signals.
Research by Golany et al. (2020) demonstrated that GANs can generate realistic single-lead ECG heart- beats using an ODF-based simulator and suggested the potential for extending this idea to multi-lead ECG generation. However, SimGAN is inherently limited to single-lead dynamics and cannot capture the essen- tial physiological interdependencies between the 12 standard ECG leads. We extend this line of work by introducing physiologically motivated constraints that enforce anatomical relationships (e.g., Einthoven's ss law) across all 12 leads. Unlike purely data-driven approachos such as SSSD-ECG (Alcaraz & Strodthoff] 2023), MultiODE-GAN explicilly embeds these biophysical laws into the training loop. Applying SimGAN independently to each lead is insufficient, as it ignores these critical inter-lead dependencies. In Section we empirically demonstrate the shortcomings of such single-lcad inference approaches.
Previous studies have explored various methods for generating 12-lead ECG data. [Lu et al. (020) suggested the use of vector quantized variational autoencoders (VQ-VAE) for generating new samples to enhance the
training of a 12-lead ECG classifier, KKog et al. (2021) introduced DiffWave, a versatile diffusion probabilistic model designed for both conditional and unconditional waveform generation.
Huang et al. 2023) proposed an unsupervised GAN-based method for generating noisy ECG signals without requiring labeled data. More recently, (Alcaraz & Strodthoff] 2023) introduced SSSD-ECG, a diffusion- based model for 12-lead ECG generation that represents the current taateoff-hee-at. We compare our approach, MultiODF-GAN, against SSSD-EEG and show that MultiODE-GAN achieves state-of-the-art, performance in generating physiologically realistic 12-lcad ECGs.
To demonstrate the impact of synthetic data, we also include 12-lead ECG classifier trained on both real and synthetic data. Research has shown that deep learning has revolutionized ECG classification, achieving state-of-tho-at results (Ribeiro ct al., 2020 Attia et al. 2019 Nejodly ct al., 2021) and, in some cases, outperforming cardiologists (Golany el al. 2022; Choi et al., 2022). However, these models often require extensive labeled datasets and do not inherently incorporate underlying cardiac physiology, gap our work aims to fill by bridging dynamic physiological models with advanced machine learning techniques.
Our work builds on prior efforts to bridge dynamic physiological modeling and generative approaches for 12- lead ECG synthesis, particularly in capturing inter-lead dependencies. We advance this direction by directly embedding an ODE-based model into the generative process, guided jointly by physiological principles and data-driven learning. We adopt a GAN backbone for its computational efficiency on high-frequency wave- forms and compatibility with differentiable ODF-based penalties, extending these constraints to diffusson models is an exciting direction for future work.
3 Physiology-Bare Based ECG Dynamical Model (EDM)
The ECG Dynamical Model (EDM), originally introduced by McSharry et al. (2003,, employs a system of three coupled ordinary differential equations (ODEs) to model the prototypical morphology of the ECG waveform. The model generates a trajectory in three-dimensional space (z((t,, x()),, where the path is modulated at key points corresponding to the P, Q, R, S, and T waves. Each of the three components, the P wave, QRS complex, and T' wave contributes to the protolypical electrical activity of the heart. Notably, the EDM can simulate synthetic ECGs with physiologically plausible PQRST morphology and prescribed heart rate dynamics, parameterized by mean/standard deviation and the spectral characteristics of heart rate variability.
3.1 Model Formulation
The EDM breaks the ECG down into three major components: the P wave, QRS complex, and T wave. These correspond distinct phases of the cardiac cycle, creating the canonical PQRST sequence. The model is defined by three coupled ODFs that evolve the trajectory ()())))()),=(())
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5 = oz wy = (,,,,)1))
(1)
ÿ = ay + wI = fu y:
(2)
a, . exp A0,
2 =
Σ
-(2 1 -zo(t))
2b
RE(P.Q.R.S.T))
= {,(,,
(3)
where:
as bi, 0n amplitude, width, and angular centers of the Gaussian-shaped P. Q, R, S, T waves.
2 is the angular velocity, where f represents the heart rate. This parameter links the trajectory's rotation around the limit cycle to the beat-to-beat heart rate.
is defined as = 1 1- 2 + y². This term drives the trajectory towards an attracting limit cycle.
A0; (0 (0-0:) 0:) mod 2n, where 0 aretan 2(y, x) E [- is the phase angle of the trajectory. As the trajectory approaches one of the fixed angular positions 0,.
=o( = Asin(2n/22) /2(): baseline wander driven by respiratory frequency f2.
We collect the ODE parameters as = {0,, a, by i (P.Q,
Each of the 12 leads, along with each type of abnormality, is parameterized by unique of EDM coefficients, available in our public code repository. The solution to the EDM system describes continuous trajectory in the chree-dimensional state space (I, 2) where the a(t) component directly corresponds to the synthesized ECG signal.
We estimate the EDM parameters The for each lead and abnormality class e by fitting the McSharry model (McSharry et al.) 2003) to real heartbeats using a least-squares optimization procedure. For each (lead, class) pair, we the parameters on training beats only and keep of fixed during GAN training.
The characteristie P, Q, R, S, and T waves are determined by the parameters 0,, which fix their positions on the unit circle. As the trajectory approaches each 9,, the component is transiently elevated or depressed, corresponding to the onset of cardiac wave. The amplitude and duration of each waveform are precisely controlled by parameters a; and b,, respectively, enabling fine-grained manipulation of ECG morphology to reflect a wide range of physiological and pathological conditions.
3.2 Numerical Integration of the EDM
The ordinary differential equations (ODEs) that form the basis of our ECG Dynamical Model are solved numerically using methods from the Runge-Kuthe Kutta family (Butcher & Butcher 1987). For this purpose, we employ the Euler discretization as a differentiable local consistency operator for enforcing ODE compliance, with fixed time step At 1, where fs the sampling frequency rather than as a high-accuracy numerical F. solver (Atkinson 2008) The time step At (1/500 Hz) is sufficiently small to avoid numerical instability or stiffness issues, in practice, higher-order solvers (e.g., Runge Kutta 4) produce trajectories that are nearly identical for this system.
The Euler method employs finite difference approximation, detailed by Milne-Thomson 2000
dt du (t) = ult + At) - u(()
At
For generic ODE expressed as F(1, u), the Euler method approximates the solution iteratively as:
u(t At) u(() F(1, u()))A)
(4)
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We implement this method by applying Eq. 44 iteratively across L time-steps, where each time-step corresponds to 4 = fAI, with A/ = I, fixed by the sampling frequency fs-
In the discrete formulation, for a time-step & corresponding to ie = lst, the iterative update is given by:
ue+1 = ue + f(l,
Applying this scheme the EDM equations (Equations (xr, yo 2) across successive time-steps:
and 3 we derive the discrete trajectories
ty lAt
J/+1 zt. + Sa (ze, (ee At.
(6)
M = + fu (Te, yes 11) A
26+1 = 21 + (at, ye, 26, te; At
This approach ensures that each step forward time l is calculated based on the known values from the previous step e, requiring the initial conditions to start the simulation.
4 MultiODE-GAN
We propose MultiODE-GAN a generative adversarial framework that incorporates physiological constraints from the ECG Dynamical Model (EDM), as described in Section 3 While traditional GANs often struggle to capture the intricate morphology and physiological consistency required for realistic medical signals, MultiODE-GAN addresses this by embedding ODE-based dynamics directly into the generation process.
4.1 MultiODE-GAN Framework
We implement an extended SimGAN baseline by applying the original single-lead ODE-based loss inde- pendently cach of the 12 leads, without modcling inter-lcad dependencies. This provides a controlled comparison to our method. SimGAN (Golany et al., 2020) was originally designed to generate only single ECG lead and, by construction, cannot model the physiological coupling that governs the full 12-lead sys- tem. In contrast, MultiODE-GAN introduces novel multi-lead ODE-constrained formulation that embeds both intra-lead dynamics and physiologically grounded inter-lcad relationships directly into the generator's loss. This enables the model to caplure the spatial-temporal structure of the 12-lead ECG by explicitly incorporating ODE-based constraints alongside Einthoven's 's law, producing substantially more realistic and clinically coherent synthetic signals.
For the underlying architectural backbone, we adopt WaveGAN (Donahue et al. 2018 one-dimensional adaptation of DCGAN (Radford et al.) 2016 well-suited for waveform generation. Although originally designed for single-channel output, we extend WaveGAN to natively produce multi-channel signals, enabling full 12-lead ECG synthesis. We further augment the architecture with an EDM-based Euler residual loss that injects biophysical structure into the generator. The key innovation of MultiODE-GAN lies in embedding these ODE-baser physiological constraints directly within the GAN training loop, allowing the model to enforce both the dynamics of individual leads and the physiological relationships between them.
We segment each ECG recording into individual cardiac cycles (heartbeats), represented fixed-length matrices h R 12 where L denotes the duration of single beat defined by the RR interval. Details of the segmentation process are described in Section 5
We adopt the Wasserstein GAN with gradient penalty (WGAN-GP) (Gulrajani et al. 2017 as the core adversarial loss formulation, chosen for its improved training stability and ability to avoid mode collapse compared to vanilla GANs, particularly beneficial in high-dimensional waveform domains.
The Wasserstein loss is defined as:
TWGAN Dw,, G) Enoppau (w (P
(7)
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Here, Pdata denotes the distribution of real ECG heartbeats, and Pz(2) represents the prior distribution of noise inputs. The discriminator Dw is trained not to classify inputs real or fake, but to estimate the Wasserstein distance between real and generated distributions. This formulation helps the generator G converge more reliably toward producing physiologically realistic ECG signals.
Backprop
Arrol
x(4)
Discriminator D.
W DMe
Lw
Total Generator
Loss
h
Generator
Zig
Condtricox
EDM Parama n p(n(c)
ECG Dynamical Model (ODE Solver)
Diagnots
Aeaan {
L
HUL-2
Figure The MultiODE-GAN generator architecture. takes random noise input to produce synthetic 12-lead ECG heartbeats. The generator's total loss combines the adversarial Wasserstein loss with two novel physiological constraints from the ECG Dynamical Model (EDM): (1) an intra-lead Euler loss enforcing alignment with the ODE- based biophysical model, and (2) inter-lead Euler loss ensuring consistency related leads.
4.2 MultiODE-GAN Generator
The MulGODE-GAN Generator enhances the WaveGAN architecture (Donalue et al. 2018 by integrating dynamics from the ECG model described in Section 3 While retaining the architectural foundation of WaveGAN, we extend the loss function to better capture the physiological accuracy required for realistic ECG signal generation.
Generator Loss Function: The generator's loss combines the Wasserstein distance with EDM-dcrived Euler Loss that enforces physiological consistency (Section 3 This Euler Loss quantifies the alignment between generated heartbeats and the dynamical model's output.
The first component of the Euler Loss quantifies how closely the temporal dynamics of each generated single lead adhere to the predictions of the EDM. For a generated heartbeat hy (for lead k),, and its corresponding EDM parameters Mh, this is defined as:
h4.441 he.e At
]
AEDM mp)
Ps (= We,e, hages lei 7e)
In this equation, hg denotes the generated heartbeat for lead k, ms includes the EDM parameters specific to lead k, and F.(-) represents the ODE from EDM (Equation 3 modeling the component (heartbeat trajectory). The variables ø and yk,, represent the trajectorics of the other two state variables, computed using the discrete Euler solution to the coupled ODEs for lead k. This intra-lead Euler Loss encourages each generated signal to remain consistent with its underlying physiological dynamics as defined by the EDM.
The intra-lead Euler Loss for the generator is defined as:
L BUL-1 (EUL)( Em-p(m)) p-p(()le)
AEDM (Ga nk)
(9)
Here, the expectation taken over noise inputs and EDM parameters The with ni sampled from class- conditional Gaussian p(n c) for lead and class c. The conditioning vector includes the diagnostic label and RR-interval duration.

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These parameters are precomputed by fitting the McSharry et al. (2033 model to real beats for each class and lead. This modeling ensures that each generated heartbeat not only exhibits realistic morphology, but also conforms to the underlying physiological dynamics prescribed by the EDM for its specific lead and condition. The Euler loss is fully differentiable, cnabling end-to-end backpropagation through the generator.
Incorporating Inter-Lead Dependencies: critical aspect of generating truly realistic 12-lead ECGs is not only accurate modeling of individual leads but also the correct representation of their physiological inter- dependencies. By leveraging standard ECG lead configurations and relationships derived from Einthoven's 's triangle and Goldberger's 's central terminal (Kenner & Sneyd 2009), our generator is further constrained to maintain these physiological relationships. These inter-lead dependencies are crucial for realistic multi-lcad ECG synthesis and clinical interpretability.
For instance, the fundamental relationships between the limb leads (1, 11, 111) and augmented limb leads (aVR, aLL, aVF) are:
1= I1-1111 III, 1
I1 I+ IIII, III), aVF= = -11+
III -
(10)
aVL = -1-111,, aVR - -1 (I+II),
To explicitly enforce these inter-lead dependencies, we introduce second component of the Euler Loss that constrains the temporal derivative of each generated dependent lead to match the weighted sum of the derivatives predicted by the EDM for its constituent leads. In practice, we approximate the time derivative of a dependent lead, such as Lead II, as the sum of the derivatives of its corresponding limb leads evaluated under their respective EDM parameters. For example, the inter-lead distance for Lead II is defined as:
(hI, TI, mun) h11.411 - hire (F2())) F.(:)))))
(11)
At
This formulation applies for all leads where such a linear combination exists, with coefficients (8, 7, etc.) determined by Equation 10 The parameters πΙ. MiI denote the EDM parameters associated with the constituent leads. This term encourages the generated signals to strictly adhere to physiological inter-lead relationships dictated by the underlying ODE model.
The inter-lead Euler Loss for the generator defined as:
LG
EUI-2 2 (Øc) = Empp(m), v>p(rle)
Σ
EDM ABDM(Gg(m) (G+(m), m)
(12)
where Ldependent denotes the set of leads that are linear combinations of others).
By leveraging inter-lead dependencies, we enforce physiological constraints within the generative model so that synthetic ECGs maintain correct relationships among leads. The generator is trained with both standard and ODF-bbsed losses, minimizing deviations from mathematically derived expectations.
The final Euler Loss is composite measure designed to guide the generator toward producing heartbeats that not only adhere to the temporal dynamics of individual leads (via L LEUI.-1) but also strictly respect the inter-lead dependencies (via LBUL-2):
L EUL (Øc) L (BUL-((GG) + (1 -1--0 8) -
L BUL-2
(13)
where å E [0, is hyperparameter that balances the contributions of the individual and inter-lead Euler Loss components (set to 0.6, see Exp. 6.2
The overall generator loss combines the WGAN term and the Fuler Loss:
To(da) EEzpp (G())) + \EUL. EUL (UL
(dG)-
(14)

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Figure illustrates the generator-discrimii loop and the two Euler-loss paths.
4.3 MultiODE-GAN Discriminator
The discriminator in the MultiODE-GAN framework retains the architecture and optimization strategy typical of WaveGAN. Its primary role is to differentiate between authentic 12-lead ECG heartbeats derived from real patients and those synthetically generated by the model. To accomplish this, the discriminator is designed to maximize the Wasserstein distance, thereby cnhancing its ability to identify discrepancies between the distributions of real and synthetic samples.
This formulation aligns with the standard WGAN discriminator loss as defined in Equation 7 optimizing the discriminator's ability to distinguish between data distributions effectively.
5
Experimental Framework
5.1
ECG Dataset
Our primary experiments use the Georgia 12-Lead ECG Challenge (G12EC) dataset (Alday et al. 2020 which comprises 10,344 12-lead ECG recordings from 7,871 patients in the southeastern United States. Fach recording is 10 seconds long, sampled at 500 Hz (5,000 time steps per lead), and may contain one or more of 27 non-mutually exclusive diagnoses, enabling comprehensive assessment of cardiac conditions.
In addition, we conduct ablation studies using the PTB-XL dataset (Wagner al., 2020), a widely recognized benchmark for ECG classification research. Both datasets are sampled at 500 Hz. Following established best practices in medical machine learning (Xu & Goodacre 2018), we perform patient-leve data splits, assigning 80% of patients to the combined training and validation sets and reserving the remaining 20% exclusively for testing.
5.2 ECG Classifier
To quantitatively evaluate the clinical utility and generative quality of the synthetic 12-load ECG data produced by MultiOEEGAAN, we assess its impact on downstream classification performance. Following standard methodologies in generative model evaluation for medical data (Golany et al. 2020 Alcaraz & Strodthoff 2023), ; compare baseline classifier trained exclusively real data against classifier trained combination of real and synthetic samples. Any significant drop in performance a held-out, unseen real test set would suggest potential distributional shift or lack of fidelity in the senthetic data, thereby serving as an indirect, yet critical, measure of generative quality.
For benchmarking MultiODE-GAN, we employ two state-of-the-art architectures widely adopted for 12-lead ECG classification:
This architecture is based on the robust ResNet model (Attia et al., 2019 Ribeiro et al. 2020), known for its effectiveness in medical signal processing. The specific implementation follows the design proposed in (Ribeiro et al. 2020). It commences with an initial convolutional layer, followed by batch normalization and ReLU activation. The core of the network consists of five residual blocks. Each block comprises three convolutional layers, interleaved with batch normalization, ReLU activation, and dropout layers, and incorporates identity skip connections to mitigate vanishing gradients. Temporal resolution is progressively downsampled via strided convolutions, applied from the sccond block onward, while the number of filters systematically increases across subsequent. blocks. A global average pooling layer aggregates temporal features before feeding into sigmoid-activated dense layer for multi-label binary classification.
ResNet Based:
Attention-Enhanced ResNet: This model incorporates ResNet backbone with a multi-head attention mechanism (Nejedly et al.) 2021). The integration of multi-head attention after the convolutional layers enables the model to effectively capture long-range temporal dependencics, which is particularly beneficial for interpreting multi-lead ECG signals. This architecture was among the top-performing models in the PhysioNet Challenge, underscoring its strong discriminative power.
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We train all classifiers using the Adam optimizer with learning rate 1 1x10=*, batch size 64, for up to 100 epochs with early stopping based on validation AUC. For each abnormality, a fixed target sensitivity is selected on the validation set and the corresponding threshold is then applied unchanged across all generative-mode augmentations when evaluating specificity on the test set.
5.3 Implementation Details
We segment each ECG recording into individual heartbeat cycles using NeuroKit2 (Makowski al., 2021) R-peaks are detected on Lead II, and all 12 leads are temporally aligned to these peaks to ensure cross-lead coherence. Each ECG x E 122x5000 is thus converted into set of heartbeat segments & E gl2xL, where L denotes the fixed beat length.
The segment length is fixed across all beats to ensure compatibility with the model architecture. In all experiments, we use a fixed length of L = 300 samples per heartbeat. Beats longer than are center-cropped, while shorter beats are zero-padded during preprocessing.
To avoid ambiguous beat-level labels, we restrict the main experiments to recordings in which the diagnosed abnormality is present throughout the entire 10-sccond signal. Consequently, all segmented heartbeats inherit the label of the original recording. Importantly, segmentation alion is performed after the patient-level data split, ensuring that no patient's recordings appear in more than one of the training, validation, or test sets, and theroby preventing data leakage.
6 Results and Comparative Analysis
Table 1: Classifier Performance Comparison Across Different Generative Models. Boldface indicates speci- ficity significantly higher than the Baseline CLS at p < 0.05 using a 5-fold cross-validated paired t-test.
Baseline CLS""
DCGAN
WaveGAN
ME-GAN
SimGAN
SSSD-ECG
MultiODE-GAN
Abuormality
Sensitivity
Specificity
Specificity
Specificity
Specificity
Specificity
Specificity
Specificity
IAVB
0.94
0.82 + 0.018
0.82 + 0.012
0.83 + 0.011
0.83 + 0.013
0.83 + 0.013
0.84 4 0.014
0.85 1 0.012
IRBBB
0.82
0.85 + 0.011
0.86 + 0.013
0.87 4 4.0111
0.86 + 0.012
0.87 + 0.014
0.87 + 0.013
0.89 1 0.013
RBBB
0.94
0.89
+ 0.013
0.89
+ 0.012
0.89 + 0.011
0.90 +: 0.010
0.90 + 0.012
0,90 + 0.012
0.92 + 0.011
LBBB
0.97
0.96
0.002
0.95
0.003
0.96 + 0.002
0.95 + 0.002
0.96 + 0.002
0.9K
0.002
0.96 + 0.003
NSIVCH
0.78
0.72 + 0.015
0.70 + 0.013
0.73 + 0.011
0.74 + 0.010
0.74 4 0.010
0.75
0.012
0.76 + 0.014
LAnFB
0.89
0.76 4: 0.006
0.76 + 0.005
0.77 + 0.006
0.77 - 0.006
0.77 + 0.007
0.77 + 0.006
0.80 + 0.008
LAD
0.89
0.88 +: 0.015
0.87 0.014
0.89 + 0.011
0.90 : 0.013
0.90 + 0.014
0.90 0.010
0.91 + 0.012
QAh
0.81
0.70
0.009
0.70 0.008
0.70 + 0.009
0.70 + 0.008
0.72 + 0.010
0.72 0.009
0.72 + 0.008
AFL
0.93
0.83 133:000 0.011
0.82 0.013
0.84 + 0.012
0.85 1 0.014
0.84 = 0.014
0.85 0.012
0.87 + 0.014
Baseline classifier follows Ribeiro al.
2020
We empirically evaluate MultiODE-GAN on 12-lead ECG classification tasks, comparing (1) baseline classifier trained exclusively on real ECG data and (2) classifiers trained on the same real data augmented with synthetic ECG samples from various generative models, including MultiODE-GAN.
6.1 Classification Performance and Generative Model Comparison:
Table summarizes the sensitivity and specificity metrics for ResNet-based ECG classifier (Ribeiro et al., 2020), evaluated across multiple cardiac conditions. In all experiments, set the classifier sensitivity to a predetermined threshold and compare the corresponding specificity. All performance metrics are assessed on a held-out test set composed entirely of unseen real samples, ensuring an unbiased evaluation of gen- eralization. Sensitivity and specificity are standard, clinically relevant metrics widely established for such
diagnostic classification tasks (Bresmaan et al. 2020 Wang et al. 2018 Golany et al. 2021).
To systematically evaluate the impact of different generative models on downstream classifier performance, we compare classifiers trained datasets augmented by synthetic data from the following methods:
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Baseline CLS (Real Data Only)
RRieirr et al. 2020 ResNet-based classifier trained only
real data.
DCGAN Radlord al.
2016):
Convolutional GAN adapted to 12-lead ECG waveforms.
Originally developed for audio waveform generation, WaveGAN captures temporal dependencies using convolutional structure, making it suitable for time-series data like ECGs.
WaveGAN Donahue et al.
2018
ME-GAN (Chen et al.
2022 A discaso-aware generative model that synthesizes multi-load ECGs
via 3D-aware panoptic representations to enforce cross-lead consistency.
This refers to our extended version of the original SimGAN frame- work. The original SimGAN was an EDM-guided GAN designed only for single-lead FCG generation and did not incorporate inter-lead dependencies. For this comparative analysis, our 'extended" Sim- GAN applics the single-Icad ODE-based loss independently cach of the 12 Ieads, without inter-lead relationships. This serves as a crucial baseline to demonstrate the necessity of our inter-lead con- straints.
SimGAN (Golany et al.
20200:
SSSD-ECG (Alcaraz & Strodthoff 2023): recent diffusion-based model specifically designed for multi-lead ECG generation, representing the current data-driven statc-of-tho-art,
MultiODE-GAN: Our proposed ODE-based generative adversarial network. EDM-guided GAN, incorporating both intra-lead temporal consistency and inter-lead physiological coherence.
Our results, as detailed in Table consistently demonstrate that augmenting training datasets with synthetic ECGs generated by MultiODE leads to significant improvements in specificity while robustly maintain- ing slightly improving sensitivity across nearly all evaluated abnormalities. These gains particularly pronounced in lower- preve-prev classes, where the limited availability of real examples often hinders the generalization capabilities of baseline models. For highly prevalent conditions, such as Left Bundle Branch Block (LBBB), where baseline models already achieve over 96% in both sensitivity and specificity, the room for further improvement is inherently limited. Sensitivity fixed via validation-set threshold selection.
Crucially, MultiODE-GAN GAN consistently outperforms all other generative models, including SSSD-ECG (AI- caraz & Strodthoff] 2023), which represents the current diffusion-based state-of-the-art, While diffuusion mod- els excel at capturing complex data distributions, their purely data-driven nature often lacks explicit mech- anisms to enforce underlying physical physiological laws. In contrast, our results show that MultiODE- GAN, by directly integrating biophysical ODE constraints and enforcing inter-lead physiological relation- ships, achieves performance that is not only comparable to, but often superior to, this strong diffusson baseline.
In all experiments, we augmented the real training data with synthetic samples at a fixed ratio of N real samples to 2N synthetic samples. This ratio was kept constant across all generative models.
Statistical Significance We performed 5-fold cross-vallis validatiin and computed specificity per fold. For each abnormality, we conducted a paired t-test comparing the performance of the bascline classifier against the classifiers augmented with synthetic data. All p-values were < 0.05, indicating that the improvements of MultiODE-GAN over the Baseline CLS are statistically significant.
6.2 Ablation Studies
We next perform ablation studies to quantify the contribution of key components of MulGODE-GAN to classifier performance. All experiments were conducted while maintaining a constant target sensitivity across different conditions, allowing us to preciscly quantify the changes in specificity achieved by cach modification. Additional experiments including quantitative analysis of inter-lead dependencies, the effect of the number of generated samples, and real-world evaluation are provided in Appendix A and Appendix B
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Impact of Base Classifier Architectures This experiment evaluates how different base classifier archi- tectures affect performance when trained with synthetic data generated by MultiODEGAAN We specifically examine the performance impact when augmenting the training data for an alternative classifier, as detailed by (Nejedly et al., 2021) from the PhysioNet/Coi /CnC Challenge. This classifier incorporates a ResNet. back- bone enhanced with multi-head attention mechanism, known for its ability to capture long-range temporal dependencies. Table 2 presents the results, showing significant improvement in the classifier's specificity when synthetic data generated by MultiODE-GAi is included in the training process.
This improvement underscores the value of our generated synthetic data in enhancing the robustness and performance of 12-lead ECG classifiers. By providing diverse and high-quality training samples, our approach helps classifiers generalize better, especially in scenarios where real-world data is scarce or imbalanced.
Table 2: Performance comparison Seanaittvitt/Spccffiiii of an attention-enhanced ResNet classifier Ne- jedly et al.) (2211) trained with and without auginentation by MultiODE- GAN synthetic data on the G12EC test set. Bold indicates statistically significant improvement.
Alt-CLS*
+MultiODE-GAN
Abuormality
Sensitivity
Specihcity
Sensitivity
Specilicity
LAVB
0.93
0.85
0.93
0.89
IRBBB
0.81
0.84
0.81
0.88
RIBB
0.93
0.90
0.93
0.92
LBBB
0.96
0.96
0,96
0.97
NSIVCB
0.80
0.73
0.80
0.77
LAnFB
0.90
0.76
0.90
0.80
LAD
0.91
0.87
0.91
0.90
QAb
0.83
0.72
0.83
0.75
AFL
0.92
0.82
0.92
0.85
Alternative Classifier architecture from Nejedly et al. (2021).
Impact of Lead-Dependent Loss This experiment investigates the influence of the novel (cad-dependent Euler loss function on both the quality of synthetic ECG data and the performance of downstream classifiers. As defined in Section 4 the Euler loss consists of complementary intra- and inter-lead components, LG EUIL-1 and LEUL-2 The objective of LEUL-1 is to ensure that the temporal dynamics of each generated lead closely follow its corresponding real lead according to the physiological dynamical model, thereby maintaining alignment with the model's equations. Conversely, T.EUL-2 captures the crucial inter-lead dependencies, ensuring that each generated lead accurately reflects its physiological relationships with other leads.
Specificity for Different 5
LAVB
0.95
IRHBB
RBDB
0.9
LBBB
Specificity
NSIVCB
0.85
LAnFB
LAD
0.8
API.
0.75
0.7
0
0.2
0.1
0.6
0.8
Figure 2: Classifier specificity as a function of the 8 weighting parameter in the composite Euler Loss (Eq. 13 The optimal value at 0.6 demonstrates the necessity of balancing individual lead temporal accuracy with inter-lead physiological consistency.
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We varied the weighting parameter 8 [0, 1], where å = 1 uses only LG L and 8 0 uses only LG EUL-2 As shown in Figure 2 the optimal performance occurs at 8 = 0.6, highlighting the critical importance of balancing individual lead accuracy with comprehensive inter-lead consistency. Notably, performance signif- icantly degrades when either component is omitted (i.c., = 0 or = 1), underscoring the complementary and essential role of both terms producing high-fidelity, physiologically realistic 12-lead ECGs.
Evaluation on the PTB-XL Dataset To assess the generalizability of MultiODE-GAN, we extended our analysis to the PTB-XL dataset (Wagner et al. 2020), well-known large-scale benchmark comprising 21,799 12-lead ECGs from 18,869 patients. Each 10-second recording includes rich metadata and diagnostic labels.
Following the experimental setup in Section 5 we compared a baseline classifier trained exclusively on real PTB-XL data with one augmented using synthetic ECGs generated by MultiODE-GAN, As shown in Table incorporating synthetic data led consistent gains classification specificity, particularly for rare complex cardiac conditions.
Table 3: PTB-XL- Classifier performance across multiple evaluation metrics. Results are shown for the baseline classifier trained on real data only and with augmentation by NultiODE-GAN synthetic data. Boldface indicates statistically significant improvement over the Baseline CLS.
Baseline CLS*
+ MutIODE-GAN
Abnormality
Seus.
Spec.
AUROC
AUPRC
Seas.
Spec.
AUROC
AUPRC
0.92
0.83
0.897
0.92
0.85
IRBBB
0.91
0.86
0.938
0.652
0.90
0.966
0.694
LBBB
0.97
0.97
0.994
0,906
0.97
0.97
0.983
0.907
NSIVCB
0.81
0.71
0.750
0.181
0.81
0.76
0.786
0.229
LAnFB
0.96
0.87
0.974
0.800
0.96
0.89
0.982
0.836
LAD
0.93
0.87
0.936
0.827
0.93
0.88
0.944
0.851
QAb
0.78
0.70
0.814
0.204
0.78
0.73
0.839
0.241
AFL
0.89
0.78
0.907
0.412
0.89
0.83
0,933
0.459
Baseline classifier follows Ribeiro et. al.
2020
6.3 Quantitative Analysis of Inter-Lead Dependencies
To quantitatively validate the ability of MultiODE-GAN to capture and enforce physiological inter-lead dependencies, we analyze the consistency between generated leads and their expected lincar combinations. Specifically, we compute the Mean Squared Error (MSE) between a generated dependent lead (e.g., Lead I) and the sum of its physiologically constituent leads (e-g., Lead IT Lead IIT), defined by Finthoven's law. This metric directly assesses how well the generated signals adhere to fundamental anatomical relationships.
Table presents the average MSE values for key dependent leads generated by MultiODE-GAN compared to baseline generative models. Lower MSE values indicate a stronger adherence to physiological inter-lead relationships. MultiODE-GAN consistently achieves significantly lower inter-lead MSE compared to other models, providing quantitative evidence that its EUT-2 component effectively guides the generator to produce physiologically coherent multi-lead ECGs.
Table 4: Mean Squared Error (MSF) for inter-lead dependencies for heartbeats generated by different models. Lower MSE indicates better physiological consistency.
Lead
Lead
Lead
aVR
a
aVF 0.5x (II+IIII
Model
(II-III)
(I+III)
(II-I)
0.5x (I+II)
0I5x(I-III
DOGAN
0.085
0.091
0.088
0.079
0.090
0.088
WaveGAN
0.078
0.083
0.080
0.071
0.083
0.080
ME-CAN
0.062
0.069
0.066
0.058
0.068
0.065
SSSD-ECG
0.055
0.060
0.058
0.050
0.060
0.057
MulHODE-GAN
0.041
0.044
0.042
0.038
0.051
0.046
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6.4 Additional Evaluation
Distributional Alignment. In addition lo classification accuracy, quantitatively assess how closely MultiODE-GAN approximates the real data distribution. We compute the Kullback Leibler (KL) divergence and Maximum Mean Discrepancy (MMD) between real and synthctic ECGs. To provide a clinically relevant comparison, these metrics are calculated in a learned feature space derived from a pre-trained ResNet-based encoder. Lower values for both KT. divergence and MMD indicate better alignment between the real and generated data distributions. MultiODE-GAN achieves consistently low KL (< 0.5) and MMD (< 0.01) scores across all abnormalities, underscoring its ability to generate realistic, high-fidelity data that closely mirror the statistical properties of real FCGs.
Qualitative Results Figure 3 provides a visual illustration of the high fidelity of ECG heartbeats generated by MultiODE-GAN, The figure displays synthetic samples for Leads II and V1, meticulously overlaid with corresponding real ECG signals.
Real Lead II
MultODE GAN
1.5
Voltage (mV)
Vottage
-1.0
0.0
-1.5
-0.5
Real Lead V1 MUtIDDE-GA Lead
-2.0
-1.0
Figure Representative synthetic ECG heartbeats generated by MultiDDE-GAN (orange) overlaid with real ECG heartbeats (blue) for Leads II and V1.
Evaluation Metrics. In addition to sensitivity and specificity at fixed operating points, we report threshold- independent metrics to provide more comprehensive assessment of downstream classificr performance. Specifically, we evaluate the area under the receiver operating characteristic curve (AUROC) and the area under the precision-ree recall curve (AUPRC), which are standard metrics for imbalanced medical classification tasks.
Table 5: Classifier performance comparison across different generative models on the G12EC datasel, eval- uated using AUROC and AUPRC.
Abuormality
Raseline CLS*
DOGAN
WaveGAN
ME-GAN
SimGAN
SSSD-EOG
MulCODE-GAN
AUROC/ / AUPRC
AUROC AUPRC
AUROC
AUFRC
AUROC/ AUPRC
AUROC
AUPRC
AUROC 7 AUFRC
AUROC / ACPEC
LAVB
0.21 0.475
0.23 / 0,187
0,228
0.501
11129 0.M8
0.33 7 0.30
0.331 0.521
0,941
0.338
IRDBD
0.902 / 0.388
0.988 / 0.40L
0.92
0.414
0.910 0.000
0.914 7 0.418
03916 0.423
0.928
0.481
RBBB
0.973 0.749
0.933 / 0,751
0.974
0.758
0,975 0.764
0.75 / 0,766
0.974 0.772
0,982 0.801
LHB3
0.880 0.851
0.988 7 0.848
0.589/
0.853
0.888 0.350
0.980 7 0.855
0.989 0.856
0.980 0.859
NSIVCB
0.831 0.140
0.838 / 0.149
0.842/1
0.158
0.810 0.166
0.841 / 0.162
0.848 0.109
0.856
0.195
LAnFB
0.921 0.298
0.922 / 0.273
0.925
0.281
0.926 / 0.283
0.926 / 0.285
0.927 0.287
0.938
0.342
LAD
0.944 0.938
0.943 / 0.641
0.946
0.653
0.948 0.861
0348 / 0.683
0.949/ 0.668
0.958
0.691
QAB
0.725 0.181
0.726 / 0.181
0.728
0.188
0,729 0.190
0.733 / 0.195
0.752 0.197
0.734 0.205
API.
0.866 0.303
0.855 / 0.306
0.858
0.316
0.861
0.321
0.860 / 0.310
0.863 0.326
0.8R1
0.377
Bascline classifier follows Ribeiro et al. (020)
7 Limitations
While MultiODE-GAN demonstrates significant advancements in piysiologically-infoeee ECG synthesis, certain limitations and avenues for future research exist. Integrating ODEs within the GAN training loop is computationally intensive, requiring further optimization for large-scale or real-time applications. Moreover, pro-computation of EDM parameters necessitates some real labeled data, posing challenge for generating entirely novel cardiac conditions. For abnormalities where baseline classifiers already achieve near-perfect performance (e.g., LBBB), the incremental gains from synthetic data augmentation naturally constrained.
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8 Conclusions
In this paper, we introduced MultiODE-GAN, novel generative adversarial framework for synthesizing highly realistic 12-lead ECG signals by incorporating domain-specifee knowledge from the ECG Dynamical Model. Our approach leverages novel Euler Loss to enforce both intra-lead temporal dynamics and critical inter-lead physiological dependencios, yiclding clinically plausible and structurally coherent ECG waveforms.
Our empirical results show that MultiODE-AA consistently perforrmms conventional generative methods, including late-of-thhearr diffusion models. This superior generative quality translates into significant gains in downstream classification performance, particularly in specificity for lower-prevalence cardiac abnormalitics especially in lower-prevaeence classes. Furthermore, we validated MultiODEGGAN's clinical utility in real- world setting, showing its potential for early detection of left ventricular systolic dysfunction (LVSD) by identifying patients potentially in need of further echocardiographic evaluation.
MultiODE- GAN holds significant promise for advancing cardiac diagnostics and research. Future work will explore diffusion-guided ODEs to combine the strengths of both generative paradigms for even higher fidelity and diversity. We also plan to extend its application to a broader spectrum of cardiac conditions and explore its utility in personalized analysis to further support clinical decision decision-making and improve patient outcomes.
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A Additional Ablation Studies
A.1 Generated Samples Number
Specificity for Different. Sample Sizes
IAVB
0.95
IRBBB
RBBB
0.9
LBBB
Specifiitt
NSIVCB
0.85
LAnFB
LAD
QAb
0.8
AFL
0.75
0.7
0
0.2N
0.5N
N
1.5N
2N
Sample Size
Figure 4: Classifier specificity across different synthetic sample sizes (0.2N, 0.5N N, 1.5N, 2N) for each abnormality class. Results show synthetic data improves specificity, with peak gains at optimal sample sizes.
17

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Published in Transactions on Machine Learning Research (02/2026)
This experiment explores how the quantity of synthetic data samples generated by the MultiODE-GAN model affects classifier performance. We analyze the effect of varying synthetic sample sizes on classifier specificity and generalization capabilities. Specifically, for each class with N existing samples, we generate illustrates that adding synthotic samples
synthetic datasets of sizes 0.2N, 0.5N, N, 1.5N, and 2N. Figure ! generally improves specificity, with certain sample sizes yielding the maximum improvement, while sensitivity remains constant.
Figure 4 illustrates that adding synthetic samples generally improves specificity, with certain sample sizes yielding the maximum improvement, while sensitivity remains constant. This analysis provides practical guidance on the optimal amount of synthetic data to augment real datasets for maximal performance gains.
B Real-World Evaluation: LVSD Prediction in Clinical Practice
To evaluate the practical utility of MultiODE-GAN in real-world clinical context, we conducted exper- imental study in collaboration with the Interventional Cardiology Unit, focusing the early detection of left ventricular systolic dysfunction (LVSD) directly from standard 12-lead ECG data.
Early detection of left ventricular systolic dysfunction (LVSD), a precursor to heart failure, is crucial for timely intervention. Current clinical practice relies on echocardiography to estimate ejection fraction (EF), with IVSD defined as EF < 50% (McDonagh et al.; F.jec.Finociion fraction is a key measurement used assess the heart's pumping efficiency, representing the percentage of blood ejected from the left ventricle. However, cchocardiograply is costly, requires specialized expertisc, and is not always accessible in primary care settings (Rutten et al., 2003 Hobbs et al.) 2000).. In contrast, 12-lead ECG is ubiquitous, non-invasive, and cost-effective making it an ideal alternative for initial LVSD screening.
B.1 Clinical Dataset
We used dataset of 13,820 paired ECG and echocardiogram records, with one ECC-echoocrddogrraply pair per patient. The data was partitioned into training (10,402), validation (2,568), and test (854) sets. The test set was balanced to include an equal proportion of normal and abnormal LV function cases, with abnormal IV function defined as EF < 50%. Each ECG was resampled to 500 Hz and formatted as a 12 x 5000 matrix (12 leads, 10 seconds). Short signals were zero-padded, longer signals truncated.
Ethics Statement: All data collection were approved by the institutional IRB the participating medical center, with informed consent obtained and all data de-identified prior to analysis.
B.2 Model Architecture
Our ResNet-based classifier, adapted from (Attia et al. 2019 Ribeiro et al. 2020 , receive a 12 x 5000 ECG matrix as input. The architecture begins with convolutional layers, followed by batch normalization and ReLU activation. It then passes through five residual blocks, each comprising three convolutional layers with batch normalization, ReLU, dropout, and skip connections to promote stable gradient flow.
After the final residual block, a global average pooling layer aggregates the learned features, which are passed to a dense layer with sigmoid activation for binary classification (EF < 50% vs. EF > 50%). The model was trained using the Adam optimizer with an initial learning rate of 0.0001, minimizing binary cross-entropy loss. We conducted iterative hyperparameter tuning across optimizer types (Adam, RMSProp, SGD), learning rates (0.01, 0.001, 0.0001), dropout rates, and residual block depths (3 7), selecting the best configuration based on validation accuracy. Training was performed for up to 100 epochs with early stopping based on validation performance
B.3 Clinical Evaluation and Impact of Synthetic Data
Six cardiologists independently annotated the 854-patient test set, inchuding three senior cardiologists (12 12-20+ years experience). Table T summarizes performance comparisons.
18

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Published in Transactions on Machine Learning Research (02/2026)
To assess the contribution of synthetic ECGs generated by MultiODE-GAN, retrained the model using combined dataset of real and synthetic signals, Synthetic ECGs were used exclusively during training for data augmentation. This augmentation improved specificity from 78% to 81% while maintaining sensitivity, particularly benefiting class balance and minority case coverage.
The ResNet classifier outperformed average physician accuracy and matched senior cardiologists sensitivity (78%). Augmenting with AlutiODE-GGNN synthetic ECGs further improved specificity to 81%, surpassing all human benchmarks.
Table 6: LNSD detection EF MultiODE-GAN data.
50%):
Comparison of physician and model performance with without
Sensitivity
Specificity
Average Physician
0.71
0.68
Senior Physician (avg.)
0.77
0.64
RosNet Classificr
0.78
0.78
RosNet + MultiODE-GAN
0.78
0.81
19


Reviews:
[{"review_id": "M26PSpVN6x", "reviewer": "Reviewer_p5B2", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper proposes MultiODE-GAN, a GAN-based model for synthesizing realistic 12-lead ECG signals. The central technical idea is to incorporate an ECG Dynamical Model (EDM) based on ordinary differential equations (ODEs) into the training objective via an Euler-based loss, with the goal of improving physiological plausibility and inter-lead consistency of generated ECGs. The authors further evaluate whether augmenting the training set with synthetic ECGs improves a downstream ResNet-based classifier for cardiac abnormality detection, reporting changes in sensitivity and specificity relative to training on real data only.\n\nStrengths:\n1. The topic is practically relevant: high-quality synthetic 12-lead ECG generation has clear value for data sharing, privacy-preserving research, and mitigating data scarcity/imbalance in clinical ML settings.\n2. The use of a downstream augmentation experiment (training a classifier with added synthetic samples) is a reasonable first step toward assessing utility beyond visual inspection, and it helps connect the generative task to an application that many readers care about.\n3. The paper is generally well written and easy to follow. The problem setup, method description, and experimental flow are presented coherently.\n\nWeaknesses:\n1. The choice of GAN as the base generative model is not sufficiently justified, and the paper does not clearly position this design choice relative to alternative generative paradigms.\n2. The experimental evidence primarily relies on downstream classification performance, which does not directly validate the paper’s key claim about improved inter-lead physiological consistency.\n3. Reporting is missing critical details (e.g., the quantity and proportion of synthetic data used), limiting interpretability and reproducibility of the stated gains.", "claims_and_evidence_yn": "No", "claims_explanation": "The paper’s main claim is that the proposed training objective enforces or improves inter-lead dependencies / physiological constraints in generated 12-lead ECGs. However, the presented experiments largely evaluate only a downstream classifier trained with augmented data. This is not a direct test of whether the generated ECGs satisfy the intended physiological or inter-lead constraints, and therefore leaves a gap between the central claim and the evidence.  \n\nIn addition, the reported improvements appear to be concentrated in specificity (as referenced in Tables 1–3), while sensitivity does not consistently improve. This makes it difficult to conclude that the synthetic data is broadly beneficial or that the method reliably improves clinically meaningful detection performance.\n\nFinally, key experimental details are unclear or missing, including (1) how many synthetic samples are added, and the synthetic-to-real ratio in training, and (2) whether the augmentation benefit is robust across multiple ratios, random seeds, or patient splits. Without these details, it is hard to assess whether the improvement is attributable to the proposed modeling contribution, or to a particular augmentation setting.", "audience_yn": "Yes", "audience_explanation": "The general direction about introducing physical information into generative modeling for biomedical time series should interest a subset of the TMLR readership, particularly those working on structured generative models, scientific ML, and healthcare signals.\n\nHowever, the paper’s current framing and evaluation limit the broader appeal. The work is built around a GAN backbone without clearly positioning this choice against more recent and widely adopted generative approaches, nor providing competitive comparisons that would help readers understand the practical relevance of the proposed contribution in today’s generative modeling landscape. It remains unclear whether the proposed “condition” integration is a general, transferable idea that would apply naturally to other modern generative backbones. Strengthening these aspects would make the paper’s findings more actionable and more compelling to a wider portion of the audience.", "requested_changes": "1. A critical improvement would be to provide direct evidence that the generated 12-lead ECGs satisfy the claimed physiological and cross-lead constraints, beyond showing downstream classifier performance. This could include quantitative analyses of inter-lead relationships on real versus synthetic ECGs, and/or Expert/clinician assessment of realism and cross-lead consistency (even a limited blinded study can be informative), or an established ECG quality protocol.\n\n2. Several additions would strengthen the work. The current presentation does not clearly specify how many synthetic samples are added, what synthetic-to-real ratio is used, and how sensitive the results are to these choices. Reporting the augmentation ratio(s), providing a small sweep over ratios, and demonstrating robustness across multiple random seeds and patient-level splits would make the reported downstream gains substantially easier to interpret.\n\n3. The empirical evaluation would benefit from a more complete set of downstream metrics (e.g., accuracy, AUROC/AUPRC, and per-class breakdown where relevant) alongside sensitivity/specificity, as well as generation-focused assessments that are appropriate for ECG time series.\n\n4. To make the contribution more actionable to the broader community, it would strengthen the paper to include a clearer justification for the choice of GAN as the generative backbone, considering other modern generative architectures. This would help readers understand the generality and relevance of the approach.", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "rvM8yeoevw", "reviewer": "Reviewer_T97b", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper proposes MultiODE-GAN, a multi-lead ECG generative model that combines a WaveGAN backbone with explicit ODE-based physiological constraints in the loss term derived from the ECG Dynamical Model. The central novel idea is an Euler-residual loss that (i) enforces intra-lead temporal dynamics to match EDM predicted derivatives, and (ii) enforces inter-lead physiological relationships using standard limb-lead dependencies via a derivative-based constraint. The authors evaluate their synthetic ECGs empirically through downstream utility: training ECG classifiers with real data plus synthetic augmentations on a held-out test set of real samples. They report consistent gains in specificity on three datasets compared to multiple baseline.\n\n## Strengths\n* Clear, clinically motivated constraint: explicitly enforcing both intra-lead ODE dynamics and inter-lead consistency is well aligned with the known challenges of 12-lead synthesis.\n* Broad baseline comparison + ablations: includes multiple generative baselines and ablations, and also checks an alternative classifier architecture.\n* Reproducibility: code is open-source.\n## Weaknesses\n* Writing quality is poor: the narrative flow is often hard to follow; several sentences are grammatically problematic, and key evaluation details are underspecified (details below).\n* Class-conditioned ODEs scale poorly: the authors discussed this in its limitation already, but I think it is worth pointing out here that this limits the practical application of this work.", "claims_and_evidence_yn": "No", "claims_explanation": "## Supported\n* Ablation evidence that both loss terms matter\n* Generalization beyond one classifier: Table 2 indicates gains also hold for an attention-enhanced ResNet classifier.\n\n## Withholding\n* Due to the lack of clarity in several aspects of the experimental protocol, I provisionally assume that the empirical evaluations are correctly implemented and that the reported results support the stated empirical conclusions. For instance, I need to know how specificity is calculated to evaluate if its improvements are fair.\n\n* The statement “This enables the model to capture the complete spatio-temporal structure of the 12-lead ECG for the first time, producing substantially more realistic and clinically coherent synthetic signals” constitutes a very strong novelty claim. However, the proposed method does not model volume conductor–based spatial effects, and prior work has already explored multi-channel ECG generation (albeit without explicit ODE constraints). In addition, the related work section does not sufficiently situate the method within this existing literature.", "audience_yn": "Yes", "audience_explanation": "it is of clinical relevance for ECG classification", "requested_changes": "A. Writing quality and organization\n\nThe manuscript would benefit from improvements in writing quality and overall organization. There are multiple instances of missing punctuation, extra spaces, and repetitive content (e.g., Section 5.3). The presentation frequently jumps back and forth between datasets and experimental procedures, and in several places concepts or data are introduced before their purpose is clearly explained. To improve readability, I suggest adding a brief overview at the beginning of each major section and revisiting the overall narrative flow of the paper.\n\nB. Clarifications needed\n\n* Inter-lead assumption (Eq. 10): Are all 12 leads modeled explicitly, or are only the 6 limb leads used to enforce inter-lead constraints?\n* Segmentation details and signal length: Is the segment length L fixed across beats, or does it vary? If fixed, please specify the chosen value and how it is determined. If variable, how does the WaveGAN architecture support variable-length signal generation?\n* Specificity computation and “predetermined threshold.”: Please clarify how specificity is computed. In particular, what target sensitivity is used, and how is the corresponding decision threshold selected?\n* Cross-validation and statistical testing: Please clarify the cross-validation protocol: is cross-validation applied to both GAN training and classifier training? In the statistical analysis, is n=5, and are the standard deviations reported in Table 1 computed across cross-validation folds?\n* Use of PTB-XL: Is the synthetic data for PTB-XL generated using a GAN trained on G12EC or a GAN trained directly on PTB-XL?", "broader_impact_concerns": "na", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "vhT9FXfBXD", "reviewer": "Reviewer_iw63", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper presents a method for 12-lead electrocardiogram (ECG) synthesis. Their method, MultiODE-GAN, is a generative adversarial network (GAN) based off WaveGAN that incorporates ordinary differential equations (ODEs) that model electrophysiological dynamics to enforce physiologically plausible ECG generation. When using MultiODE-GAN to synthetically augment ECG classification datasets, experiments show that the proposed method outperforms other GANS with respect to diagnostic specificity.\n\n**Strengths**:\n- The paper addresses an important problem, namely the lack of abundant labeled medical (specifically, ECG) data.\n- Directly enforcing physiological plausibility through ODE-based losses is clever.\n- Experiments are relatively thorough, comparing to several relevant baseline approaches.\n\n**Weaknesses**:\n- Experimental details are unclear. The authors “compare a baseline classifier trained exclusively on real data against a classifier trained on a combination of real and synthetic samples.” What combination? What is the ratio of real to synthetic samples? Why is only one ratio considered?\n- Evaluation is limited to synthetic data augmentation. There are many other ways to assess the utility of generated ECG signals—see, e.g., [1] for a more comprehensive evaluation protocol to assess diagnostic utility. It would be interesting to see how classifiers perform when trained on the entire spectrum from *only real data* to *only synthetic data*. Some of the results in the appendix, assessing physiological consistency, might benefit from being moved to the main text.\n- Evaluation is limited to specificity for reasons that are unclear. More metrics should be considered and presented. Sensitivity is not shown in Table 1 and, for reasons I do not understand, never changes throughout Tables 2 and 3.\n- More non-GAN baselines catered to ECG generation could be included for comparison (see [1] for examples).\n- Presentation quality could be improved. Many paragraphs consist of only 2-3 sentences and do not necessarily logically flow from one to the next. The use of bold text, subsections, or other formatting could help ease the reading experience.\n\n**References**\n\n[1] Núñez, José Fernando, Jamie Arjona, and Javier Béjar. \"Synthetic ECG Generation for Data Augmentation and Transfer Learning in Arrhythmia Classification.\" arXiv preprint arXiv:2411.18456 (2024).", "claims_and_evidence_yn": "No", "claims_explanation": "Experiments appear to be sound but are limited in terms of evaluation metrics, baseline methods, and evaluation settings. Evaluation in the main text is limited to specificity of ECG classifiers with synthetically augmented datasets when compared to several GAN-based methods. It seems that this experiment was only conducted with one fixed ratio/amount of synthetic data, which I could not find stated in the paper.", "audience_yn": "Yes", "audience_explanation": "ECG is a popular modality for biomedical applications of AI, and generative models for ECG synthesis have been well-studied.", "requested_changes": "**Main feedback**\n- Why is evaluation limited to specificity only?\n- Why does the sensitivity not change whatsoever (Tables 2-3)?\n- If possible, include other non-GAN baseline methods for ECG generation\n- I would suggest moving some of the physiological consistency experiments into the main text\n- Tidy up text-only sections like the Introduction so that they are more streamlined, where each paragraph is substantial and serves one purpose that links the previous paragraph to the next\n- Experimental details need to be clarified. What percentage/# samples are synthetic?\n- Experimental validation should be expanded to consider a range of augmentation (0% synthetic, 10% synthetic, …, 50% synthetic, … 100% synthetic or similar)\n- In-text citations are mostly wrong. Need the parenthetical version with “\\citep”\n- Out of curiosity, is the method deterministic? I.e., given the same conditioning information, will the method always produce the same output ECG?\n\n**Minor feedback**\n- Abstract: First 2 sentences can be improved. This should make clear what is actually motivating the need for synthesis in the first place—presumably the lack of abundant labeled data?\n- If “acquiring multi-lead ECG data at scale requires physician review”, why wouldn’t synthetically generating ECGs require the same (or even stricter) review?\n- Under eq 13: Extra space in “Exp.   6.2” can be fixed with “.\\” \n- Sec 5.1: Incorrect spacing in “We utilize the … Challenge (G12EC)dataset” … diagnoses) ,we also…” \n- Sec 5.3: How do you “restrict … main experiments to recordings where the diagnosed abnormality is present throughout the 10-second recording”?", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)