simgans: simulator-based generative adversarial networks

SimGANs: Simulator-Based Generative Adversarial Networksfor ECG Synthesis to Improve Deep ECG Classification

Tomer Golany 1 Daniel Freedman 2 Kira Radinsky 1

Abstract

Generating training examples for supervised tasksis a long sought after goal in AI. We studythe problem of heart signal electrocardiogram(ECG) synthesis for improved heartbeat classi-fication. ECG synthesis is challenging: the gen-eration of training examples for such biological-physiological systems is not straightforward, dueto their dynamic nature in which the various partsof the system interact in complex ways. However,an understanding of these dynamics has been de-veloped for years in the form of mathematicalprocess simulators. We study how to incorpo-rate this knowledge into the generative process byleveraging a biological simulator for the task ofECG classification. Specifically, we use a systemof ordinary differential equations (ODE) repre-senting heart dynamics, and incorporate this ODEsystem into the optimization process of a gener-ative adversarial network to create biologicallyplausible ECG training examples. We performempirical evaluation and show that heart simu-lation knowledge during the generation processimproves ECG classification.

1. IntroductionIn the early days of AI research, it was claimed that “thepower of an intelligent program to perform its task welldepends primarily on the quantity and quality of knowledgeit has about that task” (Buchanan & Feigenbaum, 1982).Indeed, in the last two decades it has been shown repeat-edly that access to the wealth of knowledge possessed byhumans significantly improves machine learning algorithms

1Technion - Israel Institute of Technology, Haifa, Is-rael 2Google Research. Correspondence to: Tomer Golany<[email protected]>, Kira Radinsky <[email protected]>, Daniel Freedman <[email protected]>.

Proceedings of the 37 th International Conference on MachineLearning, Online, PMLR 119, 2020. Copyright 2020 by the au-thor(s).

on various tasks (Gabrilovich & Markovitch, 2009; Zhanget al., 2016). Knowledge has been introduced into machine-learning algorithms in a host of different ways: by enrichingthe features (Gabrilovich & Markovitch, 2009), by reducingthe labeling work (Mintz et al., 2009), and by encoding con-ditional dependencies (Wainwright & Jordan, 2008). How-ever, it is less straightforward to introduce prior knowledgeabout biological and physiological systems. This is at leastpartially due to the fact that the complexity of biologicalsystems stems from their dynamic nature. Each part of thesystem interacts and behaves collectively to produce distinctphysiologic states and functionality. To fully understandthe system, one needs to model its dynamics. Consider theproblem of detecting cardiac abnormalities from an electro-cardiogram (ECG) which measures the electrical activitygenerated by the heart. In recent years, deep learning algo-rithms have been applied for the problem, yielding state-of-the-art results (Kachuee et al., 2018; Al Rahhal et al., 2016).While the performance of such algorithms is quite promis-ing, they are purely data-driven; they do not make use of thefact that at a fundamental level the interpretation of an ECGsignal involves the understanding of the electrical conduc-tion system of the heart. Intuitively, it seems reasonable thatknowledge about the physics of the heart’s electrical andmechanical activity might improve the performance of deeplearning models. This knowledge has been developed foryears in the form of mathematical process simulators. Thechallenge then becomes how to incorporate this knowledgeinto machine learning models. In this work we tackle thischallenge directly, and propose a technique which allowsus to leverage a biological simulator for the task of ECGheartbeat classification.

A simulator is a model of a phenomenon or a process cre-ated by a researcher to imitate the operation of a processor system (Banks et al., 2000). Simulation is often usedto model natural systems to better understand their behav-ior. By changing variables in the simulation, one can makepredictions about the behavior of the system over time. Sim-ulators have been developed in a variety of fields, includ-ing chemistry, biology, and physics. Developing such amodel requires deep understanding of how the system isconstructed and operates. For example, a heart simulatorrequires understanding of the heart mechanics and anatomy.

SimGANs: Simulator-Based Generative Adversarial Networks for ECG Synthesis to Improve Deep ECG Classification

Such a model can be used to mimic the pressure-volume re-lationship and simulate various conditions and pathologies.

It is common to model systems via a mathematical model,which aims to predict the system behavior given knowledgeabout its nature. We focus in this work on continuous sim-ulation, where time evolves continuously. In this setting,simulation is usually performed by numerical integrationof Ordinary Differential Equations (ODE), that capture theresearcher’s knowledge regarding the physical or biologicalnature of the problem. More specifically, since the ODEscannot generally be solved in an analytic manner, numericalapproximation procedures such as Runge-Kutta are usedto solve the equations. In this work, we leverage an ECGsimulator proposed by (McSharry et al., 2003); the modelsimulates the electrical and mechanical activity of the heartby a set of ODEs. We explore the direct integration of theseODEs into deep learning models for the purpose of furtherimproving the models, which have already reached state-of-the-art results for the task of ECG classification (Kachueeet al., 2018).

Recently, it has been shown that generating synthetic ECGsignals of different arrhythmias to train deep learning mod-els significantly improves the performance of classificationmodels in personalized settings (Golany & Radinsky, 2019).We leverage this architecture to integrate simulator knowl-edge during the generation process. We hypothesize thatthis formulation enables generation of ECGs with bettermorphological characteristic of the different arrhythmias,thus improving classification. We present empirical resultson gold-standard ECG datasets and show the superiority ofalgorithms working with simulator knowledge in the fieldof ECG heartbeat classification.

Our contribution in this work is threefold: (1) We present amethodology to leverage heart simulator ODEs to improveinstance generation by an adversarial network to improvea supervised classification. To the best of our knowledge,this is the first application where a physiological simulatorin the form of ODEs was leveraged to improve a supervisedclassification task. (2) We empirically show that utilizingthe synthetically generated ECG instances guided by theheart simulator ODEs significantly improves ECG heartbeatclassification using deep learning techniques. (3) We shareour code online for further research and experimentation.1

The structure of our paper is as follows: Section 3 describesthe ECG simulator represented by a system of ordinary dif-ferential equations. Section 4 introduces the frameworkof ECG Simulator GAN (SimGAN), a GAN-based setupwhich learns to create synthetic data by leveraging knowl-edge derived from a simulator represented by a system ofODEs (in our specific case, the ECG simulator from Section

1https://github.com/tomerGolany/sim_gan

3). Section 5 describes how the generated synthetic heart-beats from the SimGAN are used to train a deep networkthat classifies each heartbeat according to its heartbeat class.We use a ResNet convolutional model which was foundto have superior results on the ECG gold-standard dataset(Kachuee et al., 2018)). Finally, in Sections 6-7 we presentempirical evaluation comparing to state-of-the-art methods.

2. Related WorkIn imitation learning, real-world simulation has shown em-pirical gains. For example, in a racing game, an agent willhave access to a simulated environment, which is usually thegame engine. (Zadok et al., 2019) showed that autonomousvehicle driving significantly improves after training usingimitation learning in a simulated road environment. In re-inforcement learning, game simulation has been used toimprove learning (Mnih et al., 2013).

Previous work has also exploited generative models for thepurposes of creating extra training data. In the computervision realm, (Shrivastava et al., 2017) presented a GANmodel which learns to improve the realism of synthetic la-beled images using unlabeled real images. The generator’sinput is a synthetic image (instead of random noise), andthe output is a refined image. The discriminator must learnto classify images as real vs refined. They show that theimproved realism enables the training of better models forthe estimation of gaze and hand pose by adding the refinedlabeled images to the training set. In the ECG heartbeatclassification task, (Golany et al., 2020) similarly showedthat classification performance can be improved by addingsynthetic ECG heartbeats generated from a standard GANto a training set. Our work also generates synthetic ECG sig-nals that can be used in training; however, unlike previouswork, we leverage a simulator to improve the deep gener-ative model’s ability to generate synthetic labelled ECGheartbeats.

Classic ECG beat-level classification methods focused onextracting interval features and using prior knowledge onthe ECG morphology (De Chazal et al., 2004). Each ECGsignal is separated into heartbeats using heartbeat detec-tion techniques (Afonso et al., 1999). For each heartbeat,features related to the heartbeat intervals and ECG mor-phology are calculated. The combined features are thenfed into supervised machine-learning models based on lin-ear discriminants (LDs) (Nasrabadi, 2007). In recent years,the application of deep learning models to ECG classifica-tion has become popular. It has been applied to numeroustasks, such as cardiologist-level arrhythmia detection (Ra-jpurkar et al., 2017) and ECG heartbeat classification (Guler& Ubeyli, 2005; Prasad & Sahambi, 2003; Kachuee et al.,2018; Al Rahhal et al., 2016). The state-of-the-art methodfor ECG heartbeat-level classification (Kachuee et al., 2018)

https://github.com/tomerGolany/sim_gan


recently showed that superior results are reached by apply-ing a ResNet convolutional model which classifies eachheartbeat class separately. In this work, we focus on thetraining of such deep learning systems trained with addi-tional synthetic ECG heartbeats which are generated fromour proposed generative model enhanced with simulatorknowledge (Section 5).

3. The ECG SimulatorIn a normal resting heart, the physiological rhythm is re-ferred to as the normal sinus rhythm (NSR). NSR producesthe prototypical pattern of the P wave, followed by the QRScomplex, and finally the T wave. To capture this pattern,McSharry et al. represented a model of the heart by a systemof three ordinary differential equations (2003). The result-ing simulator is able to generate synthetic ECG signals withrealistic PQRST morphology, as well as prescribed heartrate dynamics. The simulator is parameterized by specificheart rate statistics, such as the mean and standard deviationof the heart rate, as well as frequency-domain characteristicsof the heart rate variability (HRV)(Malik & Camm, 1990).

The simulator is specified by three coupled ODEs, givingrise to a trajectory (x(t), y(t), z(t)). The ODEs are as fol-lows, where we have suppressed the time-dependence of x,y, and z for conciseness:

dx

dt= α(x, y)x− ωy ≡ fx(x, y; η) (1)

dy

dt= α(x, y)y + ωx ≡ fy(x, y; η) (2)

dz

dt= −

∑β∈B

aβ∆θβ(x, y)e−∆θβ(x,y)2/2b2β − (z − z0(t))

≡ fz(x, y, z, t; η) (3)

where:

α(x, y) = 1−√x2 + y2 (4)

∆θβ(x, y) = (θ(x, y)− θβ) mod 2π (5)θ(x, y) = atan2(y, x)) ∈ [−π, π] (6)

B ={P,Q,R, S, T

}(7)

The so-called “baseline wander” has been introduced by cou-pling the baseline value z0(t) to the respiratory frequencyf2 using:

z0(t) = A sin(2πf2t) (8)

where A = 0.15 mV. The baseline wander is low frequencynoise (0.05 - 3 Hz in stress tests)(Sornmo & Laguna, 2005),which is due to a variety of factors: movement duringbreathing, patient movement, poor contact between elec-trode cables and ECG recording equipment, inadequate skin

preparation where the electrode has been placed, and dirtyelectrodes. ω is a scalar representing the angular velocity ofthe trajectory as it moves around the limit cycle.

Finally, we denote the sum total of all of the parameters ofthe ODEs as η; they consist of the values θβ , aβ , and bβ forβ ∈ B =

{P,Q,R, S, T

}. Explicitly, we may write

η = (θP , θQ, θR, θS , θT , aP , aQ, aR, aS , aT ,

bP , bQ, bR, bS , bT )(9)

The solution of the equations generates a trajectory in a threedimensional state-space with coordinates (x, y, z). The gen-erated ECG signal itself is just the third dimension, i.e. thesignal z(t), which is the movement of the trajectory arounda limit cycle of unit radius in the (x, y) plane. Each revolu-tion around this circle corresponds to a single heartbeat (orcardiac-cycle).

The η parameters: Distinct points on the ECG signal, suchas the P, Q, R, S and T waves are described by events cor-responding to negative and positive attractors / repellers inthe z direction. The wave events locations are determinedby the equations parameters θP , θQ, θR, θS , θT which repre-sent fixed angles along the unit circle. When the z trajectoryapproaches one of these events, it is pushed upwards ordownwards away from the limit cycle, and then it movesback toward the limit cycle. The height and length of eachof the P, Q, R, S, T wave events is determined by the equa-tions parameters aP , aQ, aR, aS , aT and bP , bQ, bR, bS , bTrespectively. We follow the η parameters suggested by (Mc-Sharry et al., 2003) for a normal heartbeat.

Solving the ODE system is done numerically using Runge-Kutta methods (Butcher & Butcher, 1987) with a fixed timestep ∆t = 1

fswhere fs = 360 is the sampling frequency.

Specifically, for the numerical integration of the ordinarydifferential equations we leverage the simplest form of theRunge Kutta family known as the Euler method (Atkinson,2008).

The Euler method is based on the finite difference approx-imation (Milne-Thomson, 2000):

du

dt(t) ≈ u(t+ ∆t)− u(t)

∆t(10)

Given an ODE of the form du/dt = v, the above may berearranged to yield the following formula:

u(t+ ∆t) = u(t) + v(t)∆t (11)

We construct the approximate solution as follows. We runEq. 11 iteratively for L time-steps, where the `th step cor-responds to t` = `∆t; L is the number of samples thatcorresponds to a single signal. This yields

u`+1 = u` + v`∆t (12)


Notice that approximating the values at time step ` + 1requires only the values at time step `, which are alreadyknown. (Of course we require the initial conditions, u0, aswell.)

Substituting Eqs. 1, 2, 3 into Eq. 12 yields:

t` = `∆t

x`+1 = x` + fx(x`, y`; η)∆t

y`+1 = y` + fy(x`, y`; η)∆t

z`+1 = z` + fz(x`, y`, z`, t`; η)∆t

(13)

4. SimGAN Framework:Adding Simulators to GANsWe introduce the framework of ECG Simulator GAN (Sim-GAN), a GAN-based setup which has been enriched withadditional knowledge from the ECG simulator presented inSection 3. Recall that in a regular GAN setting the generatorlearns to create synthetic data based on input noise. The datais then fed to a discriminator, which attempts to discriminatebetween synthetic and real data. Although generating ECGsignals from noise allows for the creation of samples whichdo not appear in the training data, it does not yield ECGheartbeats with truly realistic morphology and characteris-tics. We therefore strive to incorporate the ECG simulatorequations directly into the generation process. Followingthe practice of (Golany & Radinsky, 2019; Golany et al.,2020; Shrivastava et al., 2017), the generated ECG signalsare then used to train a deep network (Section 5). We empir-ically show (Section 7) that the additional generated labeledexamples significantly improve the heartbeat classification.

4.1. General GAN Framework

We formulate the generative adversarial networks for ECGheartbeats generation as follows. An ECG signal taken froma patient in one lead is sliced to heartbeats (cardiac cycles),which are labelled h. Each such heartbeat may be writtenas a fixed length vector h = (h1, . . . , hL), which representsthe time evolution of the voltage values of a single heartbeat.The length L = 216; these 216 points represent a 600msrange sampled at 360 samples per second, where the rangeis from 200ms before the R-peak to 400ms after the R-peak.Our ultimate goal is to classify the heartbeats accordingto their heartbeat type c. We denote the underlying beatsdistribution of a given class as a conditional probabilityp(h|c), which reflects the distribution of heartbeat signalsgiven that the heartbeat is taken from class c.

Given a set of heartbeat signals {h(1), . . . h(K)} drawn fromclass c, our general GAN framework consists of a discrim-inator and a generator, where both are class-specific. Wedenote the discriminator by D(h;φcD), where the parame-ters of the discriminator network are given by φcD, and de-

pend on the class c. The generator is denoted by G(m;φcG),where m is a random variable with known, fixed distribution(Gaussian), and φcG are the class-specific parameters. As isstandard, G and D play the following two-player game:

minφD

LD(φD, φG) (14)

minφG

LG(φD, φG) (15)

However, note that in our case the game is not minimax,as the discriminator and generator will have objective func-tions which are not equivalent, i.e. LD 6= −LG. We nowelaborate on the specific discriminator and generators weuse, and their corresponding loss functions LD and LG.

4.2. The ECG Simulator Discriminator

Architecture The discriminator architecture and optimiza-tion remains the same as for a conventional GAN. Givenpositive ECG heartbeat samples from real patients and neg-ative ECG heartbeat samples from the generator, the objec-tive for the discriminator is to maximize the log-probabilityof assigning the correct labels to both positive and nega-tive samples. The ECG heartbeats are fed through 6 one-dimensional convolutional layers. All weights were ini-tialized from a zero-centered Gaussian distribution with astandard deviation of 0.02. After each layer we performbatch normalization and apply a Leaky ReLU activationfunction, where the slope of the leak is set 0.2. The fi-nal layer is a fully connected layer with sigmoid activationwhich classifies whether the heartbeat is from the real dataor from the generator (fake).

Loss The discriminator’s job is a typical classification prob-lem. Thus, we use a cross-entropy loss:

LD(φD, φG) =− Eh∼pdata logD(h; θD)

− Em∼m log(1−D(G(m; θG); θD))(16)

which is the standard GAN discriminator loss.

4.3. The ECG Simulator Generator

Architecture The SimGAN Generator architecture differsfrom that of the classic GAN by leveraging knowledge fromthe ECG simulator (Section 3). Figure 1 describes the archi-tecture. The generator input layer draws a length 100 vectorfrom a Gaussian distribution with zero mean and identitycovariance: m ∼ m ≡ N(0, I). This noise vector m isthen fed to 6 one-dimensional deconvolution layers. Aftereach layer batch normalization (Ioffe & Szegedy, 2015) isperformed followed by a ReLU activation function. Thefinal one-dimensional deconvolution layer of the generatorproduces an output vector of size L = 216. The outputdimensions of the generator therefore corresponds to thesame dimension of one heartbeat (cardiac-cycle).


Loss: General Considerations The generator loss functionis defined by a combination of the classical cross-entropyloss, which tries to generate cardiac-cycles that will foolthe discriminator, and a Mean Square Error (MSE) functionwhich tries to generate fake heartbeats which are morpholog-ically similar to heartbeats from the ECG Simulator (Section3). We now elaborate on each of these two loss functions.

Cross-Entropy Loss In contrast to the discriminator, thegenerator aims to minimize the log-probability that the dis-criminator correctly assigns negative labels to the samplesgenerated by G. That is, the goal is to optimize the genera-tor’s weights in such a way that the discriminator networkwill predict that the generated ECG heartbeat is real. For-mally, the cross-entropy loss function of the generator maybe written

LCEG (φD, φG) = −Em∼m logD(G(m; θG); θD) (17)

Euler Loss In an ordinary GAN setting the generator learnsto create synthetic data based on input noise which is laterfed into the discriminator. We wish to leverage our worldknowledge coming from the ECG simulator and use it togeneralize the heartbeats produced to be more realistic. Wetherefore introduce the Euler Loss function.

We begin by defining a measure of how well a given heart-beat matches the ECG simulator. If the heartbeat is givenby h and the simulator parameters are given by η, then theSimulator Distance is given by

∆sim(h, η) =

L−1∑`=1

(h`+1 − h`

∆t− fz(x`, y`, h`, t`; η)

)2

(18)where fz is defined as in Eq. 3; and the trajectories forx` and y` are given by the discrete solution to the coupledODEs in Eq. 13. In other words, the Simulator Distanceattempts to determine how closely the generated heartbeatmatches the biophysical model of a heartbeat, as specified bythe set of ODEs. In order to do so, the z trajectory – whichis the heartbeat itself – is fixed to the generated heartbeat;and the x and y trajectories are fixed to their ODE solutions.The Euler Loss then measures how closely the ODE in zholds. If the generated heartbeat is exactly reproduced bythe model, then the ODE in z will hold exactly, leading to aSimulator Distance of 0; the further the deviation betweengenerated heartbeat and the biophysical model’s prediction,the larger the Simulator Distance. Note that in order forthe Simulator Distance to be properly defined, we need anadditional item, the initial conditions for x and y. Theseare taken to be x0 = −0.41, y0 = −0.91 as suggested by(McSharry et al., 2003).

Given this definition of the Simulator Distance, we definethe Euler Loss as follows:

LEULG (φG) = Em∼m,η∼p(η|c) ∆sim(G(m; θG), η) (19)

That is, the Euler Loss is the expectation of the SimulatorDistance over realizations of both the noise vector input mto the generator, as well as the simulator parameters η. Thedistribution of the simulator parameters is approximated,for a given class c, as a Gaussian, where the mean andcovariance matrix are computed from the set of η-vectorscorresponding to that class. (For each heartbeat in the class,one may compute the η-vector which generates a simulatedheartbeat that is closest to that heartbeat.) The covariancematrix is taken as diagonal for simplicity. It is then straight-forward to sample η.

Intuitively, then, the Euler Loss tries to constrain the outputfrom the generator to lie close to the submanifold of signalswhich can produced by the simulator ODEs. Combiningthe Euler Loss with the classical cross-entropy loss enablesthe generator to create synthetic ECG heartbeats with realmorphology and characteristics which don’t exist in thetraining set, while preserving the noise which defines thereal data.

Optimization The final loss function of the generator istherefore the sum of the two loss functions:

LG(φD, φG) = LCEG (φD, φG) + LEULG (φG) (20)

The optimization is performed using Adam optimizer(Kingma & Ba, 2014) and a learning rate of 0.0002 as sug-gested for training GAN models by (Radford et al., 2015).The generator’s parameters are updated twice each iterationwhile the discriminator’s parameters are updated once.

Input100randomsam

ples

1 2,…1 255~7(9=0,;=1)

Shape100x1

100

1

Convtranspose1d1x4(2048filters)


Shape2048x4

Shape1024x7


stride2

padding1

Shape512x14


stride2

padding1

Shape256x27Convtranspose1d1x4(128filters)

stride2

padding1


stride2

padding1

Shape128x54Shape64x108

Convtranspose1d1x4(onefilter)

stride2

padding1

Generatoroutputsyntheticheart-beatShape1x216

QRSRTUVWT XWYY:

X Q = X[\]^^_`a\]bc(d Q 1 + X_fgh\(d Q 1 , i)

Figure 1. Generator model pipeline (Golany et al., 2020). Thegenerator outputs a single synthetic heartbeat with characteristicssimilar to both the heartbeats generated from the ECG simulatoras well as the real data.

5. Deep ECG ClassificationIn Section 4.1, we described the SimDCGAN – a frameworkfor generating ECG heartbeats that exhibit both a certainarrhythmia and adapt morphological characteristics from


the ECG simulator (Section 3). In this section, we describehow the generated synthetic heartbeats from the generatorare used to train a deep network that classifies each heartbeataccording to its heartbeat class.

For evaluation we use the ResNet convolutional modelwhich was found to have superior results on the ECG gold-standard dataset (Kachuee et al., 2018). The network istrained over the train portion of the dataset described inSection 6.1. The network architecture consists of a convolu-tional layer followed by five residual convolutional blocks.Each residual block contains two convolutional layers, twocorresponding ReLU activations, a residual skip connec-tion, and a pooling layer. The last residual block is fol-lowed by two fully-connected layers with 32 neurons eachand a soft-max layer to predict output class probabilities.Each convolutional layer is a one-dimensional convolutionthrough time and has 32 kernels of size 5. Once the model’sinitial training has converged, we perform further trainingusing the additional generated ECG signals from the trainedgenerator.

6. Experimental Evaluation6.1. ECG Dataset

Our framework consists of ECG recordings taken from theMIT-BIH arrhythmia database (Moody et al., 2001), whichis the most popular public dataset for discovering and clus-tering arrhythmias, and is considered the gold-standard eval-uation data for ECG heartbeat classification tasks. Thedatabase contains 48 half-hour ECG records, obtained frompatients studied by the BIH Arrhythmia Laboratory be-tween 1975 and 1979. Each record contains two 30-minuteECG lead signals digitized at 360 samples per second. Thedatabase contains annotations for both heartbeat class infor-mation and timing information verified by independent ex-perts. The database consists of a total of 109,492 heartbeats,each of which has been labelled with one of the followingfour classes: Ventricular Ectopic Beat, shortened to VEB orV; Supraventricular Ectopic Beat, shortened to SVEB or S;Fusion Beat, shortened to F; Normal beat, shortened to N.

6.2. Experimental Methodology

We follow the dataset partitioning as described by(De Chazal et al., 2004; Al Rahhal et al., 2016; Golany& Radinsky, 2019). All follow the AAMI (Associationfor the Advancement of Medical Instrumentation) recom-mendations for the ECG heartbeat classification task. Thispartition ensures that patient data is not mixed betweenthe train and the test sets. We follow the experimentalmethodology used in (De Chazal et al., 2004; Al Rahhalet al., 2016; Kutlu & Kuntalp, 2012; Jiang & Kong, 2007),again as recommended by the AAMI; namely, we compute

a precision-recall curve separately for each heartbeat class.(Note that in some of the above references, the medicalconvention is used, namely sensitivity (=recall) vs. positivepredictive value (=precision).) For example, for the case ofVEB, the precision-recall curve reports the performance intrying to distinguish V from the remaining classes S, F, andnormal heartbeat. We report similar precision-recall curvesfor SVEB and Fusion (no curve is reported for the normalheartbeat class, as it is not an arrhythmia and considereda simple problem with little clinical value). As (Kachueeet al., 2018) follows an idiosyncratic methodology differentfrom the rest of the literature, we have re-implemented themethod of (Kachuee et al., 2018) and report results for theirtechnique using the standard settings described above.

In what follows, we refer to the set of heartbeats from allpatients in the train set as the base set. We present resultsfor the following different data synthesis regimes:

No ECG Generation The ResNet convolutional modelwhich recently showed state-of-the-art results in the task ofECG heartbeat classification (Kachuee et al., 2018) – seeSection 5 – is trained on the base set, with no additionalsynthesized ECG examples.

ECG Generation using Ordinary GANs We train an or-dinary GAN without any special loss terms (e.g. no EulerLoss) to generate synthetic heartbeats. This technique hastwo flavors:(1) Vanilla GAN (VGAN) – The ResNet convolutional modelis trained on the base set with additional synthesized ECGexamples from a classical GAN model (Goodfellow et al.,2014).(2) DCGAN – The ResNet convolutional model is trainedon the base set with additional synthesized ECG examplesfrom a DCGAN (Radford et al., 2015) model. As DCGANshave shown superior performance on several tasks (Radfordet al., 2015; Yeh et al., 2017) we present results on generatedexamples from this class of generative adversarial networks.

ECG Generation using RefineGAN – Generating ECGheartbeats with a DCGAN framework where the inputs tothe generator are synthetic heartbeats that comes out fromthe ECG Simulator. We would like to compare the contri-bution of integrating the ECG formulation as a loss term(The Euler loss) to a GAN, against a GAN setting where theinput to the Generator are synthetic ECG heartbeats whichare produced from the ECG Simulator instead of randomnoise. (Same as (Shrivastava et al., 2017))

ECG Generation using ECG Simulator We would liketo test the contribution of integrating the ECG formulationto GAN as compared to directly using generated samplesfrom an ECG simulator. We therefore propose a baselinemodel which trains the ResNet convolutional model onthe base set with additional synthesized ECG examples


(a) SVEB heartbeat class (b) Fusion heartbeat class (c) VEB heartbeat class

Figure 2. Precision - Recall curves of the 3 heartbeat classes evaluated on the test-set. Each subfigure shows curves for each of the fourdata synthesis regimes described in Section 6, as well as the state of the art method of (Kachuee et al., 2018).

from the ECG Simulator (Section 3). For each of thethree type of heartbeats to be classified, we calculatethe values of their P, Q, R, S and T waves (Figure4c). The average values for each wave is taken and isinserted into the ECG simulator as the wave parameters η =(θP , θQ, θR, θS , θT , aP , aQ, aR, aS , aT , bP , bQ, bR, bS , bT )(Section 3). For each heartbeat generated from the ECGsimulator, we add additional Gaussian noise with zero meanand identity covariance to the the wave parameters η inorder to generate multiple heartbeats from the same class.

ECG Generation using SimGANs The main approach pro-posed in this paper. The ResNet convolutional model istrained on heartbeats from all patients from the trainingset with additional synthesized heartbeats from a SimGANmodel. As with ordinary GANs, two GAN architectureswith the Euler additional optimization loss are tested:(1) SimVGAN, using the VGAN architecture;(2) SimDCGAN, using the DCGAN architecture.The number of synthetically generated beats which areto be added to the training set is a parameter of themodel, and depends on the number of samples fromeach class. For a heartbeat class with n samples inthe base set, we experimented with the following values:0.1n, 0.3n, 0.5n, 0.8n, n, 1.5n, 2n.

In Appendix A we provide additional experiments whichinclude training a Wasserstein-GAN (Arjovsky et al., 2017)with and without the additional Euler loss, and classificationresults using a not deep learning based classifier.

7. ResultsComparison with State-Of-The-Art Making a fair com-parison in the task of ECG heartbeat classification is some-what challenging due to the fact that different papers use

different settings. For example, previous papers presentedresults where heartbeats from the same subjects were sharedbetween train and test sets (Jiang et al., 2006); classifierswhich used more than one lead during training (Llamedo &Martınez, 2012); classifiers which used RR intervals (Lin& Yang, 2014); and semi-supervised heartbeat classifierswhich had access to unlabeled data from the test (Golany& Radinsky, 2019). In our setting, by contrast, we onlymake use of the raw ECG signals. We therefore compareour results to techniques which are state of the art for thissetting: (Kachuee et al., 2018) and (Al Rahhal et al., 2016).

Figures 2(a) - 2(c) present precision recall curves for thetask of classifying the MIT-BIH ECG test-set over the heart-beat classes SVEB, Fusion and VEB respectively. Weshow curves for each of the four data synthesis regimesdescribed in Section 6, as well as the state of the art methodof (Kachuee et al., 2018). The SVEB and Fusion heartbeatsare quite rare and only comprise 3% and 2% of the dataset,respectively. We see that the performance of (Kachuee et al.,2018) on those heartbeat classes is quite low (Figures 2(a)and 2(b), yellow curves). Adding synthetic heartbeats fromany of the generative models improves the classificationperformance. However, the results of our proposed SimDC-GAN model are demonstrably superior to those of any othergenerative model: the green curves in Figures 2(a) and 2(b)are considerably higher than any of the other curves. Thisphenomenon is still present in the case of the much morecommon VEB heartbeats, though it is less pronounced, seeFigure 2(c). In particular, the performance of (Kachuee et al.,2018) (yellow curve) is already quite high. Nevertheless,the SimDCGAN model, again shown as the green curve,is clearly better; for example, taking two arbitrary pointson the two curves, (Kachuee et al., 2018) achieves (Recall,Precision) = (0.85, 0.75), vs. (0.88, 0.89) for SimDCGAN.


Table 1. Comparison between our method and (Al Rahhal et al., 2016). The methodology of (Al Rahhal et al., 2016) is to test against theentire MIT-BIH database. Re and Pr denote Recall and Precision. Best results are shown in bold and *.

(AL RAHHAL ET AL., 2016) VGAN DCGAN SIMVGAN SIMDCGANHEARTBEAT CLASS RE PR RE PR RE PR RE PR RE PR

SVEB (S) 0.41 0.43 0.41 0.63 0.41 0.58 0.41 0.64 *0.41 *0.80VEB (V) 0.91 0.79 0.91 0.70 0.91 0.72 0.91 0.82 *0.91 *0.84

FUSION (F) 0.60 0.04 0.60 0.10 0.60 0.20 0.60 0.25 *0.60 *0.40

In Table 1 we compare our method to another state-of-the-art deep learning technique, that of (Al Rahhal et al., 2016).We perform this comparison separately, as (Al Rahhal et al.,2016) report their results over the entire MIT-BIH dataset,that is, the combination of both train and test sets. Ascan be seen from Table 1, SimDCGAN far outperforms(Al Rahhal et al., 2016) on SVEB and Fusion heartbeats,nearly doubling the precision for the same level of recallfor SVEB, and far more than doubling the precision for thesame level of recall for Fusion. On the more common VEBheartbeats, the improvement is smaller but still present: a6% relative increase in precision for the same level of recall.

Different Generative Architectures In Figures 2(a) - 2(c)and Table 1, we demonstrate that learning to adapt to theECG simulator morphology helps for different types ofGAN architectures – both VGAN and DCGAN. Beginningwith the vanilla GANs, we observe that for all types of heart-beat, results for the model trained with additional data fromSimVGAN outperforms the standard VGAN: in Figures2(a) - 2(c), the blue curve is always significantly above thepurple curve. A similar phenomena occurs also for the DC-GAN model, in which SimDCGAN outperforms ordinaryDCGAN for all classes. Indeed, as we have already noted,SimDCGAN outperforms all generative models across allclasses, reaching state-of-the-art performance. (Table 1 sup-ports the same conclusions numerically.) We therefore con-clude that while adding synthetic ECG heartbeats alwaysimproves ECG classification, the SimDCGAN generationmethod yields the highest performance gains.

(a) SVEB heartbeat class (b) Fusion heartbeat class

Figure 3. Comparison of our method against generation of syn-thetic heartbeats directly from the ECG Simulator. Precision-Recall curves of the two more challenging heartbeat classes.

The Importance of the Generative Component By way

of an ablation study, we test whether the SimDCGAN model,which relies on the Euler Loss, brings more value as com-pared to simply adding synthetic heartbeats from the ECGsimulator to the training set directly. Figures 3(a) and 3(b)show the precision-recall curves for the two more challeng-ing classes, SVEB and Fusion. We observe that addingplain data from the ECG simulator outperforms the ResNetconvolutional model trained only with the original trainingdata for both heartbeat classes. It is therefore clear that theaddition of the simulator alone is of high value. However,SimDCGAN outperforms this model across both classes(with the exception of a small regime of very low recall inthe case of the SVEB class). We speculate this is due to thefact that the data generated by the simulator does not containthe noise that exists in the real signals. It therefore does notmodel the real ECG data quite as well as SimDCGAN.

Table 2. Comparison between our method to a RefineGAN. Bestresults are shown in bold and *.

SIMDCGAN REFINEGANHEARTBEAT CLASS RE PR RE PR

SVEB (S) *0.45 *0.7 0.4 0.47VEB (V) *0.88 *0.89 0.85 0.73

Refining ECG Simulator output We study a different ap-proach to integrate the knowledge from the ECG Simulatorinto a GAN framework. We adapt (Shrivastava et al., 2017)method, by inserting to the generator synthetic ECG heart-beats which comes directly from the ECG Simulator, in-stead of the regular setting, in which the generator receivesas input random noise. In this setting the loss terms forthe generator and discriminator are the conventional cross-entropy losses, without the additional Euler loss term. Table2 shows significant points on the precision-recall curves forthe SVEB and VEB classes. SimDCGAN outperforms thismodel across both classes.

Qualitative Examples Figure 4 presents a qualitative ex-ample of a real patient ECG wave and synthetic waves pro-duced by DCGAN and SimDCGAN. Figure 4(c) is a typicalreal heartbeat taken from the dataset. The real heartbeat isslightly noisy but the PQRST waves can be observed. Fig-ures 4(a) and 4(b) both present synthetic waves. Althoughboth look realistic, the wave 4(a), which shows a heartbeatproduced by a DCGAN model, lacks a T wave (as shown by


the red circle) and is much smoother than a real heartbeat.Figure 4b shows a heartbeat generated from the SimDC-GAN. The heartbeat contains real ECG morphology withPQRST waves and measurement noise as well.

P

Q

R

S

T

a)DCGANHeartbeat b)SimDCGAN Heartbeat c)RealHeartbeat

P

Q

R

S

T

P

Q

R

S

Figure 4. a: Heartbeat produced by DCGAN with no simulatorknowledge. The red circle shows an incorrect T wave. b: Heartbeatproduced by SimDCGAN. c: Real patient heartbeat.

8. ConclusionsWe have presented a new technique for integrating priorknowledge in the form of a mathematical process simulatorinto a deep learning pipeline. We have shown how an ODEmodel describing cardiac cycles can be incorporated into aGAN-based generative model of these cycles using a novelEuler Loss. Empirically, we have demonstrated that by usingthe synthetic cardiac cycles generated from such a model astraining data, we can improve the classification accuracy ofa standard ResNet convolutional ECG heartbeat classifier,attaining state-of-the-art results. In particular, the resultsattained are better than those when examples are addedeither (a) directly from the simulator, or (b) from ordinaryGANs which do not take the simulator into account.

The approach presented here is not specific to cardiac cy-cles; it applies equally well to any system described math-ematically by differential equations. In the future, we willinvestigate the application of the approach to other systemsdescribed by ODEs, such as calcium dynamics in neuronalmodels (De Schutter & Smolen, 1998) and chemical reac-tions; as well as systems described by PDEs.

ReferencesAfonso, V. X., Tompkins, W. J., Nguyen, T. Q., and Luo, S.

Ecg beat detection using filter banks. IEEE transactionson biomedical engineering, 46(2):192–202, 1999.

Al Rahhal, M. M., Bazi, Y., AlHichri, H., Alajlan, N., Mel-gani, F., and Yager, R. R. Deep learning approach foractive classification of electrocardiogram signals. Infor-mation Sciences, 345:340–354, 2016.

Arjovsky, M., Chintala, S., and Bottou, L. Wasserstein gan.arXiv preprint arXiv:1701.07875, 2017.

Atkinson, K. E. An introduction to numerical analysis. JohnWiley & Sons, 2008.

Banks, J., Carson, J. S., Nelson, B. L., and Nicol, D. M.Discrete-Event System Simulation (3rd Edition). PrenticeHall, 2000.

Buchanan, B. G. and Feigenbaum, E. A. Knowledge-BasedSystems in Artificial Intelligence. McGrawHill, 1982.

Butcher, J. C. and Butcher, J. The numerical analysis ofordinary differential equations: Runge-Kutta and generallinear methods, volume 512. Wiley New York, 1987.

De Chazal, P., O’Dwyer, M., and Reilly, R. B. Automaticclassification of heartbeats using ecg morphology andheartbeat interval features. IEEE transactions on biomed-ical engineering, 51(7):1196–1206, 2004.

De Schutter, E. and Smolen, P. Calcium dynamics in largeneuronal models. Methods in neuronal modeling: Fromions to networks, 2, 1998.

Gabrilovich, E. and Markovitch, S. Wikipedia-based seman-tic interpretation for natural language processing. J. Artif.Int. Res., 34(1):443–498, March 2009.

Golany, T. and Radinsky, K. Pgans: Personalized generativeadversarial networks for ecg synthesis to improve patient-specific deep ecg classification. In Proceedings of theAAAI Conference on Artificial Intelligence, volume 33,pp. 557–564, 2019.

Golany, T., Lavee, G., Yarden, S. T., and Radinsky, K. Im-proving ecg classification using generative adversarialnetworks. In Proceedings of the AAAI Conference onArtificial Intelligence, 2020.

Goodfellow, I., Pouget-Abadie, J., Mirza, M., Xu, B.,Warde-Farley, D., Ozair, S., Courville, A., and Bengio, Y.Generative adversarial nets. In NIPS. 2014.

Guler, I. and Ubeyli, E. D. Ecg beat classifier designed bycombined neural network model. Pattern recognition, 38(2):199–208, 2005.

Ioffe, S. and Szegedy, C. Batch normalization: Acceleratingdeep network training by reducing internal covariate shift.arXiv preprint arXiv:1502.03167, 2015.

Jiang, W. and Kong, S. G. Block-based neural networks forpersonalized ecg signal classification. IEEE Transactionson Neural Networks, 18(6):1750–1761, 2007.

Jiang, X., Zhang, L., Zhao, Q., and Albayrak, S. Ecg arrhyth-mias recognition system based on independent compo-nent analysis feature extraction. In TENCON 2006-2006IEEE Region 10 Conference, pp. 1–4. IEEE, 2006.

Kachuee, M., Fazeli, S., and Sarrafzadeh, M. Ecg heart-beat classification: A deep transferable representation.In 2018 IEEE International Conference on HealthcareInformatics (ICHI), pp. 443–444. IEEE, 2018.


Kingma, D. P. and Ba, J. Adam: A method for stochasticoptimization. arXiv preprint arXiv:1412.6980, 2014.

Kutlu, Y. and Kuntalp, D. Feature extraction for ecg heart-beats using higher order statistics of wpd coefficients.Computer methods and programs in biomedicine, 105(3):257–267, 2012.

Lin, C.-C. and Yang, C.-M. Heartbeat classification us-ing normalized rr intervals and morphological features.Mathematical Problems in Engineering, 2014, 2014.

Llamedo, M. and Martınez, J. P. An automatic patient-adapted ecg heartbeat classifier allowing expert assistance.IEEE Transactions on Biomedical Engineering, 59(8):2312–2320, 2012.

Malik, M. and Camm, A. J. Heart rate variability. Clinicalcardiology, 13(8):570–576, 1990.

McSharry, P. E., Clifford, G. D., Tarassenko, L., and Smith,L. A. A dynamical model for generating synthetic elec-trocardiogram signals. IEEE transactions on biomedicalengineering, 50(3):289–294, 2003.

Milne-Thomson, L. M. The calculus of finite differences.American Mathematical Soc., 2000.

Mintz, M., Bills, S., Snow, R., and Jurafsky, D. Distantsupervision for relation extraction without labeled data.In ACL ’09, 2009.

Mnih, V., Kavukcuoglu, K., Silver, D., Graves, A.,Antonoglou, I., Wierstra, D., and Riedmiller, M. Playingatari with deep reinforcement learning. arXiv preprintarXiv:1312.5602, 2013.

Moody, G. B., Mark, R. G., and Goldberger, A. L. Physionet:a web-based resource for the study of physiologic signals.IEEE Engineering in Medicine and Biology Magazine,20(3):70–75, 2001.

Nasrabadi, N. M. Pattern recognition and machine learning.Journal of electronic imaging, 16(4):049901, 2007.

Prasad, G. K. and Sahambi, J. Classification of ecg arrhyth-mias using multi-resolution analysis and neural networks.In TENCON 2003. Conference on Convergent Technolo-gies for the Asia-Pacific Region, volume 1, pp. 227–231.IEEE, 2003.

Radford, A., Metz, L., and Chintala, S. Unsupervised rep-resentation learning with deep convolutional generativeadversarial networks. arXiv preprint arXiv:1511.06434,2015.

Rajpurkar, P., Hannun, A. Y., Haghpanahi, M., Bourn, C.,and Ng, A. Y. Cardiologist-level arrhythmia detectionwith convolutional neural networks. CoRR, 2017.

Shrivastava, A., Pfister, T., Tuzel, O., Susskind, J., Wang,W., and Webb, R. Learning from simulated and unsuper-vised images through adversarial training. In Proceedingsof the IEEE conference on computer vision and patternrecognition, pp. 2107–2116, 2017.

Sornmo, L. and Laguna, P. Bioelectrical signal process-ing in cardiac and neurological applications, volume 8.Academic Press, 2005.

Wainwright, M. J. and Jordan, M. I. Graphical models,exponential families, and variational inference. Found.Trends Mach. Learn., 1(1-2):1–305, January 2008.

Yeh, R. A., Chen, C., Lim, T.-Y., Schwing, A. G., Hasegawa-Johnson, M., and Do, M. N. Semantic image inpaintingwith deep generative models. In CVPR, 2017.

Zadok, D., Hirshberg, T., Biran, A., Radinsky, K., andKapoor, A. Explorations and lessons learned in buildingan autonomous formula sae car from simulations. SI-MULTECH, 2019.

Zhang, F., Yuan, N. J., Lian, D., Xie, X., and Ma, W.-Y. Col-laborative knowledge base embedding for recommendersystems. KDD’16, 2016.

simgans: simulator-based generative adversarial networks

Documents