Fast kinematics modeling for conjunction with lensimage modeling

Matthew R. Gomer
STAR Institute
Université de Liège
4000 Liège, Belgium
mgomer@uliege.be

Luca Biggio
Eidgenössische Technische Hochschule Zürich
CH-8092 Zürich, Switzerland
luca.biggio@inf.ethz.ch

Sebastian Ertl
Max-Planck-Institut für Astrophysik
Technische Universität München
85748 Garching, Germany
ertlseb@mpa-garching.mpg.de

Han Wang
Max-Planck-Institut für Astrophysik
Technische Universität München
85748 Garching, Germany
wanghan@mpa-garching.mpg.de

Aymeric Galan
Institute of Physics, Laboratory of Astrophysics
Ecole Polytechnique Fédérale de Lausanne (EPFL)
1290 Versoix, Switzerland
aymeric.galan@epfl.ch
Dominique Sluse
STAR Institute
Université de Liège
4000 Liège, Belgium
dsluse@uliege.be

Lyne Van de VyvereSTAR Institute
Université de Liège
4000 Liège, Belgium
lyne.vandevyvere@uliege.be

Georgios Vernardos
Institute of Physics, Laboratory of Astrophysics
Ecole Polytechnique Fédérale de Lausanne (EPFL)
1290 Versoix, Switzerland
georgios.vernardos@epfl.ch

Sherry Suyu
Max-Planck-Institut für Astrophysik
Technische Universität München
85748 Garching, Germany
Academia Sinica Institute of Astronomy and Astrophysics (ASIAA)
Taipei 10617, Taiwan
suyu@mpa-garching.mpg.de

Abstract
Galaxy kinematics modeling is currently the computational bottleneck for a joint
gravitational lensing+kinematics modeling procedure. We present as a proof
of concept the Stellar Kinematics Neural Network (SKiNN), which emulates
kinematics calculations for the context of gravitational lens modeling. After a
one-time upfront training cost, SKiNN creates velocity dispersion images which
are accurate to ≲ 1% within the region of interest at a speed O(102 − 103 ) times
faster than existing kinematics modeling methods. This speedup makes it feasible
to jointly model lensing data with spatially resolved kinematic data, which corrects
for the largest source of uncertainty in the determination of the Hubble constant.

Machine Learning and the Physical Sciences workshop, NeurIPS 2022.

1

Introduction

Gravitational lensing offers a distance-ladder-independent method to directly measure the Hubble
constant, H0 , by measuring the differences in arrival time between different multiple images ofthe same distant source. Comparison of these time delays to those predicted from a model for themass distribution of the lens (typically a galaxy) gives a direct measure of distance and therefore
H0 . Unfortunately, lensing degeneracies exist which allow different mass distributions to reproduce
the same resulting image. To overcome these degeneracies and recover a precise H0 , kinematicconstraints are required to provide an independent measure of the lens mass distribution.
Until recently, the best available kinematic data were slit aperture constraints, which give a single
velocity dispersion (second moment of the velocity distribution function) for the whole system.
Recently, spatially resolved kinematics of lensing galaxies have become available through telescopes
like the James Webb Space Telescope (JWST). Lens modelers have traditionally used the spherical
Jeans model to describe single-slit kinematics [e.g., Suyu et al., 2010, Sonnenfeld et al., 2012, Wonget al., 2017, Birrer et al., 2019, Rusu et al., 2020], which in addition to lacking self-consistency
with the elliptical lens mass model, is too simplistic for spatially resolved kinematic data. The
next generalization to make from the spherical case is the axisymmetric case, for which software
such as Jeans Anisotropic MGE [JAM; Cappellari, 2008] can model galaxy kinematics (using theMulti-Gaussian Expansion method [Emsellem et al., 1994, Cappellari, 2002]), albeit at significant
computational cost.
Existing frameworks to calculate kinematics in conjunction with lens modeling come with limitations.
van de Ven et al. [2010] first compared an axisymmetric kinematics model self-consistently with alens model, but these models were fit separately. Barnabè et al. [2012] introduced a joint modelingcode designed for galaxy structure research but not used for H0 determination. Recently Yıldırım
et al. [2020, 2021] expanded the framework using JAM to be compatible with cosmological parameter
determination. However, the combined modeling procedure is still computationally expensive, withthe bottleneck being that JAM must be called for each evaluation of the likelihood, which itself
is often sampled via a Markov Chain Monte Carlo (MCMC) to estimate uncertainties. To reduce
costs, modelers often first optimize the lens model, then combine the kinematics to the likelihood
post-processing. To efficiently explore the full parameter space of both models simultaneously
requires a significant speed improvement for kinematic modeling.
Our insight is to emulate JAM using a neural network (NN), which we call SKiNN (Stellar Kinematics
Neural Network). Once trained, SKiNN is able to create the same output as JAM in the short time ittakes to call the network. This factor of 300 speed increase makes it possible to include kinematic
modeling within a joint framework with lens modeling. We note that SKiNN is trained using JAMgenerated data, and as such is designed only to emulate the JAM modeling procedure rather thanbeing applied directly to the observations. In this way, confining the machine learning aspect to JAM
emulation exploits the speed and versatility of NNs while retaining the physics of the overall model.
Figure 1 gives a summary of SKiNN’s role in a joint modeling framework.
In this paper, we focus on the machine learning aspect of the modeling framework. In an upcoming
work, we will present and discuss the full modeling procedure for joint lensing+kinematics using our
NN-based technique.

2

Related Works

The general method to speed up modeling code by replacing the expensive piece with a NN has
been applied before in adjacent fields. NNs have been used to replace expensive solver operations
in cosmological applications [Albers et al., 2019, Bonici et al., 2022], stellar population synthesisin spectral modeling [Alsing et al., 2020], and even to speed up gravitational lens modeling itself[Hezaveh et al., 2017, Perreault Levasseur et al., 2017, Pearson et al., 2019, Schuldt et al., 2021, 2022,
Park et al., 2021]. This work applies this strategy for the first time to the kinematic modeling aspect
within the framework of lens modeling.
2

Figure 1: SKiNN’s role in a joint modeling framework. Traditional lens modeling (blue) uses a model
for a source, lens mass, and lens light to construct an image which is compared against the imaging
data (top). SKiNN (orange) inputs lens mass, lens light, and kinematics parameters to produce a vrms
image which is compared against the kinematic data (bottom). The combined χ2tot can be minimized
to jointly optimize the combined model.

3

Methods

In essence, SKiNN precalculates JAM for use in lensing applications. Like JAM, the input is a vectorof parameters which describes the lensing galaxy, and the output is a vrms image, the quadratic sumof velocity dispersion and rotational velocity. More concretely, SKiNN can be seen as a functionΨθ : R8 → Rd×d , mapping an 8-dimensional vector of galaxy parameters into a d × d image of vrms .
8d×d
Given a training dataset D = {xi , yi }Nand N is the size of the dataset,
i=1 where x ∈ R , y ∈ R
the training process consists in finding an optimal set of parameters θ∗ , such that a loss function L,
measuring the performance of Ψ on D, is minimized. We use the standard mean-squared-error as theloss function, which we minimize using the Adam optimizer [Kingma and Ba, 2014]1 . The design of
the architecture of SKiNN consists of 5 blocks each comprising two 2-dimensional convolutional
layers followed by an upsampling layer and a ReLU nonlinearity.
Data set construction Our data set is constructed using JAM, which SKiNN is intended to mimic.
JAM [Cappellari, 2008] works by decomposing the sky-projected (2D) mass and light distributions ofa galaxy into MGEs, for which deprojection into 3D is calculable given an inclination of the galaxy,i. With an anisotropy, β, which describes the mixture of orbits aligned with the main ellipsoidal
axis relative to those in other directions, the axisymmetric Jeans equations are solved in cylindricalcoordinates, and the velocity is projected into the sky plane. For this proof of concept, we restrict
ourselves to a single (but widely used) class of profiles for the lens mass and light, using Power law
Elliptical Mass Distributions [PEMD Barkana, 1998] and elliptical Sérsic light profiles [Sérsic, 1963].
In total, these distributions are described with 6 parameters (see Appendix A), which, together with iand β, constitute the 8 parameters of the x vector.
From these parameters, we use the GLEE software [Gravitational Lens Efficient Explorer, Suyu
et al., 2010, 2012] to emulate the light and mass profiles which are then fed into JAM, and hence
create an injective mapping from the x vector to the vrms image. The ranges of parameters we probed
are physically motivated (see Appendix A). We create vrms images of 551 × 551 pixels at a larger
size and higher resolution than real data. Image creation through JAM is quite slow, taking ∼ 15
seconds per image.
We construct our full data set with a total of 5000 input-output pairs. Our training set is a random
subsample of N = 4000 of these, while we keep 500 instances for the validation set and 500 for
1

Overall our final model counts 7,065,451 trainable parameters. The pre-trained weights of our best model
are available upon reasonable request, with our intention being to make the final version of SKiNN open-source.

3

Figure 2: Two example vrms images created by SKiNN and compared to the JAM ground truth. The
black circle indicates the innermost two arcseconds, where real data is most constraining. Within thisregion, the error in the emulation is less than 1%, as shown by the histograms on the right.
the test set. As the training proceeds we constantly monitor the performance of the model on thevalidation set and save its weights whenever an improvement in the validation accuracy is observed.
Computational details The network is implemented in Pytorch [Paszke et al., 2019] and is trainedon 5 Tesla P100 GPUs with 16GB memory each. For efficient multi-GPU training we wrap ourmodel into Pytorch Lightning [Falcon et al., 2019], a recent Pytorch-compatible framework for large
scale GPU-based training of NNs.

4

Results

Speed Training our model in the above setting requires about one day, the stopping criterion being
the validation loss not decreasing for a certain number of consecutive epochs. Once trained, themodel can be run either on CPU or on a single GPU (the latter being roughly 10 times faster than
the former). When run on the GPU, the model infers the velocity image associated with an input
parameter vector in about 50 ms, which is roughly 300 times faster than JAM. Clearly, this advantage
comes at the price of a rather expensive training phase which, however, has to be performed only
once.
Emulator accuracy We assess the performance of our model by computing the relative error foreach pixel for the 500 input-output pairs in the test set. Two example generated vrms images withresiduals are shown in Fig. 2. Parts of the image can have error of ∼few percent for some images,
but the particularly important region to match is the innermost region of the image, where real data isable to constrain. Approximated as the black circle in Fig. 2, the error within this region is within±1% for almost all pixels. Across the 500 test images, the median absolute error averaged over anentire image is around 0.47% while the 90th percentile is approximately 1.1%. This is lower than the
typical systematic uncertainty of ≳ 2% in the velocity measurements [e.g., Collett et al., 2018]. The
model is thus capable of capturing fine details of the ground-truth velocity images while generatingthem much faster for applications to real observations of galaxies.
Limitations and outlook This is only a preliminary version of SKiNN, which we intend to updatewith other physically motivated mass models. We are also considering possible ways to speed up thetraining of the network by exploiting the symmetry of the vrms images.
In this current iteration, our result is limited to the specifics of our training set, that is, we can only
use this NN to quickly calculate kinematics for axisymmetric systems with a PEMD mass profile andSérsic light profile. In the short term, this technique can be expanded to more general mass and lightmodels; one only needs to construct new data sets using JAM and retrain the NN. In the longer term,4this general method can be reapplied using other kinematics calculation software to go beyond theaxisymmetric assumption inherent in JAM.
Joining SKiNN to the lens modeling procedure is currently undergoing implementation in thelenstronomy [Birrer and Amara, 2018, Birrer et al., 2021], GLEE [Suyu and Halkola, 2010, Suyu
et al., 2012] and GLaD2 [Chirivì et al., 2020, Yıldırım et al., 2020] software packages, consistent
with the framework of Yıldırım et al. [2020]. This joint modeling framework will efficiently tackle
the strongest source of bias in lensing studies thanks to the constraining power of spatially resolved
kinematics [Birrer et al., 2020, Yıldırım et al., 2021]. The resulting determination of cosmologicalparameters will then be more accurate.
A joint lensing+SKiNN framework will represent a methodology that likely has analogous applications in other fields. For any modeling context in which a piece of the model is calculable, but too
computationally expensive to implement and therefore excluded, a similar technique to this work
could be applied: the slow piece can be replaced by a trained NN while retaining the existing physicalaspects of the overall model.

5Conclusions

We present SKiNN, a NN created to replace the slow process of galaxy kinematics emulation which
can currently be done through JAM. The process of creating a training set and fitting the NN is asignificant computational investment, but the resulting trained network is able to create similar outputs
to JAM at greatly increased speed. The accuracy of the emulator is < 1% in the region comparablewith real data, with median error ∼ 1% across entire images. The roughly 300-fold increase in speed
makes it possible to model kinematics jointly with existing lens modeling, for which we are currently
building infrastructure, with the goal being to perform cosmological analysis accurately by using
lensing and kinematic data together to break degeneracies. We believe this method can be expandedwith more general training sets, and the general strategy likely has applications in other fields.

Impact statement
The general methodology used in this work has had applications for related fields, as discussed in
Sect. 2, and therefore likely can be applied to additional fields. Where applicable, the net effect would
be to speed up existing modeling, or perhaps add pieces to existing models which were previously
computationally unfeasible. While we do not foresee any ethical concerns with these applications, it
is possible that the general methodology could be used to speed up hypothetical modeling procedures
which are themselves unethical.

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8

1. For all authors...
(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s
contributions and scope? [Yes](b) Did you describe the limitations of your work? [Yes](c) Did you discuss any potential negative societal impacts of your work? [Yes](d) Have you read the ethics review guidelines and ensured that your paper conforms tothem? [Yes]2. If you are including theoretical results...
(a) Did you state the full set of assumptions of all theoretical results? [N/A](b) Did you include complete proofs of all theoretical results? [N/A]
3. If you ran experiments...
(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [No] We intend to
make the final version open source. The current version is available upon reasonable
request(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they
were chosen)? [Yes] Section 3(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] Specifically, errors with respect to emulation of images,
Section 4(d) Did you include the total amount of compute and the type of resources used (e.g., typeof GPUs, internal cluster, or cloud provider)? [Yes] Section 3
4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
(a) If your work uses existing assets, did you cite the creators? [Yes] JAM and GLEE
modeling codes(b) Did you mention the license of the assets? [N/A] JAM and GLEE are open-source(c) Did you include any new assets either in the supplemental material or as a URL? [N/A](d) Did you discuss whether and how consent was obtained from people whose data you’reusing/curating? [N/A](e) Did you discuss whether the data you are using/curating contains personally identifiable
information or offensive content? [N/A]
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A

AppendixLens profile parameters Here we describe the profiles used for the mass and light distributions inthis work. The mass is a PEMD, with convergence (i.e., dimensionless surface density) described as
γPL −1
κPEMD (x, y) = θE

3 − γPL
1 + qm 1−γ2PL

y2
2
,
x + 2
qm

(1)

with Einstein radius θE , axis ratio qm , and slope of the 3D mass density γPL . For the light, we use an
elliptical Sérsic profile,
1/n( q 2)x + ( qy2 )2LI(x, y) = A exp −k − 1 ,(2)reffwith effective radius reff , Sérsic index n, and axis ratio qL (the normalization A is irrelevant for the
kinematics calculation and constant k is defined uniquely through n). In total, six input parametersdescribe the mass and light distributions: θE , γPL , qm , qL , reff , and n.
These parameters together with β and i constitute the input vector x. The ranges from whichthese parameters are drawn are physically motivated. Each realization is drawn uniformly fromthe range indicated: For θE ∈ [0.5”, 2”] and γPL ∈ [1.5, 2.5], we select ranges which cover theobserved SLACS and TDCOSMO lens systems ((Gomer et al., 2022)); ranges for reff ∈ [0.5θE , θE ]and n ∈ [2, 4] and were chosen based on the priors for the Time Delay Lens Modeling Challenge((TDLMC Ding et al., 2021)); for qm ∈ [0.6, 1.0] and qL ∈ [0.6, 1.0], we have selected a wider range
than the TDLMC priors to better generalize to more elliptical galaxies; anisotropy β ∈ [−0.4, 0.4]
is motivated by the prior used by Yıldırım et al. ((2020)), but slightly widened as motivated by
JAM models of SAURON observations of early-type galaxies ((Cappellari et al., 2007)); Inclinationi ∈ [arccos(qmin ), π/2] has a lower bound set by the flattest Gaussian component of the MGE
decomposition (with axis ratio qmin ), below which the deprojection becomes nonphysical.

10