Domain Adaptation for Simulation-Based Dark
Matter Searches with Strong Gravitational Lensing

Stephon Alexander and Michael W. Toomey
Department of Physics and Brown Theoretical Physics Center
Brown University
Providence, RI 02912
Sergei Gleyzer
Department of Physics & Astronomy
University of Alabama
Tuscaloosa, AL 35401
Pranath Reddy
University of Florida
Gainesville, FL 32611
Marcos Tidball
Universidade Federal do Rio Grande do Sul
Porto Alegre, Brazil

Abstract
The application of machine learning for quantifying dark matter substructure is
growing in popularity. However, due to the differences with the real instrumental
data, machine learning models trained on simulations are expected to lose accuracy
when applied to real data. Here, domain adaptation can serve as a crucial bridge
between simulations and real data applications. In this work, we demonstrate thepower of domain adaptation techniques applied to strong gravitational lensing datawith dark matter substructure. We show with simulated data sets representative of
Euclid and Hubble Space Telescope (HST) observations that domain adaptation
can significantly mitigate the losses in the model performance when applied to new
domains.

1

Introduction

One of the great achievements of astrophysics in the last century was the realization by Zwicky,
Rubin and others that the observed baryonic mass (stars, galaxies, etc.) was not consistent with the
dynamics of galaxies and clusters. A natural solution to this problem was to consider unseen dark
matter compensating for this discrepancy. Presently, all efforts aimed at extracting a non-gravitationalsignature of dark matter have come up empty. While this does not mean that dark matter can notcommunicate with Standard Model (SM) particles, as its SM couplings may be strongly suppressed,
there is also the possibility that such interactions do not exist. Since its discovery, subsequent evidence
for particle dark matter from its coupling to gravity is almost irrefutable [1, 2, 3]. A particularly
sensitive probe is strong gravitational lensing [4, 5, 6] for which we restrict ourselves in this article.
Strong gravitational lensing has already seen some promising success in extracting information
about dark matter substructure. More recently, there has been a plethora of applications of machine
Machine Learning and the Physical Sciences workshop, NeurIPS 2022.

learning to this challenge, ranging from classification [7, 8, 9], regression [10], segmentation analysis[11, 12], and anomaly detection [13]. To date, all works have exclusively focused on the applicationof these techniques to simulations, in large part due to the limited availability of strong lensing data;something that is anticipated to change in the near future with the commissioning of the Vera C.
Rubin Observatory and the launch of Euclid [14, 15]. However, naively applying models trained on
simulations to real data will not likely to be successful, as the data idiosyncrasies will significantly
diminish the accuracy of the model. A promising method to bridge the gap between a model trainedon simulations and real data is based on the technique of domain adaptation (DA) [16]. A subset oftransfer learning, domain adaptation is focused on the generalization of the model across different
domains or data sets drawn from different underlying distributions. The goal of domain adaptation isto adapt a model trained on one data set (source) by generalizing it to another domain (target), where
the objective of the model is unchanged. In practice, domain adaptation can be realized in several
ways, including supervised, semi-supervised, and unsupervised approaches [17, 18, 19].
In this work, we consider domain adaptation for dark matter searches in strong gravitational lensing.
With the present lack of sufficient real data, we use two data sets to realize mock observations withdifferent surveys, HST and Euclid, of galaxy-galaxy strong gravitational lensing to carefully test
the performance of domain adaptation prior to its applications to real data. We evaluate the models
trained on the source data set to identify various types of dark matter substructure on the target data
set. We compare the performance of two domain adaptation algorithms based on convolutional neuralnetworks and equivariant neural networks that incorporate a known group symmetry to enhance
performance.

2

Dark Matter Detection and Strong Gravitational Lensing

The Λ Cold Dark Matter (ΛCDM) model envisions near-scale invariant density fluctuations, present
in the early universe, serving as seeds of large scale structure via hierarchical structure formation.
Structures such as dark matter halos are formed from the coalescence of smaller halos [20]. Evidence
for such merges has been observed in our Galaxy [21, 22, 23] and is a general prediction of Nbody simulations where evidence of mergers should remain largely intact. Comparison between
simulation and observation indicates good agreement with ΛCDM on large-scales [1, 2, 3]. However,discrepancies begin to arise on smaller, sub-galactic scales. These include the core-vs-cusp, too big
to fail, missing satellite, and diversity problems.
Similar to [7, 13] we consider data sets of three substructure classes; no substructure, NFW subhalosof cold dark matter, and vortex substructure of superfluid type (axion) dark matter. We construct thesimulations with lenstronomy [24] to mimic the characteristics of HST and Euclid using the default
instrument and observational settings. For background sources we use images of galaxies from the
Galaxy10 DECals data set [25] processed with a Gaussian mask and convolved with a Gaussian ofsize 2 pixels. This prevents the lensing of unwanted foreground sources and noise in the image. We
choose the apparent magnitude of the background galaxy such that the the signal-to-noise ratio (SNR)of the lensing arcs are consistent with real lensing data – SN R ∼ 20 [26].
Domain adaptation requires at least two data sets, the source and the target. In this work we will
demonstrate domain adaptation between models trained on mock HST observations and Euclid (andvice versa). Thus, it will be our goal to successfully adapt and evaluate the algorithms trained on agiven source data set to the target data set.

3

Domain Adaptation

The goal of our work is to train a supervised model on a source data set and adapt it to a target data
set. For this task we use a Convolutional Neural Network (CNN), specifically EfficientNet [27], as
our base architecture. This is the same type of architecture that has achieved top performance inprevious applications to lensing data sets [7, 13, 28]. More generally, CNNs are known to outperform
other methods of classification for strong gravitational lenses [29], nonetheless, as noted by [30], a
model trained on simulations can perform poorly on real data.
To improve the performance of models trained on simulated data, we use unsupervised domain
adaptation, which attempts to mitigate the effects of the domain shift between the source and thetarget domains. It enables a transfer of knowledge gained from a labeled source data set to a distinct
2

Figure 1: Example of lensing simulation of mock HST (top) and Euclid (bottom) data sets. Imageof real galaxy from the Galaxy10 DECals data set (left), image processed with a Gaussian filter toremove unwanted background (center), and final lensed image (right).

unlabeled target data set, within the constraint that the objective remains the same [31]. Examples
from each data set are shown in Figure. 1.
We utilize Adversarial Discriminative Domain Adaptation (ADDA) [32], an adversarial adaptation
method with the goal of minimizing the domain discrepancy distance through an adversarial objectivewith respect to a discriminator. Ideally the discriminator will be unable to distinguish between thesource and the target distributions. We consider that we have access to source images Xs and labels
Ys that come from a source distribution ps (x, y) and also target images Xt from a target distribution
pt (x, y). Our objective is to learn a target encoder Mt and classifier Ct that classifies Xt into K
classes.
In addition to the baseline CNN models, we also consider an Equivariant Neural Network (ENN)[33] for substructure classification. ENNs can be thought of a generalization of a CNN that encode
the representation of a useful symmetry, both global or local, such that its group convolutions areinvariant symmetries present in the data. This is useful if there is a known symmetry. As we expect
lensing images to have symmetries beyond simple translation, for example rotations, the flexibility ofchoosing different group representations is expected to improve the performance.
The ENN we use consists of a group equivariant convolutional neural network [34] with six equivariant
convolution blocks. We utilize the dihedral group D2 , whose symmetry mappings include the identity,
rotations by ±π and horizontal/vertical reflections. Each block is composed of a convolutional
layer, a batch normalization layer and a ReLU activation function. After each pair of layers we
perform channel-wise average-pooling and in the end we use a fully connected layer for multiclassclassification.

4

Results

We test the applicability of unsupervised domain adaptation using ADDA between mock HST and
Euclid observations in the context of multi-class classification of three types of substructure: no
substructure, NFW subhalos of CDM, and superfluid DM vortices. We employ two different baseclassifiers - EfficientNet and an ENN. For training we use 30, 000 images for the source domain
and 30, 000 images for the target domain; in both cases there are 10, 000 images per class. For
3

Euclid to HST

1.0

0.6

True Positive Rate

0.8

True Positive Rate

0.8

0.6

0.4

0.4

ADDA (ENet), 0.924
Target (ENet), 0.685
ADDA (ENN), 0.999
Target (ENN), 0.915

0.2
0.0

HST to Euclid

1.0

0.0

0.2

0.4

0.6

False Postive Rate

0.8

ADDA (ENet), 0.991
Target (ENet), 0.789
ADDA (ENN), 0.999
Target (ENN), 0.973

0.2
1.0

0.0

0.0

0.2

0.4

0.6

False Postive Rate

0.8

1.0

Figure 2: Application of ADDA for domain adaptation between two different mock simulations ofstrong lensing surveys: from Euclid to HST (left) and HST to Euclid (right). Dashed lines represent
the naive application of classifier trained on source data set to target and solid represents result after
training with ADDA. Red curves represent models based on EfficientNet and blue based on an ENN.

validation we use 7, 500 images for the source domain and 7, 500 images for the target; in bothcases there are 2, 500 images per class. We use the Adam optimizer [35] to minimize the loss. We
trained both EfficientNet and the ENN for 200 epochs, training with a patience of 15 epochs, suchthat if the accuracy of the model does not improve in 15 epochs, the training is stopped. Learning
rate, weight decay and other hyperparameters were optimized through a hyperparameter search.
We have set the learning rate to 2 × 10−3 and weight decay to 1 × 10−5 . We utilize the area
under the ROC curve (AUC) on the target validation set as the metric for classifier performancefor all the models. All quoted AUC values are macro-averaged. All machine learning models were
implemented using PyTorch [36] and are run on a single NVIDIA Tesla P100 GPU. The ROC curves
for both combinations of source/target in addition to results for different architectures are presented
in Figure 2.
We first train EfficientNet on the source data sets where it achieved a macro-averaged AUC ≈ 0.999
for both data sets and an accuracy of ≈ 99.6% for Euclid and ≈ 99.5% for HST. Applying thesemodels to the corresponding target data sets naively, i.e. without domain adaptation, results in an
AUC of ≈ 0.685 (0.789) when the Euclid (HST) model is applied to the HST (Euclid) data set andan accuracy of ≈ 50% (51%), a significantly degraded performance. This degradation is anticipated,even though the underlying physics is identical, due to different instrument systematics.
Following the application of unsupervised domain adaptation with ADDA, we observe a significant
improvement in the application to the target data set. With the EfficientNet based algorithm weachieve a ≈ 35% improvement in AUC, at ≈ 0.924, and accuracy of ≈ 87% adapting from Euclid
to HST data sets. Going the other direction the AUC is improved to 0.991 and reaches an accuracyof 0.88%. This is a remarkable improvement in performance which can be further appreciated bycomparing the ROC curves in Figure. 2. The red dashed curves correspond to the naive application of
the EfficientNet and the solid curves correspond to the performance with ADDA. While there is stillsome room for improvement in DA from Euclid to HST (the left figure), DA from HST to Euclid
results in near-perfect classification.
While EfficientNet saw a great improvement with ADDA, it is clear there is still a lot on the table asstandard binary classification has an AUC near unity. This informs us that one should be able to further
improve on the UDA performance, particularly in the case of going from Euclid to HST. A natural
improvement to consider is changing the baseline classifier to an equivariant neural network. When
we leverage extra, known symmetries of lensing we should see an appreciable bump in performanceas our model does not need to learn this unimportant symmetry. In principal one could also consider
the inclusion of other UDA but we will not consider that in this work.
Training and testing on source data sets the ENN achieves an AUC of ≈ 0.999 on both data sets
and an accuracy of 99.4% and 99.7% for Euclid and HST respectively. The naive application to the4test data set (i.e. no domain adaption) again results in degraded performance, realized with an AUC
for the Euclid (HST) trained model applied to HST (Euclid) data of ≈ 0.915 (0.973). This realizes
a remarkable performance simply in the naive application of the ENN to the target data set. After
training with ADDA we find that our models are then able to achieve effectively perfect classification
with AUCs of ≈ 0.999 for both combinations of source/target and accuracy an of 99.1% (97.5%)
from Euclid (HST) to HST (Euclid). We see that the performance with our ENN is near optimal,
achieving a significant performance bump over the CNN. This is truly impressive as our DA algorithm
is unsupervised – it never saw the labels from the target data set, yet was nearly perfect at adapting to
the new domain. This is, of course, exactly the kind of transfer of knowledge one would hope to beable to do between simulations and real data sets.

5

Discussion & Conclusion

With the upcoming arrival of strong gravitational lensing data from Euclid and the Vera Rubin
Observatory, it is imperative to assess how algorithms trained on simulations can be applied to realworld data. In this work, we studied how unsupervised domain adaptation algorithms can be used
to adapt a model trained on one set of data (the source) to another set of gravitational lensing data(the target). To make a quantitative comparison, we based our work on two sets of realistic lensing
simulations, a mock data set from Euclid and another for HST. While we have restricted ourselves tosubstructure classification in this work, domain adaptation techniques can be additionally useful inthe broader context of studying dark matter, from regression to image segmentation, in applications
to real world strong gravitational lensing data sets.

Broader Impact
This work serves to augment the understanding and application of machine learning in cosmology which is still very much in its initial stages. This work serves to increase the accessibility to thoseinterested in applications of machine learning for strong lensing applications around the globe as our
simulation data set and analysis pipeline is open sourced. Given the computational requirements of
our implementation, those who have limited access to computing power may be at a disadvantage.

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