imprints of cosmological parameters (Villaescusa-Navarro et al. 2022). The question however is: can this information be retrieved under the uncertainties of the baryonic model?

In Villaescusa-Navarro et al. (2020), the authors illustrated on an analytically tractable toy model that a Neural Network can be trained to optimally marginalise over baryonic effects as long as the training data left the associated parameters free to vary according to a given prior. While promising, this result didn't necessarily imply that this implicit marginalisation would be robust to a change in baryonic model. To investigate precisely this question, Villaescusa-Navarro et al. (2020) trained a CNN on 2D projected density fields from the CAMELS dataset to regress ($\Omega_{m}$, $\sigma_{8}$). They showed that models trained on IllustrisTNG lead to almost unbiased results on SIMBA and vice-versa, implying that in the process of learning a summary statistic that marginalises over baryonic effects, the neural networks are discarding the part of the signal affected by baryons, and are therefore no longer extremely sensitive to the details of the modelling of those effects. This result remains of course limited, but is very encouraging for the analysis of data.

## 6.2.2. Dark matter substructures from strong gravitational lensing

In previous sections we have described how deep learning has significantly impacted the detection and characterisation of strong gravitational lenses. Strong lensing systems can be used as well to constrain the substructure of dark matter on extended arcs which contains a wealth of information about the properties and distribution of dark matter on small scales and, consequently, about the underlying nature of the dark matter particle. The information can therefore be used to distinguish between various dark matter models—warm or cold dark matter for example.

However, probing this effect is challenging since the likelihood function for realistic simulations of population-level parameters is intractable. Alexander et al. (2020) followed a simple approach which consists in converting the inference problem into a classification of CNNs in classification mode to distinguish various types of dark matter models. They show they can reach AUC scores above 90%, for images with no substructure, spherical subhalos, and vortices on idealised simulations. Varma, Fairbairn, & Figueroa (2020) also used a CNN for multi-class classification in seven different categories corresponding to different lower mass cut-offs of the sub-halo mass function. They report being able to correctly identify the lower mass cut-off within an order of magnitude to better than ~90% accuracy.

Other works have attempted to go a step beyond by estimating the parameters describing the dark matter substructure in a regression mode using simulation based inference with deep learning. The first work exploring this is by Brehmer et al. (2019). The authors characterise substructure with a set of parameters and show, in a proof-of-concept application to simulated data, that neural networks can be trained to accurately estimate likelihood ratios associated to the dark matter substructure parameters (Figure 44). They conclude that ~100 strong lenses might be enough for characterising the abundance of substructure down to ~10%. Coogan, Karchev, & Weniger (2020) also used a likelihood-free approach to infer posterior distribution of the mass and positions of sub-halos in simulated systems. Vernardos, Tsagkatakis, & Pantazis (2020) applied a similar approach combining a simulator based on Gaussian Random Fields for the potential but combined

with images of real galaxies for the lensed source and show they can also constrain the substructure parameters.

### 6.2.3. Reconstructing cosmological fields

Applications in cosmology also go beyond cosmological parameter estimation, and an increasingly large number of works explore applications of deep learning for inferring latent cosmological fields from observations.

Weak-Lensing Mass-Mapping One active research question in weak gravitational lensing is the reconstruction of the matter distribution that gives rise to the measured lensing effect, a task known as mass-mapping. This problem is made particularly difficult by the noisy nature of the observations (intrinsic galaxy ellipticities being much larger than the weak gravitational shear) and the need to invert a linear operator mapping shear to projected mass (also known as convergence) which becomes ill-posed in the presence of survey masks. This is therefore an instance of an ill-posed inverse problem, which does not have any unique solution, in the sense that different mass-maps can lead to a shear signal equally compatible with the data. For these problems, the hope of Deep Learning approaches is that they can learn, implicitly or explicitly, a prior on the signal to recover from training data, and use that prior to solve the inverse problem in an optimal fashion.

The first class of methods, explored in Shirasaki, Yoshida, & Ikeda (2019), Shirasaki et al. (2021), used a conditional adversarial network adapted from the pix2pix model (Isola et al. 2016) for image-to-image translation. In this approach, a first network with a U-Net structure is tasked with taking a noisy convergence obtained by a rough direct inversion of the shear field as an input, and outputting an estimate of the true convergence map. To train this denoiser, a second network is introduced to act as a discriminator, taking as an input both noisy and denoised convergence maps, coming either from the denoiser or from the training set, and outputting a probability between 0 and 1 of the denoised image being real. The model is then trained with a combination of a standard adversarial loss and an l1 loss between the recovered denoised mass-map and the truth from simulations. It is to be noted here that this adversarial model is not a generative model, the denoiser does not take random variables as an input and is therefore deterministic. Instead the adversarial loss can be understood as a learned similarity metric to compare recovered to true map. Training such a model typically requires a set of ray-traced lensing simulations, that are corrupted to include the same noise properties and masks as present in the data. Shirasaki et al. (2021) generated mock HSC observations including photometric redshift uncertainties, shape measurement uncertainties, realistic galaxy ellipticity noise and distribution on the sky, and actual HSC survey masks. On simulations the authors find that about 60% of the peaks identified on the denoised maps have significant clusters counterparts, against about 85% of positive matches on true maps, highlighting that the recovered map still correlate well with real structures, and the authors further show that the 1-point statistics of the recovered map shows stronger cosmological dependence than the noisy maps, hinting at interesting applications in constraining cosmological parameters. While this approach provides empirically good results, one drawback of this pix2pix training is that the recovered map does not have a clear Bayesian interpretation. As we will see below, subsequently developed techniques abandon this effective adversarial training
imprints of cosmological parameters (Villaescusa-Navarro et al. 2022). The question however is: can this information be retrieved under the uncertainties of the baryonic model?

In Villaescusa-Navarro et al. (2020), the authors illustrated on an analytically tractable toy model that a Neural Network can be trained to optimally marginalise over baryonic effects as long as the training data left the associated parameters free to vary according to a given prior. While promising, this result didn't necessarily imply that this implicit marginalisation would be robust to a change in baryonic model. To investigate precisely this question, Villaescusa-Navarro et al. (2020) trained a CNN on 2D projected density fields from the CAMELS dataset to regress ($\Omega_{m}$, $\sigma_{8}$). They showed that models trained on IllustrisTNG lead to almost unbiased results on SIMBA and vice-versa, implying that in the process of learning a summary statistic that marginalises over baryonic effects, the neural networks are discarding the part of the signal affected by baryons, and are therefore no longer extremely sensitive to the details of the modelling of those effects. This result remains of course limited, but is very encouraging for the analysis of data.

## 6.2.2. Dark matter substructures from strong gravitational lensing

In previous sections we have described how deep learning has significantly impacted the detection and characterisation of strong gravitational lenses. Strong lensing systems can be used as well to constrain the substructure of dark matter on extended arcs which contains a wealth of information about the properties and distribution of dark matter on small scales and, consequently, about the underlying nature of the dark matter particle. The information can therefore be used to distinguish between various dark matter models—warm or cold dark matter for example.

However, probing this effect is challenging since the likelihood function for realistic simulations of population-level parameters is intractable. Alexander et al. (2020) followed a simple approach which consists in converting the inference problem into a classification of CNNs in classification mode to distinguish various types of dark matter models. They show they can reach AUC scores above 90%, for images with no substructure, spherical subhalos, and vortices on idealised simulations. Varma, Fairbairn, & Figueroa (2020) also used a CNN for multi-class classification in seven different categories corresponding to different lower mass cut-offs of the sub-halo mass function. They report being able to correctly identify the lower mass cut-off within an order of magnitude to better than ~90% accuracy.

Other works have attempted to go a step beyond by estimating the parameters describing the dark matter substructure in a regression mode using simulation based inference with deep learning. The first work exploring this is by Brehmer et al. (2019). The authors characterise substructure with a set of parameters and show, in a proof-of-concept application to simulated data, that neural networks can be trained to accurately estimate likelihood ratios associated to the dark matter substructure parameters (Figure 44). They conclude that ~100 strong lenses might be enough for characterising the abundance of substructure down to ~10%. Coogan, Karchev, & Weniger (2020) also used a likelihood-free approach to infer posterior distribution of the mass and positions of sub-halos in simulated systems. Vernardos, Tsagkatakis, & Pantazis (2020) applied a similar approach combining a simulator based on Gaussian Random Fields for the potential but combined

with images of real galaxies for the lensed source and show they can also constrain the substructure parameters.

### 6.2.3. Reconstructing cosmological fields

Applications in cosmology also go beyond cosmological parameter estimation, and an increasingly large number of works explore applications of deep learning for inferring latent cosmological fields from observations.

Weak-Lensing Mass-Mapping One active research question in weak gravitational lensing is the reconstruction of the matter distribution that gives rise to the measured lensing effect, a task known as mass-mapping. This problem is made particularly difficult by the noisy nature of the observations (intrinsic galaxy ellipticities being much larger than the weak gravitational shear) and the need to invert a linear operator mapping shear to projected mass (also known as convergence) which becomes ill-posed in the presence of survey masks. This is therefore an instance of an ill-posed inverse problem, which does not have any unique solution, in the sense that different mass-maps can lead to a shear signal equally compatible with the data. For these problems, the hope of Deep Learning approaches is that they can learn, implicitly or explicitly, a prior on the signal to recover from training data, and use that prior to solve the inverse problem in an optimal fashion.

The first class of methods, explored in Shirasaki, Yoshida, & Ikeda (2019), Shirasaki et al. (2021), used a conditional adversarial network adapted from the pix2pix model (Isola et al. 2016) for image-to-image translation. In this approach, a first network with a U-Net structure is tasked with taking a noisy convergence obtained by a rough direct inversion of the shear field as an input, and outputting an estimate of the true convergence map. To train this denoiser, a second network is introduced to act as a discriminator, taking as an input both noisy and denoised convergence maps, coming either from the denoiser or from the training set, and outputting a probability between 0 and 1 of the denoised image being real. The model is then trained with a combination of a standard adversarial loss and an l1 loss between the recovered denoised mass-map and the truth from simulations. It is to be noted here that this adversarial model is not a generative model, the denoiser does not take random variables as an input and is therefore deterministic. Instead the adversarial loss can be understood as a learned similarity metric to compare recovered to true map. Training such a model typically requires a set of ray-traced lensing simulations, that are corrupted to include the same noise properties and masks as present in the data. Shirasaki et al. (2021) generated mock HSC observations including photometric redshift uncertainties, shape measurement uncertainties, realistic galaxy ellipticity noise and distribution on the sky, and actual HSC survey masks. On simulations the authors find that about 60% of the peaks identified on the denoised maps have significant clusters counterparts, against about 85% of positive matches on true maps, highlighting that the recovered map still correlate well with real structures, and the authors further show that the 1-point statistics of the recovered map shows stronger cosmological dependence than the noisy maps, hinting at interesting applications in constraining cosmological parameters. While this approach provides empirically good results, one drawback of this pix2pix training is that the recovered map does not have a clear Bayesian interpretation. As we will see below, subsequently developed techniques abandon this effective adversarial training
<figure><img src="image_46.png" /><figcaption>**Figure 44.** Illustration of lensed systems and the corresponding likelihood ratio maps estimated with simulation based inference and deep learning. The black crosses show the true values. Figure from Brehmer et al. (2019).</figcaption></figure>

but gain a proper Bayesian interpretation of the output of the models.

In Jeffrey et al. (2020), the authors introduce a method, called DeepMass, using a similar Unet architecture but trained under a simple Mean Squared Error loss between true convergence map and output from the Unet. As highlighted by the authors, a regression model trained under an MSE implicitly learns to predict the mean of the posterior distribution of the target given the input. In the present case, the authors generate a suite of ray-tracing lensing simulations matching the DES Science Verification setting, the noiseless convergence maps from these simulations provide an implicit prior, while the simulated shear observations (including realistic noise and masks) provide an implicit likelihood. By training the model to reconstruct the true convergence given simulated shear data under an MSE loss, the model will therefore learn to output the mean posterior convergence map, under the implicit prior and implicit likelihood that are provided by the training set. In this DES SV setting, the authors demonstrate that this approach leads to an 11% improvement in MSE evaluated on simulations compared to a standard Wiener filter approach.

While Jeffrey et al. (2020) had the benefit of providing a Bayesian understanding of the network output, it did not provide any sort of uncertainty on the recovered map, which makes the interpretation and scientific exploitation of these results difficult. To overcome these limitations, Remy et al. (2022) introduced an approach allowing to sample from the full posterior distribution of the mass-mapping problem. They proposed to use a similar U-Net architecture, not to directly estimate the convergence map, but to learn from simulations a generative prior on convergence maps using a Denoising Score Matching technique (Song & Ermon 2019). With this approach, it can be shown that a neural network trained as a Gaussian denoiser under a simple MSE loss will actually learn the score of the data distribution, that is, the gradient of the log likelihood of the data. Once trained on simulations, this U-Net gives explicit access to the prior. The authors show that it is possible to combine this learned prior with an explicit data likelihood in an Hamiltonian Monte Carlo sampling procedure to sample from the full posterior distribution of the problem. Figure 45 illustrates on the bottom row posterior samples achieved with this method on a simulation of the HST/ACS COSMOS field, compared to the ground truth (top left). Most interestingly, it is

shown that the mean of the posterior samples indeed converge to the same solution as the DeepMass estimate.

*Initial Conditions Reconstructions* One particularly interesting problem for the analysis of galaxy surveys is the reconstruction of the initial density field from the observed Large-Scale Structure. This can for instance be used to refine Baryonic Acoustic Oscillations (BAO) measurements (e.g. Schmittfull, Baldauf, & Zaldarriaga 2017), or part of a Bayesian forward modelling inference scheme (Seljak et al. 2017).

In a first example of applying deep learning to this problem, Mao et al. (2020) proposed a 3D CNN trained on N-body simulations to recover the initial density field at $z=10$ given the final density field at $z=0$ under a density weighted mean squared error loss. They find that their convolution model is capable of beating a standard linear reconstruction on scales smaller than $k \le 0.2h \text{ Mpc}^{-1}$, but under performs on larger scales. Interestingly they find that their learned inversion can extrapolate to some extent to other cosmological parameters; a model trained on WMAP7 cosmology is capable of reconstructing initial conditions on WMAP5 simulations that lead to a slightly biased BAO signal, but still significantly different from the WMAP7 signal of the training data.

Going beyond a direct inversion method, Modi et al. (2021a) proposed an iterative reconstruction scheme based on Recurrent Inference Machines (RIM, Putzky & Welling 2017). This approach can be thought of as a learned iterative reconstruction algorithm. At each iteration a recurrent neural network proposes an update of the current reconstruction based on the knowledge of previous iterations and on the gradient of an explicit data likelihood term. In the absence of this neural network, the algorithm would result in a standard gradient descent scheme leading to a Maximum Likelihood Estimation of the initial conditions. By training the Neural Network to minimise at each iteration the Mean Squared Error between the current solution and the true initial conditions, the network will learn both an implicit prior, and a fast inference scheme to minimise the number of updates needed. The result will therefore be a fast convergence towards the mean posterior solution. Most interestingly, in order to compute this explicit likelihood, a differentiable forward model is needed, that is, in this case an N-body simulation. The authors make use of the
<figure><img src="image_46.png" /><figcaption>**Figure 44.** Illustration of lensed systems and the corresponding likelihood ratio maps estimated with simulation based inference and deep learning. The black crosses show the true values. Figure from Brehmer et al. (2019).</figcaption></figure>

but gain a proper Bayesian interpretation of the output of the models.

In Jeffrey et al. (2020), the authors introduce a method, called DeepMass, using a similar Unet architecture but trained under a simple Mean Squared Error loss between true convergence map and output from the Unet. As highlighted by the authors, a regression model trained under an MSE implicitly learns to predict the mean of the posterior distribution of the target given the input. In the present case, the authors generate a suite of ray-tracing lensing simulations matching the DES Science Verification setting, the noiseless convergence maps from these simulations provide an implicit prior, while the simulated shear observations (including realistic noise and masks) provide an implicit likelihood. By training the model to reconstruct the true convergence given simulated shear data under an MSE loss, the model will therefore learn to output the mean posterior convergence map, under the implicit prior and implicit likelihood that are provided by the training set. In this DES SV setting, the authors demonstrate that this approach leads to an 11% improvement in MSE evaluated on simulations compared to a standard Wiener filter approach.

While Jeffrey et al. (2020) had the benefit of providing a Bayesian understanding of the network output, it did not provide any sort of uncertainty on the recovered map, which makes the interpretation and scientific exploitation of these results difficult. To overcome these limitations, Remy et al. (2022) introduced an approach allowing to sample from the full posterior distribution of the mass-mapping problem. They proposed to use a similar U-Net architecture, not to directly estimate the convergence map, but to learn from simulations a generative prior on convergence maps using a Denoising Score Matching technique (Song & Ermon 2019). With this approach, it can be shown that a neural network trained as a Gaussian denoiser under a simple MSE loss will actually learn the score of the data distribution, that is, the gradient of the log likelihood of the data. Once trained on simulations, this U-Net gives explicit access to the prior. The authors show that it is possible to combine this learned prior with an explicit data likelihood in an Hamiltonian Monte Carlo sampling procedure to sample from the full posterior distribution of the problem. Figure 45 illustrates on the bottom row posterior samples achieved with this method on a simulation of the HST/ACS COSMOS field, compared to the ground truth (top left). Most interestingly, it is

shown that the mean of the posterior samples indeed converge to the same solution as the DeepMass estimate.

*Initial Conditions Reconstructions* One particularly interesting problem for the analysis of galaxy surveys is the reconstruction of the initial density field from the observed Large-Scale Structure. This can for instance be used to refine Baryonic Acoustic Oscillations (BAO) measurements (e.g. Schmittfull, Baldauf, & Zaldarriaga 2017), or part of a Bayesian forward modelling inference scheme (Seljak et al. 2017).

In a first example of applying deep learning to this problem, Mao et al. (2020) proposed a 3D CNN trained on N-body simulations to recover the initial density field at $z=10$ given the final density field at $z=0$ under a density weighted mean squared error loss. They find that their convolution model is capable of beating a standard linear reconstruction on scales smaller than $k \le 0.2h \text{ Mpc}^{-1}$, but under performs on larger scales. Interestingly they find that their learned inversion can extrapolate to some extent to other cosmological parameters; a model trained on WMAP7 cosmology is capable of reconstructing initial conditions on WMAP5 simulations that lead to a slightly biased BAO signal, but still significantly different from the WMAP7 signal of the training data.

Going beyond a direct inversion method, Modi et al. (2021a) proposed an iterative reconstruction scheme based on Recurrent Inference Machines (RIM, Putzky & Welling 2017). This approach can be thought of as a learned iterative reconstruction algorithm. At each iteration a recurrent neural network proposes an update of the current reconstruction based on the knowledge of previous iterations and on the gradient of an explicit data likelihood term. In the absence of this neural network, the algorithm would result in a standard gradient descent scheme leading to a Maximum Likelihood Estimation of the initial conditions. By training the Neural Network to minimise at each iteration the Mean Squared Error between the current solution and the true initial conditions, the network will learn both an implicit prior, and a fast inference scheme to minimise the number of updates needed. The result will therefore be a fast convergence towards the mean posterior solution. Most interestingly, in order to compute this explicit likelihood, a differentiable forward model is needed, that is, in this case an N-body simulation. The authors make use of the
<figure><img src="image_47.png" /><figcaption>**Figure 45.** Illustration of weak-lensing mass-map reconstructions in a simulated COSMOS survey setting with the posterior sampling method of Remy et al. (2022) and the DeepMass direct posterior mean estimation method of Jeffrey et al. (2020). As can be seen, individual posterior samples (bottom row) are visually similar to a real convergence map (e.g. ground truth at the top left) but exhibit variability on structures not strongly constrained by data (e.g. outside of the survey region marked by the white contours). The top row illustrates that the DeepMass estimate indeed recovers the Bayesian posterior mean.</figcaption></figure>

FlowPM TensorFlow-based fast N-body code (Modi, Lanusse, & Seljak 2021b) for this likelihood which becomes just a layer within a neural network (Figure 46). The authors propose to initialise the reconstruction at the standard linear reconstruction, and show that in only 10 iterations this method yields a better solution than an iterative reconstruction based on 400 iterations of an LBFGS minimiser.

## Summary of Deep learning for Cosmology

### 1. Emulation

* The main motivation for using deep learning in simulations is to bypass some of the expensive computational steps needed to generate large volume and high-resolution simulations needed to model modern surveys. Applications have targeted in particular: emulating N-body simulations, enhancing the resolution of existing simulations (super-resolution), and learning mappings between the 3D dark matter distribution and dark matter halos or a range of hydrodynamical fields.

* Despite many impressive results, these methods have not yet been used for scientific applications. The main difficulty is that deep models can only be used within their training regimes (defined by specific training resolution, specific sets of cosmological parameters, or specific hydrodynamical run), and thus it is unclear whether they will bring concrete computational gains when the cost of the training sets are taken into account.

<figure><img src="image_48.png" /><figcaption>**Figure 46.** CosmicRIM initial conditions reconstruction technique of Modi et al. (2021a) based on a 3D LSTM recurrent neural network and includes explicitly at each iteration the gradients of the data likelihood. The plot shows a given ground truth initial condition field (top left) and associated final field (bottom left), along with the reconstructed initial conditions (top right) and reconstructed final field (bottom right).</figcaption></figure>

* An alternative class of models, which so far as attracted limited attention, aims instead for a minimal set of parameters, building upon known symmetries and physical insight, greatly reducing the amount of training simulations needed
<figure><img src="image_47.png" /><figcaption>**Figure 45.** Illustration of weak-lensing mass-map reconstructions in a simulated COSMOS survey setting with the posterior sampling method of Remy et al. (2022) and the DeepMass direct posterior mean estimation method of Jeffrey et al. (2020). As can be seen, individual posterior samples (bottom row) are visually similar to a real convergence map (e.g. ground truth at the top left) but exhibit variability on structures not strongly constrained by data (e.g. outside of the survey region marked by the white contours). The top row illustrates that the DeepMass estimate indeed recovers the Bayesian posterior mean.</figcaption></figure>

FlowPM TensorFlow-based fast N-body code (Modi, Lanusse, & Seljak 2021b) for this likelihood which becomes just a layer within a neural network (Figure 46). The authors propose to initialise the reconstruction at the standard linear reconstruction, and show that in only 10 iterations this method yields a better solution than an iterative reconstruction based on 400 iterations of an LBFGS minimiser.

## Summary of Deep learning for Cosmology

### 1. Emulation

* The main motivation for using deep learning in simulations is to bypass some of the expensive computational steps needed to generate large volume and high-resolution simulations needed to model modern surveys. Applications have targeted in particular: emulating N-body simulations, enhancing the resolution of existing simulations (super-resolution), and learning mappings between the 3D dark matter distribution and dark matter halos or a range of hydrodynamical fields.

* Despite many impressive results, these methods have not yet been used for scientific applications. The main difficulty is that deep models can only be used within their training regimes (defined by specific training resolution, specific sets of cosmological parameters, or specific hydrodynamical run), and thus it is unclear whether they will bring concrete computational gains when the cost of the training sets are taken into account.

<figure><img src="image_48.png" /><figcaption>**Figure 46.** CosmicRIM initial conditions reconstruction technique of Modi et al. (2021a) based on a 3D LSTM recurrent neural network and includes explicitly at each iteration the gradients of the data likelihood. The plot shows a given ground truth initial condition field (top left) and associated final field (bottom left), along with the reconstructed initial conditions (top right) and reconstructed final field (bottom right).</figcaption></figure>

* An alternative class of models, which so far as attracted limited attention, aims instead for a minimal set of parameters, building upon known symmetries and physical insight, greatly reducing the amount of training simulations needed
<table><caption>**Table 1.** Overview of the different deep learning techniques used in the fields of galaxy formation and cosmology, divided by type of application (see text for details).</caption><thead><tr><th colspan="2">Model</th><th></th><th>CNNs</th><th>Enc.</th><th>Gene</th><th>BNN</th><th>RNN</th><th>Trans.</th><th>GNN</th></tr></thead><tbody><tr><td rowspan="4">1. Computer Vision</td><td rowspan="3">Classification</td><td>Morphology</td><td>✓</td><td>✓</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Strong Lenses</td><td>✓*</td><td>✓*</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Transients</td><td></td><td></td><td></td><td></td><td>✓*)</td><td>✓*)</td><td></td></tr><tr><td>Segmentation</td><td></td><td></td><td>✓*</td><td>✓*</td><td></td><td></td><td></td><td></td></tr><tr><td rowspan="6">2. Galaxy Properties</td><td>Photoz</td><td>✓</td><td></td><td></td><td></td><td>✓</td><td></td><td></td><td></td></tr><tr><td>Structure</td><td>✓*)</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Stellar Populations</td><td>✓*</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Lensing</td><td>✓*</td><td></td><td></td><td></td><td>✓*</td><td></td><td></td><td></td></tr><tr><td>Physical Processes</td><td>✓*</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Dark Matter</td><td>✓*</td><td></td><td></td><td></td><td>✓*</td><td></td><td></td><td>✓*</td></tr><tr><td rowspan="3">3. Discovery</td><td>Visualization</td><td></td><td>✓</td><td>✓</td><td>✓</td><td></td><td></td><td></td><td></td></tr><tr><td>Outliers</td><td></td><td>✓</td><td>✓</td><td>✓</td><td></td><td>✓</td><td></td><td></td></tr><tr><td>Laws</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>✓*</td></tr><tr><td rowspan="2">4. Cosmology</td><td>Emulation</td><td>✓*</td><td>✓*</td><td>✓*</td><td>✓*</td><td></td><td></td><td></td><td></td></tr><tr><td>Cosmological inference</td><td>✓*</td><td></td><td></td><td>✓*</td><td>✓*</td><td></td><td></td><td></td></tr></tbody></table>

CNNs: Standard classification and regression Convolutional Neural Networks including modern architectures such as ResNets. Enc: Encoder-Decoder networks and variants. Gene: Generative Models. BNNs: Bayesian Neural Networks; we also include Mixture Density Networks. RNNs: Recursive Neural Networks. Trans: Transformers. GNNs; Graph Neural Networks. A blue (red) background indicates supervised (unsupervised) learning. The star symbol highlights applications which require simulations to train the neural networks. The bracket after the star symbol indicates that the use of simulations is not always mandatory.

and even opening the possibility of inferring these parameters from the data itself.

## 2. Cosmological Inference

* Deep learning is opening a new way of comparing observations to theory: 1. it allows for the automatic extraction of cosmological information from high-dimensional data without requiring analytic summary statistics; 2. Neural Density Estimation makes it possible to perform Bayesian inference by leveraging numerical simulations.

* Although in theory a deep learning approach is statistically sound, it assumes that the simulators provide an accurate physical model of the observations. Any unaccounted for systematics may result in biases, which due to the black-box nature of deep neural networks are difficult to test/detect.

* A number of papers are starting to apply these methodology on data. In weak-lensing, the gains compared to a more standard power spectrum analysis have remained limited on current generation surveys when systematics are included and marginalised over in the analysis.

* A few applications have proposed to perform high-dimensional inference of cosmological fields (e.g. dark matter maps, reconstructing initial conditions). These works however do not yet attempt joint inference of fields and cosmological parameters.

## 7. Final thoughts: Assessing the present and future of deep learning for galaxy surveys

This final section is devoted to extract some indicators about the impact that deep learning has had in the analysis of galaxy surveys. We also attempt to highlight some of the key challenges these methods are facing, which in some cases might prevent or delay the general deployment of deep learning for scientific analysis. Some of these challenges have already been highlighted in the previous sections, but this section tries to extract the most commonly encountered

### 7.1. On the penetration of deep learning techniques in astronomy

We start by questioning what are the deep learning techniques most commonly used in astronomy and how efficiently the rapid progresses made in the ML community reach our community. In the previous sections, we have described different applications of deep learning making use of a variety of techniques. We summarise in Table 1, the broad type of neural network architectures used in the four categories of scientific applications we defined in this review. We have divided the neural network models in seven big groups. CNNs encapsulates all variants of convolutional neural networks, from Vanilla to more complex Residual Networks. The second group contains, generally speaking, image to image networks such as Encoder-Decoders, Autoencoders but also segmentation specific networks such as Mask RCNNs or YOLO.
<table><caption>**Table 1.** Overview of the different deep learning techniques used in the fields of galaxy formation and cosmology, divided by type of application (see text for details).</caption><thead><tr><th colspan="2">Model</th><th></th><th>CNNs</th><th>Enc.</th><th>Gene</th><th>BNN</th><th>RNN</th><th>Trans.</th><th>GNN</th></tr></thead><tbody><tr><td rowspan="10">Application</td><td rowspan="10">1. Computer Vision</td><td rowspan="3">Classification</td><td>Morphology</td><td>✓</td><td>✓</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Strong Lenses</td><td>✓*</td><td>✓*</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Transients</td><td></td><td></td><td></td><td></td><td>✓*)</td><td>✓*)</td><td></td></tr><tr><td rowspan="2">Segmentation</td><td></td><td></td><td>✓*</td><td>✓*</td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td rowspan="5">2. Galaxy Properties</td><td>Photoz</td><td>✓</td><td></td><td></td><td>✓</td><td></td><td></td><td></td></tr><tr><td>Structure</td><td>✓*)</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Stellar Populations</td><td>✓*</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Lensing</td><td>✓*</td><td></td><td></td><td>✓*</td><td></td><td></td><td></td></tr><tr><td>Physical Processes</td><td>✓*</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td rowspan="4">3. Discovery</td><td>Dark Matter</td><td>✓*</td><td></td><td></td><td>✓*</td><td></td><td></td><td>✓*</td></tr><tr><td rowspan="3">Visualization</td><td></td><td>✓</td><td>✓</td><td>✓</td><td></td><td></td><td></td><td></td></tr><tr><td>Outliers</td><td>✓</td><td>✓</td><td>✓</td><td></td><td>✓</td><td></td><td></td></tr><tr><td>Laws</td><td></td><td></td><td></td><td></td><td></td><td></td><td>✓*</td></tr><tr><td rowspan="2">4. Cosmology</td><td rowspan="2">Emulation</td><td></td><td>✓*</td><td>✓*</td><td>✓*</td><td>✓*</td><td></td><td></td><td></td></tr><tr><td>Cosmological inference</td><td>✓*</td><td></td><td></td><td>✓*</td><td>✓*</td><td></td><td></td></tr></tbody></table>

CNNs: Standard classification and regression Convolutional Neural Networks including modern architectures such as ResNets. Enc: Encoder-Decoder networks and variants. Gene: Generative Models. BNNs: Bayesian Neural Networks; we also include Mixture Density Networks. RNNs: Recursive Neural Networks. Trans: Transformers. GNNs; Graph Neural Networks. A blue (red) background indicates supervised (unsupervised) learning. The star symbol highlights applications which require simulations to train the neural networks. The bracket after the star symbol indicates that the use of simulations is not always mandatory.

and even opening the possibility of inferring these parameters from the data itself.

## 2. Cosmological Inference

* Deep learning is opening a new way of comparing observations to theory: 1. it allows for the automatic extraction of cosmological information from high-dimensional data without requiring analytic summary statistics; 2. Neural Density Estimation makes it possible to perform Bayesian inference by leveraging numerical simulations.

* Although in theory a deep learning approach is statistically sound, it assumes that the simulators provide an accurate physical model of the observations. Any unaccounted for systematics may result in biases, which due to the black-box nature of deep neural networks are difficult to test/detect.

* A number of papers are starting to apply these methodology on data. In weak-lensing, the gains compared to a more standard power spectrum analysis have remained limited on current generation surveys when systematics are included and marginalised over in the analysis.

* A few applications have proposed to perform high-dimensional inference of cosmological fields (e.g. dark matter maps, reconstructing initial conditions). These works however do not yet attempt joint inference of fields and cosmological parameters.

## 7. Final thoughts: Assessing the present and future of deep learning for galaxy surveys

This final section is devoted to extract some indicators about the impact that deep learning has had in the analysis of galaxy surveys. We also attempt to highlight some of the key challenges these methods are facing, which in some cases might prevent or delay the general deployment of deep learning for scientific analysis. Some of these challenges have already been highlighted in the previous sections, but this section tries to extract the most commonly encountered

### 7.1. On the penetration of deep learning techniques in astronomy

We start by questioning what are the deep learning techniques most commonly used in astronomy and how efficiently the rapid progresses made in the ML community reach our community. In the previous sections, we have described different applications of deep learning making use of a variety of techniques. We summarise in Table 1, the broad type of neural network architectures used in the four categories of scientific applications we defined in this review. We have divided the neural network models in seven big groups. CNNs encapsulates all variants of convolutional neural networks, from Vanilla to more complex Residual Networks. The second group contains, generally speaking, image to image networks such as Encoder-Decoders, Autoencoders but also segmentation specific networks such as Mask RCNNs or YOLO.
The third family of models are Generative Models which include Variational Autoencoders, Generative Adversarial Networks and also Autoregressive models. We then include Bayesian Neural Networks which allow for uncertainty quantification (Mixture Density Networks and Flow models are also included in this category), Recursive Neural Networks and Transformers mostly suited for sequences. The last group is made of Graph Neural Networks. The table first shows that applications in astronomy cover a wide range of deep learning techniques. Although standard CNNs are the most commonly used—probably because it is the most established approach and because imaging is the most common type of data—other more recent models are regularly applied to astronomical data. On the one side, this reflects the fact that astronomical data is rather diverse—including images, but also spectra, time sequences, simulations and observations. On the other hand, it suggests that the penetration of new ML techniques is efficient. State-of-the-art methods are rapidly applied to astronomy. This is likely a consequence of the fact that, even advanced ML methods are becoming increasingly easy to use for non-experts. It is almost straightforward to test a new technique on an astrophysical problem with current high level implementations. The downside is that, generally speaking, the methods are often applied *blindly* off the shelf, with little domain specific adaptation. Consequently, a feature that is still lacking in a fair amount of the applications of deep learning to astronomy is the inclusion of previous physical knowledge into the data-driven models. This can be done by adapting the loss functions or by modifying the neural network architectures to incorporate known symmetries (see work by Villar et al. 2021a; Bowles et al. 2021). It obviously requires deeper knowledge of the machine learning aspects which is something that will likely take more time.

Another interesting feature that emerges from Table 1 is that training on simulations is the most common approach in astronomy. Almost all supervised approaches rely at some stage on simulated data. It reflects that the samples of labelled data remain small and/or that the measurements in observations are noisy. Relying on simulations to train the models adds however an important element of uncertainty to all applications. Machine learning approaches are indeed very sensitive to domain shift issues. According to Table 1, almost all recent applications are affected by those at some extent. We will discuss this further in Subsection 7.3.

## 7.2. Measuring the impact of deep learning

We now move to measuring the impact of works using deep learning in the astronomical literature. We have seen in the introduction that the number of papers making use of neural networks has increased exponentially over the past half decade. In this subsection, we try to measure the impact of these works with some standard metrics. Figure 47 shows the evolution of the number of papers, number of citations and average citations per paper in the period 2015–2021. The publications are divided in the four different categories defined in this review, that is, computer vision, galaxy properties, discovery and cosmology. We have only included in the figure the works explored for this work. As a consequence, it is very likely that the figure is not complete and that the numbers presented are closer to a lower limit. However, it should provide a good overview of the general trends and represents a more controlled experiment than a purely automatic search. We also emphasise that the division in categories is a choice by the

authors of this review. Therefore, there is some obvious overlap between the different types of applications.

Nevertheless, the figure reveals some interesting behaviours. We first confirm the global increasing trend of the number of papers using deep learning for galaxy surveys. Since 2015 there is a clear exponential increase. In 2021, there are at least 70 papers using deep learning in the context of galaxy surveys, while there were less than 5 in 2015. This is factor of ~15 increase and implies more than a paper per week on average. If we look at the division per type of application, we see that computer vision type of applications (i.e. classification, segmentation) concentrate the largest fraction of publications. All the other remaining classes share similar fractions. However, there is clearly a decreasing trend of the relative importance of classification and segmentation applications. While the fraction of these papers was around 70–80% in 2015–2017, it is only of ~20% for papers published in 2021. The trends seem to suggest a diversification of the applications of deep learning to astronomy moving from computer vision tasks—mainly classification and segmentation—to a large variety of different applications. The number of yearly works for data processing seems to flatten indeed after 2019, while other applications like data exploration (*discovery*) rapidly rise.

A similar behaviour is observed in terms of citations. In 2016, roughly 80% of the citations are for papers using deep learning for computer vision tasks. The fraction is steadily decreasing, but still remains close to ~50% in 2021 even though the number of papers is only 20%. It probably reflects a delay between the publication time and the time papers start to be cited.

We attempt to remove this effect in Figure 48. We plot the number of papers per year as a function of the number of citations per paper and per year, averaged over the time elapsed since the first citation in a given group. That way, for papers focusing on computer vision, the time window is set to 6 years, while for the others, we consider 4 yr. We also show in the figure, for reference, the location of publications flagged with the keywords *galaxy evolution*, according to the NASA ADS search engine. Using this normalisation, we observe several interesting trends. Works using deep learning for galaxy surveys, represent roughly >5% of all works focusing on galaxies. This is not a large fraction but it is still remarkable given the relatively recent emergence of deep learning. They receive on average ~1.5 times less citations per publication. It suggests that the impact of deep learning papers remains moderate as compared to the average. We might speculate about possible reasons. One possibility is that most of the works making use of deep learning are thought in preparation of future big data surveys which have not arrived yet (e.g. Euclid, LSST). The works are therefore more a demonstration of feasibility. It could be argued that existing surveys such as DES or SDSS for example are already good targets for data-driven science. This is certainly true, and as we have seen in this review, there are many works targeting these surveys. However, the sizes and dimensionality of current surveys still allow one to use relatively optimised codes and leave some room for some manual checks. This is also supported by the trends observed when the deep learning papers are divided by topic. The highest citation rate is measured for works focusing on simulation (i.e. emulation of cosmological simulations), which by definition do not require new observational data. Publications focused on unsupervised discovery, which strongly rely on new data being available, present the lowest impact, although they show a strong increase in numbers (Figure 47). It could also be that the technology is still too young,
The third family of models are Generative Models which include Variational Autoencoders, Generative Adversarial Networks and also Autoregressive models. We then include Bayesian Neural Networks which allow for uncertainty quantification (Mixture Density Networks and Flow models are also included in this category), Recursive Neural Networks and Transformers mostly suited for sequences. The last group is made of Graph Neural Networks. The table first shows that applications in astronomy cover a wide range of deep learning techniques. Although standard CNNs are the most commonly used—probably because it is the most established approach and because imaging is the most common type of data—other more recent models are regularly applied to astronomical data. On the one side, this reflects the fact that astronomical data is rather diverse—including images, but also spectra, time sequences, simulations and observations. On the other hand, it suggests that the penetration of new ML techniques is efficient. State-of-the-art methods are rapidly applied to astronomy. This is likely a consequence of the fact that, even advanced ML methods are becoming increasingly easy to use for non-experts. It is almost straightforward to test a new technique on an astrophysical problem with current high level implementations. The downside is that, generally speaking, the methods are often applied *blindly* off the shelf, with little domain specific adaptation. Consequently, a feature that is still lacking in a fair amount of the applications of deep learning to astronomy is the inclusion of previous physical knowledge into the data-driven models. This can be done by adapting the loss functions or by modifying the neural network architectures to incorporate known symmetries (see work by Villar et al. 2021a; Bowles et al. 2021). It obviously requires deeper knowledge of the machine learning aspects which is something that will likely take more time.

Another interesting feature that emerges from Table 1 is that training on simulations is the most common approach in astronomy. Almost all supervised approaches rely at some stage on simulated data. It reflects that the samples of labelled data remain small and/or that the measurements in observations are noisy. Relying on simulations to train the models adds however an important element of uncertainty to all applications. Machine learning approaches are indeed very sensitive to domain shift issues. According to Table 1, almost all recent applications are affected by those at some extent. We will discuss this further in Subsection 7.3.

## 7.2. Measuring the impact of deep learning

We now move to measuring the impact of works using deep learning in the astronomical literature. We have seen in the introduction that the number of papers making use of neural networks has increased exponentially over the past half decade. In this subsection, we try to measure the impact of these works with some standard metrics. Figure 47 shows the evolution of the number of papers, number of citations and average citations per paper in the period 2015–2021. The publications are divided in the four different categories defined in this review, that is, computer vision, galaxy properties, discovery and cosmology. We have only included in the figure the works explored for this work. As a consequence, it is very likely that the figure is not complete and that the numbers presented are closer to a lower limit. However, it should provide a good overview of the general trends and represents a more controlled experiment than a purely automatic search. We also emphasise that the division in categories is a choice by the

authors of this review. Therefore, there is some obvious overlap between the different types of applications.

Nevertheless, the figure reveals some interesting behaviours. We first confirm the global increasing trend of the number of papers using deep learning for galaxy surveys. Since 2015 there is a clear exponential increase. In 2021, there are at least 70 papers using deep learning in the context of galaxy surveys, while there were less than 5 in 2015. This is factor of ~15 increase and implies more than a paper per week on average. If we look at the division per type of application, we see that computer vision type of applications (i.e. classification, segmentation) concentrate the largest fraction of publications. All the other remaining classes share similar fractions. However, there is clearly a decreasing trend of the relative importance of classification and segmentation applications. While the fraction of these papers was around 70–80% in 2015–2017, it is only of ~20% for papers published in 2021. The trends seem to suggest a diversification of the applications of deep learning to astronomy moving from computer vision tasks—mainly classification and segmentation—to a large variety of different applications. The number of yearly works for data processing seems to flatten indeed after 2019, while other applications like data exploration (*discovery*) rapidly rise.

A similar behaviour is observed in terms of citations. In 2016, roughly 80% of the citations are for papers using deep learning for computer vision tasks. The fraction is steadily decreasing, but still remains close to ~50% in 2021 even though the number of papers is only 20%. It probably reflects a delay between the publication time and the time papers start to be cited.

We attempt to remove this effect in Figure 48. We plot the number of papers per year as a function of the number of citations per paper and per year, averaged over the time elapsed since the first citation in a given group. That way, for papers focusing on computer vision, the time window is set to 6 years, while for the others, we consider 4 yr. We also show in the figure, for reference, the location of publications flagged with the keywords *galaxy evolution*, according to the NASA ADS search engine. Using this normalisation, we observe several interesting trends. Works using deep learning for galaxy surveys, represent roughly >5% of all works focusing on galaxies. This is not a large fraction but it is still remarkable given the relatively recent emergence of deep learning. They receive on average ~1.5 times less citations per publication. It suggests that the impact of deep learning papers remains moderate as compared to the average. We might speculate about possible reasons. One possibility is that most of the works making use of deep learning are thought in preparation of future big data surveys which have not arrived yet (e.g. Euclid, LSST). The works are therefore more a demonstration of feasibility. It could be argued that existing surveys such as DES or SDSS for example are already good targets for data-driven science. This is certainly true, and as we have seen in this review, there are many works targeting these surveys. However, the sizes and dimensionality of current surveys still allow one to use relatively optimised codes and leave some room for some manual checks. This is also supported by the trends observed when the deep learning papers are divided by topic. The highest citation rate is measured for works focusing on simulation (i.e. emulation of cosmological simulations), which by definition do not require new observational data. Publications focused on unsupervised discovery, which strongly rely on new data being available, present the lowest impact, although they show a strong increase in numbers (Figure 47). It could also be that the technology is still too young,
<figure><img src="image_49.png" /></figure>

<figure><img src="image_50.png" /></figure>

<figure><img src="image_51.png" /></figure>

<figure><img src="image_52.png" /><figcaption>**Figure 47.** Impact of works using deep learning for galaxy surveys. Each symbol shows a different class of application as labelled (see text for details). The top left and right panels show the number of papers and number of citations as a function of time respectively. The bottom left and right panels show the fraction of papers and citations in each class of application.</figcaption></figure>

and it is just a matter of time that the impact increases. The majority of the works are still at the proof-of-concept stage and have not reached the deployment stage. This could also mean eventually that there are some challenges/limitations which have not been deeply explored yet and that prevent these new methods to be fully adopted by the community. We explore these challenges more carefully in the following section.

## 7.3. Challenges

We list in Table 2 what we think are some of the major challenges that deep learning works face and which need to be addressed

in the coming years by the community based on the works reported in this review. Some of these challenges are not specific to the astronomical community and can benefit from solutions arising from the field of Machine Learning. However, in some cases, the requirements are more strict in astronomy. The table also provides some possible solutions along with some list of—non-exhaustive—references which have explored these solutions.

### 7.3.1. Small (and biased) labelled datasets

A major challenge in applications of deep learning for astronomy is the lack of large enough labelled data sets to train supervised
<figure><img src="image_49.png" /></figure>

<figure><img src="image_50.png" /></figure>

<figure><img src="image_51.png" /></figure>

<figure><img src="image_52.png" /><figcaption>**Figure 47.** Impact of works using deep learning for galaxy surveys. Each symbol shows a different class of application as labelled (see text for details). The top left and right panels show the number of papers and number of citations as a function of time respectively. The bottom left and right panels show the fraction of papers and citations in each class of application.</figcaption></figure>

and it is just a matter of time that the impact increases. The majority of the works are still at the proof-of-concept stage and have not reached the deployment stage. This could also mean eventually that there are some challenges/limitations which have not been deeply explored yet and that prevent these new methods to be fully adopted by the community. We explore these challenges more carefully in the following section.

## 7.3. Challenges

We list in Table 2 what we think are some of the major challenges that deep learning works face and which need to be addressed

in the coming years by the community based on the works reported in this review. Some of these challenges are not specific to the astronomical community and can benefit from solutions arising from the field of Machine Learning. However, in some cases, the requirements are more strict in astronomy. The table also provides some possible solutions along with some list of—non-exhaustive—references which have explored these solutions.

### 7.3.1. Small (and biased) labelled datasets

A major challenge in applications of deep learning for astronomy is the lack of large enough labelled data sets to train supervised
<table><caption>**Table 2.** Major challenges that deep learning works applied to astronomy might suffer and that will need to be addressed in the coming years. We also provide elements of solutions already being explored along with the corresponding references.</caption><thead><tr><th></th><th></th></tr></thead><tbody><tr><td><strong>Challenge 1 Small (and biased) labelled datasets</strong></td><td></td></tr><tr><td>Solution 1.A Transfer Learning</td><td>Domínguez Sánchez et al. (2019), Samudre et al. (2022), Lukic et al. (2019)</td></tr><tr><td>Solution 1.B Simulated dataset</td><td>Jacobs et al. (2017), Vega-Ferrero et al. (2021)</td></tr><tr><td>Solution 1.C Self-supervised learning</td><td>Hayat et al. (2021)</td></tr><tr><td>Solution 1.D Active Learning and similar</td><td>Walmsley et al. (2020)</td></tr><tr><td><strong>Challenge 2 Uncertainty</strong></td><td></td></tr><tr><td>Solution 2.A Bayesian approximations</td><td>Walmsley et al. (2020), Perreault Levasseur et al. (2017)</td></tr><tr><td>Solution 2.B Density Estimators</td><td>Kodi Ramanah et al. (2020)</td></tr><tr><td><strong>Challenge 3 Interpretability</strong></td><td></td></tr><tr><td>Solution 3.A Saliency maps and similar</td><td>Huertas-Company et al. (2018), Bowles et al. (2021), Bhambra et al. (2022)</td></tr><tr><td>Solution 3.B Symbolic regression</td><td>Cranmer et al. (2020)</td></tr><tr><td>Solution 3.C Physics informed</td><td>Scaife & Porter (2021), Villar et al. (2021a), Charnock et al. (2019)</td></tr><tr><td><strong>Challenge 4 Domain shift</strong></td><td></td></tr><tr><td>Solution 4.A Transfer Learning</td><td>Tuccillo et al. (2018), Domínguez Sánchez et al. (2019), Ghosh et al. (2020)</td></tr><tr><td>Solution 4.A Domain Adaptation</td><td>Ćiprijanović et al. (2021b)</td></tr><tr><td><strong>Challenge 5 Benchmarking</strong></td><td></td></tr><tr><td>Solution 5.A Standardised datasets</td><td>PLAsTiCC, SKA data challenge, Galaxy Zoo</td></tr></tbody></table>

<figure><img src="image_53.png" /><figcaption>**Figure 48.** Number of citations normalised by the number of papers and years (from the first publication in that category) as a function of the number of papers per year. Each symbol shows a different category as defined in this work (see text for details). The ‘all galaxy evolution’ group includes an automatic search of all publications in the field of galaxy formation (with or without deep learning).</figcaption></figure>

deep learning models. This issue has been highlighted multiple times during the description of works. We have seen in Section 3 that deep learning was first applied to classification of galaxy morphologies. This was mainly triggered by the fact that large samples of labelled images existed from previous citizen science efforts. A similar behaviour is observed in the radio domain. However, the vast majority of the data are unlabelled preventing the deployment of supervised deep learning methods. Although this issue is common to many disciplines, it is particularly critical in astronomy since the properties of the images change depending on the instrument used. Therefore, a sample of labelled images from one observational dataset cannot directly be exported to another survey, hampering the use of deep learning. We notice however, that large dataset are not always needed. Some works (e.g. Ackermann et al. 2018; Walmsley et al. 2019) have demonstrated that reliable results can be obtained with small training samples. This is probably a consequence of the relatively limited complexity of astronomical images compared to natural images.

Nevertheless, reducing the amount of labels is usually desirable for most supervised applications. There exist several types of solutions listed in Table 2. An obvious one is to manually label more data. It is however very time consuming and cannot be done for every new survey. Active learning approaches allow to label only the more informative examples for the deep learning model, hence reducing the needed time. Active learning has been particularly explored in the framework of galaxy morphology (e.g. Walmsley et al. 2020). Another fairly straightforward solution is Transfer Learning. This is usually done by training a model in a similar dataset to the target dataset for which
<table><caption>**Table 2.** Major challenges that deep learning works applied to astronomy might suffer and that will need to be addressed in the coming years. We also provide elements of solutions already being explored along with the corresponding references.</caption><thead><tr><th></th><th></th></tr></thead><tbody><tr><td><strong>Challenge 1 Small (and biased) labelled datasets</strong></td><td></td></tr><tr><td>Solution 1.A Transfer Learning</td><td>Domínguez Sánchez et al. (2019), Samudre et al. (2022), Lukic et al. (2019)</td></tr><tr><td>Solution 1.B Simulated dataset</td><td>Jacobs et al. (2017), Vega-Ferrero et al. (2021)</td></tr><tr><td>Solution 1.C Self-supervised learning</td><td>Hayat et al. (2021)</td></tr><tr><td>Solution 1.D Active Learning and similar</td><td>Walmsley et al. (2020)</td></tr><tr><td><strong>Challenge 2 Uncertainty</strong></td><td></td></tr><tr><td>Solution 2.A Bayesian approximations</td><td>Walmsley et al. (2020), Perreault Levasseur et al. (2017)</td></tr><tr><td>Solution 2.B Density Estimators</td><td>Kodi Ramanah et al. (2020)</td></tr><tr><td><strong>Challenge 3 Interpretability</strong></td><td></td></tr><tr><td>Solution 3.A Saliency maps and similar</td><td>Huertas-Company et al. (2018), Bowles et al. (2021), Bhambra et al. (2022)</td></tr><tr><td>Solution 3.B Symbolic regression</td><td>Cranmer et al. (2020)</td></tr><tr><td>Solution 3.C Physics informed</td><td>Scaife & Porter (2021), Villar et al. (2021a), Charnock et al. (2019)</td></tr><tr><td><strong>Challenge 4 Domain shift</strong></td><td></td></tr><tr><td>Solution 4.A Transfer Learning</td><td>Tuccillo et al. (2018), Domínguez Sánchez et al. (2019), Ghosh et al. (2020)</td></tr><tr><td>Solution 4.A Domain Adaptation</td><td>Ćiprijanović et al. (2021b)</td></tr><tr><td><strong>Challenge 5 Benchmarking</strong></td><td></td></tr><tr><td>Solution 5.A Standardised datasets</td><td>PLAsTiCC, SKA data challenge, Galaxy Zoo</td></tr></tbody></table>

<figure><img src="image_53.png" /><figcaption>**Figure 48.** Number of citations normalised by the number of papers and years (from the first publication in that category) as a function of the number of papers per year. Each symbol shows a different category as defined in this work (see text for details). The ‘all galaxy evolution’ group includes an automatic search of all publications in the field of galaxy formation (with or without deep learning).</figcaption></figure>

deep learning models. This issue has been highlighted multiple times during the description of works. We have seen in Section 3 that deep learning was first applied to classification of galaxy morphologies. This was mainly triggered by the fact that large samples of labelled images existed from previous citizen science efforts. A similar behaviour is observed in the radio domain. However, the vast majority of the data are unlabelled preventing the deployment of supervised deep learning methods. Although this issue is common to many disciplines, it is particularly critical in astronomy since the properties of the images change depending on the instrument used. Therefore, a sample of labelled images from one observational dataset cannot directly be exported to another survey, hampering the use of deep learning. We notice however, that large dataset are not always needed. Some works (e.g. Ackermann et al. 2018; Walmsley et al. 2019) have demonstrated that reliable results can be obtained with small training samples. This is probably a consequence of the relatively limited complexity of astronomical images compared to natural images.

Nevertheless, reducing the amount of labels is usually desirable for most supervised applications. There exist several types of solutions listed in Table 2. An obvious one is to manually label more data. It is however very time consuming and cannot be done for every new survey. Active learning approaches allow to label only the more informative examples for the deep learning model, hence reducing the needed time. Active learning has been particularly explored in the framework of galaxy morphology (e.g. Walmsley et al. 2020). Another fairly straightforward solution is Transfer Learning. This is usually done by training a model in a similar dataset to the target dataset for which
labelled data exist. The neural network weights are then refined
with a small sample of labelled examples from the target sample.
Transfer learning has also been explored for galaxy morphology
(e.g. Domínguez Sánchez et al. 2019). Other works use simulations
to overcome the lack of labels. As seen in the previous section,
using simulations is a fairly common approach in astronomy.
This is probably because the complexity of astronomical objects—
especially images—allows one to obtain quite realistic simulations
with fairly simple approaches. This is the case, for instance, for
the classification of strong lenses, which exclusively rely on sim-
ulated training sets. Although this is a very efficient approach,
it also implies some potential issues related to the change of
domain. This is a general challenge which we have grouped into
Challenge 4 in Table 2. Finally, the ML community has recently
started to explore the so-called self-supervised approaches to
reduce the amount of labelled examples. The underlying idea is
to first compute some meaningful representations of the data in
an unsupervised way and then use the obtained representations
to train a supervised network. Because the self-supervised step fil-
ters out non-informative features, the amount of needed examples
for training is reduced. Hayat et al. (2021) and Sarmiento et al.
(2021) have recently explored self-supervised learning for galaxy
morphology, photometric redshifts and galaxy kinematics respec-
tively. Contrastive self-supervised learning can also be potentially
employed to reduce the gap between simulations and observations
as well as Autoencoder representations.

In addition to the size of the labelled datasets, a major challenge is how representative are those. A common problem in astronomy is that the sample for which labels are available does not always overlap with the inference datasets. This typically could happen because it is easier to obtain labels for a sub population of objects. The extrapolation of the trained neural network results to a dataset which was not exactly used for training is usually a problem. This issue has become very obvious for photometric redshift estimation (see Section 4) in which ML approaches tend to fail for examples poorly represented in the training. The only solution adopted by the community has been to obtain more training data.

7.3.2. *Uncertainty*

Uncertainty quantification is a major challenge for applications
of deep learning to physical sciences. This is a major differ-
ence with respect to standard computer vision applications on
natural imaging, which usually do not require well calibrated
uncertainties. Therefore, standard deep learning methods directly
exported to astronomy do not quantify uncertainties and this is
generally not acceptable for scientific applications. Some tasks
such as classification can sometimes be accepted without pre-
cise uncertainty and rely on statistical measurements (i.e. ROC,
Precision-Recall curves). However, if deep learning is intended
to be used for accurate measurements of galaxy properties or to
constrain models, they need to incorporate error measurements.
We see a changing trend in the community. The early efforts
did not include uncertainty estimation; however more and more
works are introducing at least some quantification of errors. The
community has explored several solutions, but the problem is far
from being solved. A promising approach are Bayesian Neural
Networks (BNNs) which aim at measuring posterior distributions
from a Bayesian perspective. Several approaches have been pro-
posed over the past years and the implementation has also become
more straightforward, which favours the use by non-experts. For

example BNNs are the common approach for the modelling of
strong lenses for example (e.g. Perreault Levasseur et al. 2017).
However, one need to keep in mind that BNNs compute an
approximation of the true posterior distributions which some-
times might not be accurate enough (see review by Charnock
et al. 2020). Density Estimator networks such as Regressive Flows
or Mixture Density Networks are another approach to sample
from complex posterior distributions. Although several works in
astronomy pay careful attention to the quantification of uncertain-
ties, they represent still a minority of the deep learning literature
in astronomy. We believe this is a major challenge for the future
deployment of deep learning in the analysis of deep surveys.

7.3.3. *Interpretability*

Related to the problem of uncertainty quantification is inter-
pretability. By moving to a data-driven approach to data analysis,
we unavoidably loose some control on what type of information
is extracted and used by the neural network models. This effect
is sometimes referred as the *black-box effect*. Deep learning mod-
els are in general opaque black boxes which perform complex
non-linear mappings, difficult to unveil. Although this might be
a problem for most applications, it is particularly worrying for sci-
entific ones, and therefore constitutes a major challenge for the
acceptance of deep learning by the astronomical community. For
example, not properly understanding the information used can
generate some biases, as demonstrated by Dhar & Shamir (2022)
which show that deep learning based classifications are sensitive
to the location of the galaxy in the sky. The field of interpretabil-
ity of deep neural networks is even less developed than the one
of uncertainty estimation and in general the techniques employed
provide a limited amount of information. A common approach
is to identify the regions of the input data that provide most of
the information for the network decisions. These methods can be
generally useful to identify biases—for example, the neural net-
work model focuses on background noise—but are still far from
providing any physical interpretation of what is being measured.
Some works have looked at ways of enhancing the explainabil-
ity (Huertas-Company et al. 2018; Bhambra, Joachimi, & Lahav
2022) but the amount of extracted information typically consists
on the identification of pixels in the input images which con-
tribute most to the decision. Although this is certainly valuable
information to identify biases in particular, it does not provide a
true explainability in terms of physical meaning. Interpreting the
results is easier when the inputs are parameters instead of raw
pixel values. In such cases, there exists the possibility of perform-
ing symbolic regressions to try to dig into the relations learned by
the neural networks (e.g. Cranmer et al. 2020; Villaescusa-Navarro
et al. 2022). An interesting research line to ease interpretability is
the inclusion of prior physical constraints in the neural network
model. Architectures that preserve known symmetries of the phys-
ical problem are for example an interesting way to keep a control
on what the networks are extracting (e.g. Scaife & Porter 2021;
Villar et al. 2021a; Bowles et al. 2021).

7.3.4. *Domain shift*

Another key challenge faced by deep learning applications to
astronomy is related to the change of domains between the train-
ing and the inference steps. As highlighted in Table 1, a majority
of the applications of deep learning to astronomy rely on simu-
lations for training the models. This is typically justified because
labelled data exist. The neural network weights are then refined
with a small sample of labelled examples from the target sample.
Transfer learning has also been explored for galaxy morphology
(e.g. Domínguez Sánchez et al. 2019). Other works use simulations
to overcome the lack of labels. As seen in the previous section,
using simulations is a fairly common approach in astronomy.
This is probably because the complexity of astronomical objects—
especially images—allows one to obtain quite realistic simulations
with fairly simple approaches. This is the case, for instance, for
the classification of strong lenses, which exclusively rely on sim-
ulated training sets. Although this is a very efficient approach,
it also implies some potential issues related to the change of
domain. This is a general challenge which we have grouped into
Challenge 4 in Table 2. Finally, the ML community has recently
started to explore the so-called self-supervised approaches to
reduce the amount of labelled examples. The underlying idea is
to first compute some meaningful representations of the data in
an unsupervised way and then use the obtained representations
to train a supervised network. Because the self-supervised step fil-
ters out non-informative features, the amount of needed examples
for training is reduced. Hayat et al. (2021) and Sarmiento et al.
(2021) have recently explored self-supervised learning for galaxy
morphology, photometric redshifts and galaxy kinematics respec-
tively. Contrastive self-supervised learning can also be potentially
employed to reduce the gap between simulations and observations
as well as Autoencoder representations.

In addition to the size of the labelled datasets, a major challenge is how representative are those. A common problem in astronomy is that the sample for which labels are available does not always overlap with the inference datasets. This typically could happen because it is easier to obtain labels for a sub population of objects. The extrapolation of the trained neural network results to a dataset which was not exactly used for training is usually a problem. This issue has become very obvious for photometric redshift estimation (see Section 4) in which ML approaches tend to fail for examples poorly represented in the training. The only solution adopted by the community has been to obtain more training data.

7.3.2. Uncertainty

Uncertainty quantification is a major challenge for applications of deep learning to physical sciences. This is a major difference with respect to standard computer vision applications on natural imaging, which usually do not require well calibrated uncertainties. Therefore, standard deep learning methods directly exported to astronomy do not quantify uncertainties and this is generally not acceptable for scientific applications. Some tasks such as classification can sometimes be accepted without precise uncertainty and rely on statistical measurements (i.e. ROC, Precision-Recall curves). However, if deep learning is intended to be used for accurate measurements of galaxy properties or to constrain models, they need to incorporate error measurements. We see a changing trend in the community. The early efforts did not include uncertainty estimation; however more and more works are introducing at least some quantification of errors. The community has explored several solutions, but the problem is far from being solved. A promising approach are Bayesian Neural Networks (BNNs) which aim at measuring posterior distributions from a Bayesian perspective. Several approaches have been proposed over the past years and the implementation has also become more straightforward, which favours the use by non-experts. For

example BNNs are the common approach for the modelling of
strong lenses for example (e.g. Perreault Levasseur et al. 2017).
However, one need to keep in mind that BNNs compute an
approximation of the true posterior distributions which some-
times might not be accurate enough (see review by Charnock
et al. 2020). Density Estimator networks such as Regressive Flows
or Mixture Density Networks are another approach to sample
from complex posterior distributions. Although several works in
astronomy pay careful attention to the quantification of uncertain-
ties, they represent still a minority of the deep learning literature
in astronomy. We believe this is a major challenge for the future
deployment of deep learning in the analysis of deep surveys.

7.3.3. Interpretability

Related to the problem of uncertainty quantification is inter-
pretability. By moving to a data-driven approach to data analysis,
we unavoidably loose some control on what type of information
is extracted and used by the neural network models. This effect
is sometimes referred as the *black-box effect*. Deep learning mod-
els are in general opaque black boxes which perform complex
non-linear mappings, difficult to unveil. Although this might be
a problem for most applications, it is particularly worrying for sci-
entific ones, and therefore constitutes a major challenge for the
acceptance of deep learning by the astronomical community. For
example, not properly understanding the information used can
generate some biases, as demonstrated by Dhar & Shamir (2022)
which show that deep learning based classifications are sensitive
to the location of the galaxy in the sky. The field of interpretabil-
ity of deep neural networks is even less developed than the one
of uncertainty estimation and in general the techniques employed
provide a limited amount of information. A common approach
is to identify the regions of the input data that provide most of
the information for the network decisions. These methods can be
generally useful to identify biases—for example, the neural net-
work model focuses on background noise—but are still far from
providing any physical interpretation of what is being measured.
Some works have looked at ways of enhancing the explainabil-
ity (Huertas-Company et al. 2018; Bhambra, Joachimi, & Lahav
2022) but the amount of extracted information typically consists
on the identification of pixels in the input images which con-
tribute most to the decision. Although this is certainly valuable
information to identify biases in particular, it does not provide a
true explainability in terms of physical meaning. Interpreting the
results is easier when the inputs are parameters instead of raw
pixel values. In such cases, there exists the possibility of perform-
ing symbolic regressions to try to dig into the relations learned by
the neural networks (e.g. Cranmer et al. 2020; Villaescusa-Navarro
et al. 2022). An interesting research line to ease interpretability is
the inclusion of prior physical constraints in the neural network
model. Architectures that preserve known symmetries of the phys-
ical problem are for example an interesting way to keep a control
on what the networks are extracting (e.g. Scaife & Porter 2021;
Villar et al. 2021a; Bowles et al. 2021).

7.3.4. Domain shift

Another key challenge faced by deep learning applications to
astronomy is related to the change of domains between the train-
ing and the inference steps. As highlighted in Table 1, a majority
of the applications of deep learning to astronomy rely on simu-
lations for training the models. This is typically justified because
the availability of labelled samples is limited (see Subsection 7.3.1), because we aim at accessing information that is only available on simulations, for example, galaxy mergers, dark matter or information about the cosmological model or because the likelihood is intractable but we have an idea on how to simulate the data (see review by Cranmer, Brehmer, & Louppe 2019). For example, the work by Bottrell et al. (2019) examines very carefully the impact of using more or less realistic simulations for training. The community has explored several solution to mitigate the impact of training on simulations and apply to data. A simple approach is to use transfer learning. This is only possible when there exist some measurements in observations which can be used to fine tune the weights from the neural network model trained on simulations (e.g. Tuccillo et al. 2018). This is not always possible though, especially when we try to infer the parameters of a model and has also the problem of propagating the biases of any existing method previously applied to observations. Domain adaptation techniques are another alternative approach which attempt to make the features learned by the model agnostic to the differences between domains. As opposed to transfer learning, this is done during training so that no domain specific features are learned. Ćiprijanović et al. (2021a) have recently quantify the gain of such techniques for the identification of galaxy mergers. It remains however an open issue for the future.

### 7.3.5. Benchmarking and deployment

A final challenge which has not been discussed much in the literature so far is related to how the different approaches can be robustly compared. As we have thoroughly described in this review, the past years have witnessed an emergence of a large number of deep learning methods applied to a diversity of scientific topics. In many cases, the results are shown for a specific dataset, with a specific configuration, which makes it hard to compare with existing approaches. The ML community has been using since many years, what is called standardised datasets. These are common datasets which are publicly shared and on which any new approach is usually tested. This benchmarking approach has been an important channel for progress in the community. The astronomical one is not used to this type of approach and therefore, with some noticeable exceptions (e.g. PLAsTiCC, Galaxy Zoo for classification), we lack of a coherent way of comparing methods. We argue that this is an important aspect on which to work as a community to boost progress. Having standardised datasets on which test models can not only help comparing methods but also identify pitfalls and biases and therefore contribute to make the neural network models more robust. This is an important step towards a full deployment of these approaches into scientific pipelines.

## 8. Summary

This work reviews the use of modern deep learning techniques for the analysis of deep galaxy surveys. Although machine learning has been used in astronomy for several decades, the recent deep learning revolution as induced an unprecedented number of new works exploring the use of these novel techniques in astronomy. The purpose of this review is to assess how deep learning has been used in astronomy and what are the key achievements and challenges. We do not describe however the technical aspects of deep learning techniques.

We have divided deep learning applications in four broad categories defined by the type of application: 1—computer vision, 2—galaxy properties, 3—discovery and 4—cosmology. The first sections of these review (Sections 3–6) describe the most relevant works in each category. A summary of the main points for each type of application is included at the end of the corresponding section. The first category (Section 3) includes general computer vision applications such as classifications and object detection. The second (Section 4) is related to measure galaxy properties, deep learning acts as a fast emulator and as universal approximator. The third category (Section 5) illustrates all efforts related to visualisation and identification on new types of objects. The fourth group (Section 6) contains publications which use deep learning for cosmology. Namely we include two main applications: more efficient simulation and cosmological inference.

The last section (Section 7) focuses on extracting some lessons about the use of deep learning techniques in astronomy, on the impact they have had so far and on what are—in our humble opinion—the key challenges that will need to be addressed in the near future. We list below key take away messages from this analysis:

* The first work using deep learning in astronomy is from 2015. Since then, the number of works using deep learning for galaxy surveys has increased exponentially. There is factor of ~15 increase between 2015 and 2021.

* The most common deep learning method used are sequential Convolutional Neural Networks with different degrees of complexities. However, there is a good variety of techniques which have been tested for astronomy including recent developments such as Transformers or self-supervised approaches. This reflects a democratisation of these techniques which are becoming increasingly easy to use. However, the methods are often applied with a limited amount of physically driven modifications. The combination of previous physical knowledge with data-driven models is still an open issue, even if it is a rapidly changing field.

* The majority of the works (>50%) focus on what we call computer vision applications—which essentially include classification and segmentation. This is also the field in which deep learning has brought the most important breakthroughs. These are applications which are more prone to a direct import from the ML community. However, we measure a diversification of the applications which span a variety of topics such as the acceleration of cosmological simulations, the inference of galaxy properties or constraints on cosmology.

* The works using deep learning represent ~5% of all works on galaxy formation which is remarkable. The receive however ~3 citations per paper and per year on average. This is roughly ~1.5 times less citations than publications on galaxies for example, although computer vision applications also perform better in this front. It suggests a moderate impact of deep learning so far which might be explained because most of the works are still at the exploratory stage.

* We have identified a set of 5 major challenges which frequently appear in deep learning applications and that we believe need to be addressed in the nearby future. 1—Small labelled datasets; 2—Uncertainty estimation; 3—Interpretability; 4—Domain shift and 5—Benchmarking.
the availability of labelled samples is limited (see Subsection 7.3.1), because we aim at accessing information that is only available on simulations, for example, galaxy mergers, dark matter or information about the cosmological model or because the likelihood is intractable but we have an idea on how to simulate the data (see review by Cranmer, Brehmer, & Louppe 2019). For example, the work by Bottrell et al. (2019) examines very carefully the impact of using more or less realistic simulations for training. The community has explored several solution to mitigate the impact of training on simulations and apply to data. A simple approach is to use transfer learning. This is only possible when there exist some measurements in observations which can be used to fine tune the weights from the neural network model trained on simulations (e.g. Tuccillo et al. 2018). This is not always possible though, especially when we try to infer the parameters of a model and has also the problem of propagating the biases of any existing method previously applied to observations. Domain adaptation techniques are another alternative approach which attempt to make the features learned by the model agnostic to the differences between domains. As opposed to transfer learning, this is done during training so that no domain specific features are learned. Ćiprijanović et al. (2021a) have recently quantify the gain of such techniques for the identification of galaxy mergers. It remains however an open issue for the future.

### 7.3.5. Benchmarking and deployment

A final challenge which has not been discussed much in the literature so far is related to how the different approaches can be robustly compared. As we have thoroughly described in this review, the past years have witnessed an emergence of a large number of deep learning methods applied to a diversity of scientific topics. In many cases, the results are shown for a specific dataset, with a specific configuration, which makes it hard to compare with existing approaches. The ML community has been using since many years, what is called standardised datasets. These are common datasets which are publicly shared and on which any new approach is usually tested. This benchmarking approach has been an important channel for progress in the community. The astronomical one is not used to this type of approach and therefore, with some noticeable exceptions (e.g. PLAsTiCC, Galaxy Zoo for classification), we lack of a coherent way of comparing methods. We argue that this is an important aspect on which to work as a community to boost progress. Having standardised datasets on which test models can not only help comparing methods but also identify pitfalls and biases and therefore contribute to make the neural network models more robust. This is an important step towards a full deployment of these approaches into scientific pipelines.

## 8. Summary

This work reviews the use of modern deep learning techniques for the analysis of deep galaxy surveys. Although machine learning has been used in astronomy for several decades, the recent deep learning revolution as induced an unprecedented number of new works exploring the use of these novel techniques in astronomy. The purpose of this review is to assess how deep learning has been used in astronomy and what are the key achievements and challenges. We do not describe however the technical aspects of deep learning techniques.

We have divided deep learning applications in four broad categories defined by the type of application: 1—computer vision, 2—galaxy properties, 3—discovery and 4—cosmology. The first sections of these review (Sections 3–6) describe the most relevant works in each category. A summary of the main points for each type of application is included at the end of the corresponding section. The first category (Section 3) includes general computer vision applications such as classifications and object detection. The second (Section 4) is related to measure galaxy properties, deep learning acts as a fast emulator and as universal approximator. The third category (Section 5) illustrates all efforts related to visualisation and identification on new types of objects. The fourth group (Section 6) contains publications which use deep learning for cosmology. Namely we include two main applications: more efficient simulation and cosmological inference.

The last section (Section 7) focuses on extracting some lessons about the use of deep learning techniques in astronomy, on the impact they have had so far and on what are—in our humble opinion—the key challenges that will need to be addressed in the near future. We list below key take away messages from this analysis:

* The first work using deep learning in astronomy is from 2015. Since then, the number of works using deep learning for galaxy surveys has increased exponentially. There is factor of ~15 increase between 2015 and 2021.

* The most common deep learning method used are sequential Convolutional Neural Networks with different degrees of complexities. However, there is a good variety of techniques which have been tested for astronomy including recent developments such as Transformers or self-supervised approaches. This reflects a democratisation of these techniques which are becoming increasingly easy to use. However, the methods are often applied with a limited amount of physically driven modifications. The combination of previous physical knowledge with data-driven models is still an open issue, even if it is a rapidly changing field.

* The majority of the works (>50%) focus on what we call computer vision applications—which essentially include classification and segmentation. This is also the field in which deep learning has brought the most important breakthroughs. These are applications which are more prone to a direct import from the ML community. However, we measure a diversification of the applications which span a variety of topics such as the acceleration of cosmological simulations, the inference of galaxy properties or constraints on cosmology.

* The works using deep learning represent ~5% of all works on galaxy formation which is remarkable. The receive however ~3 citations per paper and per year on average. This is roughly ~1.5 times less citations than publications on galaxies for example, although computer vision applications also perform better in this front. It suggests a moderate impact of deep learning so far which might be explained because most of the works are still at the exploratory stage.

* We have identified a set of 5 major challenges which frequently appear in deep learning applications and that we believe need to be addressed in the nearby future. 1—Small labelled datasets; 2—Uncertainty estimation; 3—Interpretability; 4—Domain shift and 5—Benchmarking.
## A. Acronyms

We summarise in this appendix the acronyms used for designating types of deep learning methods, galaxy surveys as well as simulated datasets. For every method we also indicate a reference where more details can be obtained.

Machine Learning:

* **ANN:** Artificial Neural Network

* **ARF:** Auto Regressive Flow—Papamakarios, Pavlakou, & Murray (2017)

* **BNN:** Bayesian Neural Network—Charnock et al. (2020), Goan & Fookes (2020)

* **CAE:** Convolutional Autoencoder

* **CNN:** Convolutional Neural Network

* **DT:** Decision Tree

* **(W)GAN:** (Wasserstein) Generative Adversarial Network—Goodfellow et al. (2014), Arjovsky, Chintala, & Bottou (2017)

* **GNN:** Graph Neural Network

* **Mask R-CNN:** Mask Region Convolutional Neural Network—He et al. (2017)

* **MLP:** Multi-Layer Perceptron

* **MDN:** Mixture Density Network—Bishop (1994)

* **RF:** Random Forest

* **RNN:** Recursive Neural Network

* **SOM:** Self-Organising Map

* **SVM:** Support Vector Machines

* **VAE:** Variational Autoencoder—Pu et al. (2016)

* **YOLO:** You Only Look Once—Redmon et al. (2015)

Deep galaxy surveys where deep neural networks have been applied:

* **CANDELS:** Cosmic Assembly Near-Infrared Deep Extragalactic Legacy Survey; Koekemoer et al. (2011)

* **DECaLS:** The Dark Energy Camera Legacy Survey; Dey et al. (2019)

* **DES:** The Dark Energy Survey; Dark Energy Survey Collaboration et al. (2016)

* **Euclid:** Laureijs et al. (2011)

* **HSC:** Hyper Suprime Cam; Aihara et al. (2018)

* **MaNGA (SDSS IV):** Mapping Nearby Galaxies at APO; Bundy et al. (2015)

* **Pan-STARRS:** Panoramic Survey Telescope and Rapid Response System; Chambers et al. (2016)

* **PAU:** Physics of the Accelerating Universe;

* **SDSS I / II Legacy Surveys:** Sloan Digital Sky Survey;

* **LSST:** Legacy Survey of Space and Time; Ivezić et al. (2019)

* **S-PLUS:** Southern Photometric Local Universe Survey; Mendes de Oliveira et al. (2019)

* **GAMA:** Galaxy and Mass Assembly; Driver et al. (2011)

* **ZTF:** Zwicky Transient Facility; Bellm (2014)

Simulated datasets used to train deep neural networks:
**CAMELS:** Villaescusa-Navarro et al. (2021b)

**PLAsTiCC:** Photometric LSST Astronomical Time-Series Classification Challenge

**IllustrisTNG:** Pillepich et al. (2018)

**EAGLE:** Schaye et al. (2015)

**SIMBA:** Davé et al. (2019)

**VELA:** Ceverino et al. (2015)

**Horizon-AGN:** Dubois et al. (2014)

**Acknowledgements.** MHC acknowledges financial support from the State Research Agency (AEI-MCINN) of the Spanish Ministry of Science and Innovation under the grants “Galaxy Evolution with Artificial Intelligence” with reference PGC2018-100852-A-100, from the ACIISI, Consejería de Economía, Conocimiento y Empleo del Gobierno de Canarias and the European Regional Development Fund (ERDF) under grant with reference PROID2020010057, and from IAC project P/301802, financed by the Ministry of Science and Innovation, through the State Budget and by the Canary Islands Department of Economy, Knowledge and Employment, through the Regional Budget of the Autonomous Community.

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## A. Acronyms

We summarise in this appendix the acronyms used for designating types of deep learning methods, galaxy surveys as well as simulated datasets. For every method we also indicate a reference where more details can be obtained.

Machine Learning:

* **ANN:** Artificial Neural Network

* **ARF:** Auto Regressive Flow—Papamakarios, Pavlakou, & Murray (2017)

* **BNN:** Bayesian Neural Network—Charnock et al. (2020), Goan & Fookes (2020)

* **CAE:** Convolutional Autoencoder

* **CNN:** Convolutional Neural Network

* **DT:** Decision Tree

* **(W)GAN:** (Wasserstein) Generative Adversarial Network—Goodfellow et al. (2014), Arjovsky, Chintala, & Bottou (2017)

* **GNN:** Graph Neural Network

* **Mask R-CNN:** Mask Region Convolutional Neural Network—He et al. (2017)

* **MLP:** Multi-Layer Perceptron

* **MDN:** Mixture Density Network—Bishop (1994)

* **RF:** Random Forest

* **RNN:** Recursive Neural Network

* **SOM:** Self-Organising Map

* **SVM:** Support Vector Machines

* **VAE:** Variational Autoencoder—Pu et al. (2016)

* **YOLO:** You Only Look Once—Redmon et al. (2015)

Deep galaxy surveys where deep neural networks have been applied:

* **CANDELS:** Cosmic Assembly Near-Infrared Deep Extragalactic Legacy Survey; Koekemoer et al. (2011)

* **DECaLS:** The Dark Energy Camera Legacy Survey; Dey et al. (2019)

* **DES:** The Dark Energy Survey; Dark Energy Survey Collaboration et al. (2016)

* **Euclid:** Laureijs et al. (2011)

* **HSC:** Hyper Suprime Cam; Aihara et al. (2018)

* **MaNGA (SDSS IV):** Mapping Nearby Galaxies at APO; Bundy et al. (2015)

* **Pan-STARRS:** Panoramic Survey Telescope and Rapid Response System; Chambers et al. (2016)

* **PAU:** Physics of the Accelerating Universe;

* **SDSS I / II Legacy Surveys:** Sloan Digital Sky Survey;

* **LSST:** Legacy Survey of Space and Time; Ivezić et al. (2019)

* **S-PLUS:** Southern Photometric Local Universe Survey; Mendes de Oliveira et al. (2019)

* **GAMA:** Galaxy and Mass Assembly; Driver et al. (2011)

* **ZTF:** Zwicky Transient Facility; Bellm (2014)

Simulated datasets used to train deep neural networks:
**CAMELS:** Villaescusa-Navarro et al. (2021b)

**PLAsTiCC:** Photometric LSST Astronomical Time-Series Classification Challenge

**IllustrisTNG:** Pillepich et al. (2018)

**EAGLE:** Schaye et al. (2015)

**SIMBA:** Davé et al. (2019)

**VELA:** Ceverino et al. (2015)

**Horizon-AGN:** Dubois et al. (2014)

**Acknowledgements.** MHC acknowledges financial support from the State Research Agency (AEI-MCINN) of the Spanish Ministry of Science and Innovation under the grants “Galaxy Evolution with Artificial Intelligence” with reference PGC2018-100852-A-100, from the ACIISI, Consejería de Economía, Conocimiento y Empleo del Gobierno de Canarias and the European Regional Development Fund (ERDF) under grant with reference PROID2020010057, and from IAC project P/301802, financed by the Ministry of Science and Innovation, through the State Budget and by the Canary Islands Department of Economy, Knowledge and Employment, through the Regional Budget of the Autonomous Community.

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