- Research
- Open Access
Fast cosmic web simulations with generative adversarial networks
- Andres C. Rodríguez^{1}Email authorView ORCID ID profile,
- Tomasz Kacprzak^{2},
- Aurelien Lucchi^{3},
- Adam Amara^{2},
- Raphaël Sgier^{2},
- Janis Fluri^{2},
- Thomas Hofmann^{3} and
- Alexandre Réfrégier^{2}
https://doi.org/10.1186/s40668-018-0026-4
© The Author(s) 2018
- Received: 31 July 2018
- Accepted: 15 November 2018
- Published: 23 November 2018
Abstract
Dark matter in the universe evolves through gravity to form a complex network of halos, filaments, sheets and voids, that is known as the cosmic web. Computational models of the underlying physical processes, such as classical N-body simulations, are extremely resource intensive, as they track the action of gravity in an expanding universe using billions of particles as tracers of the cosmic matter distribution. Therefore, upcoming cosmology experiments will face a computational bottleneck that may limit the exploitation of their full scientific potential. To address this challenge, we demonstrate the application of a machine learning technique called Generative Adversarial Networks (GAN) to learn models that can efficiently generate new, physically realistic realizations of the cosmic web. Our training set is a small, representative sample of 2D image snapshots from N-body simulations of size 500 and 100 Mpc. We show that the GAN-generated samples are qualitatively and quantitatively very similar to the originals. For the larger boxes of size 500 Mpc, it is very difficult to distinguish them visually. The agreement of the power spectrum \(P_{k}\) is 1–2% for most of the range, between \(k=0.06\) and \(k=0.4\). For the remaining values of k, the agreement is within 15%, with the error rate increasing for \(k>0.8\). For smaller boxes of size 100 Mpc, we find that the visual agreement to be good, but some differences are noticable. The error on the power spectrum is of the order of 20%. We attribute this loss of performance to the fact that the matter distribution in 100 Mpc cutouts was very inhomogeneous between images, a situation in which the performance of GANs is known to deteriorate. We find a good match for the correlation matrix of full \(P_{k}\) range for 100 Mpc data and of small scales for 500 Mpc, with ∼20% disagreement for large scales. An important advantage of generating cosmic web realizations with a GAN is the considerable gains in terms of computation time. Each new sample generated by a GAN takes a fraction of a second, compared to the many hours needed by traditional N-body techniques. We anticipate that the use of generative models such as GANs will therefore play an important role in providing extremely fast and precise simulations of cosmic web in the era of large cosmological surveys, such as Euclid and Large Synoptic Survey Telescope (LSST).
Keywords
- Methods: numerical
- Cosmology
- Large-scale structure of Universe
1 Introduction
The large scale distribution of matter in the universe takes the form of a complicated network called the cosmic web (Bond et al. 1996; Coles and Chiang 2000; Forero-Romero et al. 2009; Dietrich et al. 2012; Libeskind et al. 2017). The properties of this distribution contain important cosmological information used to study the nature of dark matter, dark energy, and the laws of gravity (DES Collaboration 2017; Hildebrandt et al. 2017; Joudaki et al. 2017), as different cosmological models give rise to dark matter distributions with different properties. Simulations of these cosmic structures (Springel 2005; Potter et al. 2017) play a fundamental role in understanding cosmological measurements (Fosalba et al. 2015; Busha et al. 2013). These simulations are commonly computed using N-body techniques, which represent the matter distribution as a set of particles that evolve throughout cosmic time according to the underlying cosmological model and the laws of gravity. Creating a single N-body simulation requires the use of large computational resources for a long period of time such as days or weeks (Teyssier et al. 2009; Boylan-Kolchin et al. 2009). Furthermore, reliable measurements of cosmological parameters typically require a large number of simulations of various cosmological models (Harnois-Déraps and van Waerbeke 2015; Kacprzak et al. 2016). This creates a strong need for fast, approximate approaches for generating simulations of cosmic web (Heitmann et al. 2010a; Heitmann et al. 2009b; Lawrence et al. 2010; Lin and Kilbinger 2015; Howlett et al. 2015).
Here we demonstrate the possibility of using deep generative models to synthesize samples of the cosmic web. Deep generative models (Kingma and Welling 2014; Goodfellow et al. 2014) are able to learn complex distributions from a given set of data, and then generate new, statistically consistent data samples. Such a deep generative model can be trained on a set of N-body simulations. Once the training is complete, the generative model can create new, random dark matter distributions that are uncorrelated to the training examples. A practical advantage of using a generative model is that the generation process is extremely fast, thus giving us the ability to generate a virtually unlimited number of samples of the cosmic web. Having access to such a large amount of simulations can potentially enable more reliable scientific studies and would therefore enhance our ability to understand the physics of the Universe.
In the last decade, deep learning approaches have achieved outstanding results in many fields, especially for computer vision tasks such as image segmentation or object detection (Krizhevsky et al. 2012). Deep convolutional neural networks (DCNN) have also recently been used as data generating mechanisms. Here a latent random vector, typically a high-dimensional Gaussian, is passed through a DCNN in order to output images. Generative Adversarial Networks (GAN) create such a model by adopting an adversarial game setting between two DCNN players, a generator and a discriminator. The goal of the generator is to produce samples resembling the originals while the discriminator aims at distinguishing the originals from the fake samples produced by the generator. The training process ends when a Nash equilibrium is reached, that is when no player can do better by unilaterally changing his strategy.
The rise of deep generative models has sparked a strong interest in the field of astronomy. Deep generative models have been used to generate astronomical images of galaxies (Regier et al. 2015; Ravanbakhsh et al. 2017; Schawinski et al. 2017) or to recover certain features out of noisy astrophysical images (Schawinski et al. 2017). GANs were recently applied to generating samples of projected 2D mass distribution, called convergence (Mustafa et al. 2017). This approach can generate random samples of convergence maps, which are consistent with the original simulated maps according to several summary statistics. The projection process, however, washes out the complex network structures present in the dark matter distribution. Here, we instead focus on generating the structure of the cosmic web without projection, therefore preserving the ability of the generative model to create halos, filaments, and sheets. We accomplish our goal by synthesizing thin slices of dark matter distribution which have been pixelised to create 2D images that serve as training data for a GAN model.
A demonstration of this method on 2D slices presents a case for the development of deep learning methods able to generate full, 3D dark matter distributions. For cosmological applications, it may be more efficient to work with the full 3D matter distributions generated by a GAN, rather then 2D convergence maps. For gravitational lensing, the convergence map depends on the input distribution of background galaxies (see (Refregier 2003), for review); the 3D matter distribution is projected onto the sky plane by integrating the mass in radial direction against a lensing kernel, which depends on distribution \(n(z)\) of redshifts z of background galaxies. For most lensing studies, the uncertainty on \(n(z)\) is large and is effectively marginalised over. If the 3D distributions are simulated, then the projection can be done analytically, for a given \(n(z)\) (Harnois-Déraps et al. 2012; Sgier et al. 2018). For a 2D generative model, a separate GAN would have to be trained for each \(n(z)\) distribution. This may be particularly important for analyses beyond the power spectrum, such as peak statistics (Dietrich and Hartlap 2010; Kacprzak et al. 2016; Martinet et al. 2018) or deep learning (Schmelzle et al. 2017; Gupta et al. 2018), which use simulations to predict both the signal and its uncertainty. In this paper we demonstrate the feasibility of GAN-based methods for capturing the type of matter distributions characteristic for in N-body simulations. As the development of 3D generative methods for N-body data is likely to be a very challenging due to scalability issues and memory requirements, we consider this to be an important step in asserting that this approach is worth pursuing further.
In learning the cosmic web structures, which are more feature-rich than projected convergence maps, we encountered and addressed several important challenges. The first was to handle data with very large dynamic range of the data; the density in the images created from slices of N-body simulations span several orders of magnitude. Secondly, we explored how mode collapse, a feature of GANs causing the model to focuses on a single local minimum, affects the quality of results (Tolstikhin et al. 2017; Metz et al. 2016; Salimans et al. 2016). As mode collapse is expected to depend on the degree of homogeneity between samples, we tested the performance of GANs for both large and small cosmological volumes, of size 500 and 100 Mpc; the matter density distributions in large boxes are considerably more homogeneous than in small boxes.
Finally, expanding on the work of (Mustafa et al. 2017), we additionally evaluate the cross-correlations of the GAN-generated data with itself and the training set. A high cross-correlation would be an indication of lack of independence between the generated samples, a feature which we would judge to be undesirable in this task.
The paper is organised as follows. In Sect. 2 we describe the Generative Adversarial Networks. Section 3 contains the information on N-body simulations used. Our implementation of the algorithm is described in Sect. 4 and diagnostics used to evaluate its performance are detailed in Sect. 5. We present the results in Sect. 6 and conclude in Sect. 7.
2 Generative adversarial networks
3 N-body simulations data
We created N-body simulations of cosmic structures in boxes of size 100 Mpc and 500 Mpc with 512^{3} and 1024^{3} particles respectively. We used L-PICOLA (Howlett et al. 2015) to create 10 independent simulation boxes for both box sizes. The cosmological model used was ΛCDM (Cold Dark Matter) with Hubble constant \(H_{0}=100\), \(h=70\) km s^{−1} Mpc^{−1}, dark energy density \(\varOmega_{\varLambda} = 0.72\) and matter density \(\varOmega _{m} = 0.28\). We used the particle distribution at redshift \(z=0\). We cut the boxes into thin slices to create grayscale, two-dimensional images of the cosmic web. This is accomplished by dividing the x-coordinates into uniform intervals to create 1000 segments. We then selected 500 non-consecutive slices and repeated this process for the y and z axes, which gave us 1500 samples from each of the 10 realizations, yielding a total of \(15{,}000\) samples as our training dataset. We pixelised these slices into \(256 \times256\) pixel images. The value at each pixel corresponded to its particle count. After the pixelisation, the images are smoothed with a Gaussian kernel with standard deviation of one pixel. This step is done to decrease the particle shot noise.
In this paper we used L-PICOLA: a faster, but approximate simulator. For a real application of our method a more precise simulator would be used, such as GADGET-2 (Springel 2005) or PkdGrav3 (Potter et al. 2017). Nevertheless, for the purpose of demonstration of performance of GANs, we consider L-PICOLA simulations to be sufficient. We do not expect the results to differ much if GANs were trained on simulations generated using more precise codes.
4 Implementation and training
We use a slightly modified version of the standard DCGAN architecture (Radford et al. 2015), which was shown to achieve good results on natural images, including various datasets such as LSUN-Bedrooms (3 million indoor bedrooms images) (Yu et al. 2015) or the celebrity face dataset (CelebA, 200000 28 × 28 pixel celebrity faces) (Liu et al. 2015).
Architecture used in the discriminator and generator networks. We used a batch size of \(m = 16\) samples. The neural network has ∼32 million trainable parameters. Parameters for our Wasserstein-1 distance implementation are shown in brackets
Layer | Operation | Output | Dimension |
---|---|---|---|
Discriminator | |||
X | m × 256 × 256 × 1 | ||
\(h_{0}\) | conv | LeakyRelu–BatchNorm | m × 128 × 128 × 64 |
\(h_{1}\) | conv | LeakyRelu–BatchNorm | m × 64 × 64 × 128 |
\(h_{2}\) | conv | LeakyRelu–BatchNorm | m × 32 × 32 × 256 |
\(h_{3}\) | conv | LeakyRelu–BatchNorm | m × 16 × 16 × 512 |
\(h_{4}\) | linear | sigmoid (identity) | m × 1 |
Generator | |||
z | m × 200 (m × 100) | ||
\(h_{0}\) | linear | Relu–BatchNorm | m × 16 × 16 × 512 |
\(h_{1}\) | deconv | Relu–BatchNorm | m × 32 × 32 × 256 |
\(h_{2}\) | deconv | Relu–BatchNorm | m × 64 × 64 × 128 |
\(h_{3}\) | deconv | Relu–BatchNorm | m × 128 × 128 × 64 |
\(h_{4}\) | deconv | tanh | m × 256 × 256 × 1 |
A commonly faced problem when training GANs is a phenomenon called mode collapse (Tolstikhin et al. 2017; Metz et al. 2016; Salimans et al. 2016), where the network focuses on a subset of the modes of the underlying data distribution. In these regions where the generator is fooling the discriminator well, the gradient signal becomes weak and the discriminator might be unable to properly lead the generator to the right target distribution. the generator might converge to them, leaving out parts of regions of the target distribution. Wasserstein-1 loss, has some empirical evidence to prevent mode collapse but still suffers from it.
Hyper-parameters used in our GAN implementations. Adam (Kingma et al. 2014) is the algorithm used to estimate the gradient in our models
Hyperparameter | GAN | Description | |
---|---|---|---|
Standard | Wasserstein-1 | ||
Batch size | 16 | 16 | Number of training samples used to compute the gradient at each update |
z dimension | 200 | 100 | Dimension of the gaussian prior distribution |
Learning rate D | 1⋅10^{−5} | 1⋅10^{−5} | Discriminator learning rate used by the Adam optimizer |
\(\beta_{1}\) | 0.5 | 0.5 | Exponential decay for the Adam optimizer |
\(\beta_{2}\) | 0.999 | 0.999 | Exponential decay for the Adam optimizer |
Learning rate G | 1⋅10^{−5} | 1⋅10^{−8} | Generator learning rate used by the Adam optimizer |
Gradient penalty | - | 1000 | Gradient penalty applied for Wasserstein-1 |
a | 4 | 4 | Parameter in s(x) to obtain the scaled images |
5 Diagnostics
The diagnostic measures used in this work are: average histogram of pixel values in the images, average histogram of values of maxima (“peaks”), average auto power spectrum and the average cross-power spectrum of pairs of images within the sample.
We compute both auto and cross power spectrum from 2D images using a discrete Fourier transform, followed by averaging over angles.
One of the popular alternatives to power spectrum for analysing matter density distribution is the peak statistics. These statistics capture non-Gaussian features present in the cosmic web and are commonly used on weak lensing data (Martinet et al. 2017; Kacprzak et al. 2016). A “peak” is a pixel in the density map that is higher than all its immediate 24 neighbours. The peaks are then counted as a function of their height.
6 Results
We focused our study on two simulation regimes: large-scale distribution, simulated in boxes of size 500 Mpc, and small-scale distribution, with boxes of size 100 Mpc. For both configurations we ran 10 independent simulations. From these boxes, we cut out a total of 15,000 thin, 2D slices for each box size. We design a GAN model where both the discriminator and generator are deep convolutional neural networks. These networks consists of 5 layers, with 4 convolutional layers using filter sizes of \(5 \times5\) pixels.
We trained the model parameters using ADAM, a gradient based optimizer (Kingma et al. 2014), which yields a model that can generate new, random cosmic web images. We assessed the performance of the generative model in several ways. First, we performed a visual comparison of the original and synthetic images. A quantitative assessment of the results was performed based on summary statistics commonly used in cosmology, described in Sect. 5. The angular power spectrum is a standard measure used for describing the matter distribution (Kilbinger 2015). Another important statistic used for cosmological measurements is the distribution of maxima in the density distribution, often called “peak statistics” (Dietrich and Hartlap 2010; Kacprzak et al. 2016). This statistic compares the number of maxima in the maps as a function of their values. We also assessed the statistical independence of samples generated by GANs, as real cosmic structures are expected to be independent due to isotropy and homogeneity of the universe, unless they are physically close to each other. To assess the independence of generated cosmic web distributions, we compare the cross-correlations between pairs of images. Another statistic we used was the histogram of pixel values of N-body and GAN-generated images. Finally, we calculated the covariance between the power spectrum values at different k.
6.1 Large images of size 500 Mpc
6.2 Small images of size 100 Mpc
7 Conclusion
We demonstrated the ability of Generative Adversarial Models to learn the distribution describing the complex structures of the cosmic web. We implemented a generative model based on deep convolutional neural networks, trained it on 2D images of cosmic web produced from N-body simulations, and used it to generate a synthetic cosmic web. Our GAN-generated images are visually very similar to the ones from N-body simulations: the generative model managed to capture the complex structures of halos, filaments and voids. We compared the GAN-generated images to the N-body originals using several summary statistics and found a good agreement. Most notably, for 500 Mpc, the power agreement on power spectrum was very good: between \(k=0.06\) and \(k=0.4\) the level of 1–2% is close to the requirements for precision cosmology (Schneider et al. 2016). The correlation matrices of \(P(k)\) values had similar structures and agreed to around 5% at small scales, but the GANs did not reproduce the large scale correlations well, with ∼20% difference. While more work would be needed to improve this agreement further, this result is promising for using GANs as emulators of mass density distributions for practical applications.
For 100 Mpc images the error on the power spectrum was larger, reaching 20%. We attribute this feature to the fact that images in the 100 Mpc sample are much more inhomogeneous than the 500 Mpc sample: some images contain dense regions with halos, and some relatively empty regions with few features. We have seen empirically that this tends to induce a known phenomenon in GANs called mode collapse, where the training algorithm focuses on a subset of the target distribution. This results in the model generating a few specific types of images, for example the ones with empty regions, more often than others. We conclude that the application of GANs is suitable for large, homogeneous datasets. For the type of inhomogeneous distributions appearing in the 100 Mpc sample, some techniques addressing mode-collapse (Srivastava et al. 2017; Grnarova et al. 2017) might be required if high-quality statistics are required.
An important advantage of the approach we presented here is that, once trained, it generates new samples in a fraction of a second on a modern Graphics Processing Unit (GPU). Compared to a classical N-body technique, this constitutes a gain of several orders of magnitude in terms of simulation time. The availability of this approach has the potential to dramatically reduce the computational burden required to acquire the data needed for most cosmological analyses. Examples of such analyses include the computation of covariance matrices for cosmology with large scale structure (Harnois-Déraps and van Waerbeke 2015) or analyses using weak lensing shear peak statistics (Dietrich and Hartlap 2010). Generative methods may become even more important in the future; the need for fast N-body simulations is anticipated to grow in the era of large cosmological datasets obtained by the Euclid^{1} and LSST^{2} projects. The need for fast simulations will be amplified further by the emergence of new analysis methods, which can be based on advanced statistics (Petri et al. 2013) or deep learning (Schmelzle et al. 2017). These methods aim to extract more information from cosmological data and often use large simulation datasets. While we demonstrated the performance of GANs for 2D images using training on a single GPU, this approach can naturally be extended to generate 3D mass distributions (Ravanbakhsh et al. 2016) for estimating cosmological parameters from dark matter simulations.
Finally, it would be interesting to explore how many simulations are needed to train a GAN model for a given precision requirement. Another future direction would be to further explore the agreement between the original and GAN-generated images in terms of advanced statistics, such as for example 3-pt functions or Minkowski functionals. Going beyond the cross-correlations to further tests for independence of the GAN-generated samples could also be of interest. We leave this exploration to future work.
Declarations
Acknowledgements
We acknowledge the support of the IT service of the Leonhard and Euler clusters at ETH Zurich.
Availability of data and materials
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Funding
This work was support in part by grant number 200021_169130 from the Swiss National Science Foundation. The funding body had no involvement in the design of the study, collection, analysis, and interpretation of data, or writing the manuscript.
Authors’ contributions
AR performed the experiment design and the full analysis. TK and AL performed the experiment design and provided direct guidance and supervision. JF and RS prepared the N-body simulations dataset. AA, AR and TH provided guidance, supervision and resources necessary to faciliate the project. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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