Cosmological constraints with deep learning from KiDS-450 weak lensing maps

Convolutional neural networks (CNNs) have recently been demonstrated on synthetic data to improve upon the precision of cosmological inference. In particular, they have the potential to yield more precise cosmological constraints from weak lensing mass maps than the two-point functions. We present the cosmological results with a CNN from the KiDS-450 tomographic weak lensing dataset, constraining the total matter density ${\mathrm{\Omega }}_{\mathrm{m}}$, the fluctuation amplitude ${\sigma }_8$, and the intrinsic alignment amplitude $A_{\mathrm{IA}}$. We use a grid of N-body simulations to generate a training set of tomographic weak lensing maps. We test the robustness of the expected constraints to various effects, such as baryonic feedback, simulation accuracy, a different value of $H_0$, or the light cone projection technique. We train a set of ResNet-based CNNs with varying depths to analyze sets of tomographic KiDS mass maps divided into 20 flat regions, with applied Gaussian smoothing of $\sigma =2.34\mathrm{arc}\min$. The uncertainties on shear calibration and $n(z)$ error are marginalized in the likelihood pipeline. Following a blinding scheme, we derive constraints on $S_8={\sigma }_8({\mathrm{\Omega }}_{\mathrm{m}}/0.3{)}^{0.5}=0.777_{-0.036}^{+0.038}$ with our CNN analysis, with $A_{\mathrm{IA}}=1.398_{-0.724}^{+0.779}$. We compare this result to the power spectrum analysis on the same maps and likelihood pipeline and find an improvement of about 30% for the CNN. We discuss how our results offer excellent prospects for the use of deep learning in future cosmological data analysis.

Figure 1

The number of objects per pixel summed over all four redshift bins for 4 of our 20 projected $5\times 5{\mathrm{deg}}^2$ patches.

Figure 2

Overview of the training pipeline. The different blocks are described in Sec. 4.

Figure 3

The grid of simulated cosmologies. The dotted lines show the sampled values of ${\mathrm{\Sigma }}_8={\sigma }_8({\mathrm{\Omega }}_{\mathrm{m}}/0.3{)}^{0.6}$. The red contours show the 68%, 95%, and 99.9% confidence contours of the fiducial analysis of KiDS-450 (HV16). The green contours show the 68%, 95%, and 99.9% confidence contours of baseline results ($\mathrm{\Lambda }\mathrm{CDM},\mathrm{TT},\mathrm{TE},\mathrm{EE}+\mathrm{lowE}+\mathrm{lensing}$) of Planck 2018 [51].

Figure 4

The average power spectrum of 100 convergence maps (ten per simulation in our training set, no masking, no noise) for each redshift bin and with intrinsic alignment ($A_{\mathrm{IA}}=1$) for a cosmology with ${\mathrm{\Omega }}_{\mathrm{m}}=0.29$ and ${\sigma }_8=0.78$. The theoretical predictions were obtained using the nICAEA code [59]. The maps were generated with an increased resolution of $256\times 256\mathrm{pixels}$ to avoid power loss through pixelization effects.

Figure 5

Sketch of the network architecture: The input is downsampled twice, which is followed by the residual blocks (see Fig. 6) and the final output layer. We examined architectures with 10, 15, and 25 residual blocks. Three values of the output are used as predictions for ${\mathrm{\Omega }}_{\mathrm{m}}$, ${\sigma }_8$, and $A_{\mathrm{IA}}$, and the other six are used to build the covariance matrix of the loss in Eq. (17).

Figure 6

A residual layer used in the networks. The input is convolved with a weight kernel that does not change the channel dimension, and padding is applied such that the image dimension is conserved. This is followed by an activation function. This process is then repeated, and the original input is added to the output before the final activation. We do not apply batch normalization in our residual blocks, as it did not improve our results.

Figure 7

Impact of various potential systematic effects on the $S_8$ constraints. The blue constraints correspond to different mock observations obtained with our fiducial inference pipeline. The first three constraints correspond to the mock observation with identical random seeds, but with different inference pipeline settings. The black constraints were obtained using a tomographic power spectrum analysis. The black solid line corresponds to the true $S_8$ value of the mock observation.

Figure 8

Left panel: The 68% and 95% confidence contours from our tomographic power spectrum analysis and the constraints obtained with the CNN. Right panel: The 68% and 95% confidence contours from our CNN, the fiducial results from KiDS-450 (HV16), the KiDS-450 power spectrum analysis with three redshift bins [65], and the baseline results from Planck 2018 [51]. The convex hull of our simulated cosmologies (prior) is drawn in black.

Figure 9

The $S_8={\sigma }_8\sqrt{{\mathrm{\Omega }}_{\mathrm{m}}/0.3}$ constraints of our analysis compared to the KiDS-450 two-point-correlation function analysis HV16 the KiDS-450 power spectrum analysis [65], the first year results of the Dark Energy Survey [8], the Subaru Hyper Suprime-Cam first year results [85], and Planck 2018 [51].

Figure 10

The 68% and 95% confidence contours of our fiducial mock observation obtained by combining the predictions of all our six considered networks (blue) and a single network (green). The convex hull of our simulated cosmologies (prior) is drawn in black. The black cross in the middle shows the cosmological parameters used to generate the mock observations.

Figure 11

The train and validation loss of the network architecture with 15 residual blocks. The black solid line indicates the switch from asynchronous training to synchronous training.

Figure 12

Evaluations of the test set from a network with 15 residual blocks. The blue points correspond to our fiducial training with the loss described in Eq. (17). For the red points, we add the loss of Eq. (c1) with $\lambda =1000$.

Figure 13

The constraints on the intrinsic alignment amplitude $A_{\mathrm{IA}}$ and the degeneracy parameter $S_8$ for our fiducial analysis, a nontomographic network analysis, and our power spectrum analysis using our fiducial mock observation.

Figure 14

Difference of the measured power spectrum of the KiDS-450 dataset and the average auto and cross-correlations of the all simulated cosmologies from the simulated surveys used for the inference with intrinsic alignment amplitude $A_{\mathrm{IA}}=1.5$. The number of the correlated redshift bins is indicated in each panel. The spectrum includes the applied noise, mask, and smoothing. The error bars are obtained from the diagonal entries of the used covariance matrix (see Fig. 15).

Figure 15

Correlation matrix obtained from the covariance matrix used for the power spectrum analysis.