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Using CNNs for Precision Shear Measurement in Stage IV Surveys

Published online by Cambridge University Press:  05 October 2026

L. M. Voigt Open the ORCID record for L. M. Voigt [Opens in a new window]
L. M. Voigt*
Affiliation:
University of Essex
*
email: lv18675@essex.ac.uk
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Abstract

Weak gravitational lensing of background galaxies traces the growth of dark matter and is key to constraining the cosmological model. This proceedings paper provides an overview of the analysis presented in Voigt (2024), applying convolutional neural networks (CNNs) to the challenge of precision shear estimation. Using a shallow CNN, we investigate the sensitivity of shear recovery to point spread function (PSF) misestimation, with reference to the accuracy requirements of Euclid. To mitigate noise bias we employ a non-standard loss function, and assess performance assuming a linear relation between predicted and true shears.

Keywords

methods: data analysisgravitational lensing: weakcosmology: observations

Information

Type
Contributed Paper
Information
Proceedings of the International Astronomical Union , Volume 21 , Symposium S397: Exploring the Universe with Artificial Intelligence (UniversAI) , August 2025 , pp. 8 - 12
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Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution and reproduction, provided the original article is properly cited.
Copyright
© The Author(s), 2026. Published by Cambridge University Press on behalf of The International Astronomical Union

1. Introduction

Cosmic shear, the subtle distortion of distant galaxy shapes by intervening large-scale structure, provides a sensitive probe of dark energy and is one of the main science drivers for upcoming Stage IV surveys, including Euclid (Laureijs et al. Reference Laureijs2011), LSST (LSST Dark Energy Science Collaboration 2012), and Roman (Spergel et al. Reference Spergel2015). Achieving the full potential of these surveys requires unprecedented control of systematic effects, including, for example, noise, model-fitting, and selection biases (see Mandelbaum Reference Mandelbaum2018). Machine learning (ML) methods offer a promising approach to shear estimation (e.g. Tewes et al. Reference Tewes, Kuntzer, Nakajima, Courbin, Hildebrandt and Schrabback2019; Ribli et al. Reference Ribli, Dobos and Csabai2019). However, all ML methods depend critically on training set fidelity - mismatches between training simulations and real survey data introduce systematic biases. This study investigates CNNs for precision shear measurement, focusing on hyperparameter optimization and sensitivity to point spread function (PSF) misestimation in the training data.

2. Methodology

2.1. Gravitational shear and bias requirements

The intrinsic shape of a galaxy can be described by its complex ellipticity,

(1) e Superscript normal i normal n normal t Baseline equals left parenthesis StartFraction a minus b Over a plus b EndFraction right parenthesis e Superscript 2 i phi Baseline equals e 1 Superscript normal i normal n normal t Baseline plus i e 2 Superscript normal i normal n normal t Baseline comma \begin{equation} e^{\mathrm{int}} = \left(\frac{a-b}{a+b}\right)e^{2i\phi} = e^{\mathrm{int}}_1 + i e^{\mathrm{int}}_2,\end{equation}

where a and b denote the major and minor axes and phi $\phi$ is the position angle. In the weak lensing limit, lensing adds a small shear, so that

(2) e 1 Superscript normal l normal e normal n Baseline almost equals e 1 Superscript normal i normal n normal t Baseline plus gamma 1 comma e 2 Superscript normal l normal e normal n Baseline almost equals e 2 Superscript normal i normal n normal t Baseline plus gamma 2 comma \begin{equation} e^{\mathrm{len}}_1 \approx e^{\mathrm{int}}_1 + \gamma_1, \quad e^{\mathrm{len}}_2 \approx e^{\mathrm{int}}_2 + \gamma_2,\end{equation}

with gamma 1 $\gamma_1$ and gamma 2 $\gamma_2$ the shear components.

Shear measurement methods are commonly assessed using a linear model for the recovered shear,

(3) ModifyingAbove gamma With caret Subscript i Baseline equals left parenthesis 1 plus m Subscript i Baseline right parenthesis gamma Subscript i Baseline plus c Subscript i Baseline comma \begin{equation} \hat{\gamma}_i = (1+m_i)\gamma_i + c_i,\end{equation}

where m Subscript i $m_i$ and c Subscript i $c_i$ quantify multiplicative and additive biases. For Euclid, the performance targets are StartAbsoluteValue m Subscript i Baseline EndAbsoluteValue less than 2 times 10 Superscript negative 3 $|m_i| < 2\times10^{-3}$ and StartAbsoluteValue c Subscript i Baseline EndAbsoluteValue less than 2 times 10 Superscript negative 4 $|c_i| < 2\times10^{-4}$ (Amara & Réfrégier Reference Amara and Réfrégier2008).

2.2. CNN architecture and loss function

We train two independent CNNs, one for each ellipticity component ModifyingAbove e With caret Subscript i $\hat{e}_i$ . The networks follow a shallow design (Figure 1) with a single convolutional layer consisting of 3 times $\times$ 3 pixel filters, followed by a dense layer with hyperbolic tangent activation function.

Figure 1.

Schematic of the CNN used for shear estimation (from Voigt Reference Voigt2024). Each input is a 15 times 15 $15\times15$ pixel stamp containing a PSF-convolved galaxy. The network predicts galaxy ellipticities ModifyingAbove e With caret Subscript i $\hat{e}_i$ , which are then averaged across the sample to obtain shear estimates ModifyingAbove gamma With caret Subscript i $\hat{\gamma}_i$ .

A schematic of a convolutional neural network for shear estimation.

To reduce the impact of noise bias we employ a mean square bias (MSB) loss function (Gruen et al. Reference Gruen, Seitz, Koppenhoefer and Riffeser2010):

(4) normal upper M normal upper S normal upper B equals StartFraction 1 Over n Subscript normal g normal a normal l Baseline EndFraction sigma summation Underscript n equals 1 Overscript n Subscript normal g normal a normal l Baseline Endscripts left bracket StartFraction 1 Over n Subscript normal r normal e normal a normal l Baseline EndFraction sigma summation Underscript m equals 1 Overscript n Subscript normal r normal e normal a normal l Baseline Endscripts left parenthesis e Subscript i Superscript normal l normal e normal n Baseline minus ModifyingAbove e With caret Subscript i Baseline right parenthesis right bracket squared comma \begin{equation} \mathrm{MSB} = \frac{1}{n_{\mathrm{gal}}}\sum_{n=1}^{n_{\mathrm{gal}}}\left[\frac{1}{n_{\mathrm{real}}}\sum_{m=1}^{n_{\mathrm{real}}}\left(e_i^{\mathrm{len}}-\hat{e}_i\right)\right]^2,\end{equation}

where n Subscript normal g normal a normal l $n_{\mathrm{gal}}$ is the number of unique galaxy images in the training set, and n Subscript normal r normal e normal a normal l $n_{\mathrm{real}}$ is the number of noisy versions generated for each.

2.3. Galaxy and PSF simulation

We create simulated Euclid-like images with a pixel scale of 0.1 arcsec and an elliptical Gaussian PSF with FWHM 0.17 arcsec. The galaxy population consists of exponential disks and de Vaucouleurs ellipticals in a 4:1 ratio, with sizes, ellipticities and signal-to-noise ratios drawn from realistic distributions (Fig. 2, see Voigt Reference Voigt2024).

Figure 2.

Simulation workflow. Starting from analytic galaxy (1) and PSF (2) models, images are convolved on a large, fine grid (3), binned up to 0.1 arcsec pixels (4), cropped (5), and combined with noise (6) to produce the final 15 times 15 $15\times15$ pixel stamps (7). Figure from Voigt (Reference Voigt2024).

A diagram illustrating the simulation workflow for galaxy images.

3. Results

3.1. CNN optimisation

We investigate how the CNN performance depends on the model hyperparameters. As shown in Figure 3, we find that using 10 Superscript 5 $10^{5}$ galaxies in the training sets with 30 filters and 100 epochs is sufficient for the multiplicative and additive biases to fall within the Euclid requirements. For models trained on noisy images, the number of noisy realisations of each unique galaxy image is critical. With n Subscript normal r normal e normal a normal l Baseline equals 1 $n_{\mathrm{real}}=1$ (equivalent to a mean square error loss) the multiplicative biases remain large ( StartAbsoluteValue m Subscript i Baseline EndAbsoluteValue greater than or equivalent to 0.1 $|m_i|\gtrsim 0.1$ ). Using greater than or equivalent to 500 $\gtrsim500$ noise realisations suppresses noise bias by nearly two orders of magnitude, bringing the results in line with the requirements.

Figure 3.

Dependence of multiplicative ( m Subscript i $m_i$ ; top) and additive ( c Subscript i $c_i$ ; bottom) biases on training set size (left), epochs (middle left), number of filters (middle right), and number of noisy realisations (right). Crosses (squares) denote the first (second) shear component. Dark shaded regions show Euclid requirements. Figure from Voigt (Reference Voigt2024).

Multiple line graphs depict biases based on different training parameters.

3.2. PSF misestimation

We assess how shear estimates are affected when the PSF used for training differs from that in the test data. The dependence of the multiplicative and additive biases can be described by

(5) m Subscript i Baseline equals beta Subscript 0 comma i Baseline plus beta Subscript 1 comma i Baseline left parenthesis StartFraction delta r Subscript normal h Superscript normal upper P normal upper S normal upper F Baseline Over r Subscript normal h Superscript normal upper P normal upper S normal upper F Baseline EndFraction right parenthesis comma c Subscript i Baseline equals alpha Subscript 0 comma i Baseline plus alpha Subscript 1 comma i Baseline delta e Subscript i Superscript normal upper P normal upper S normal upper F Baseline comma \begin{equation}m_i = \beta_{0,i}+\beta_{1,i}\left(\frac{\delta r_\mathrm{h}^{\mathrm{PSF}}}{r_\mathrm{h}^\mathrm{PSF}}\right), \qquad c_i = \alpha_{0,i} + \alpha_{1,i} \,\delta e_i^{\mathrm{PSF}},\end{equation}

where delta r Subscript normal h Superscript normal upper P normal upper S normal upper F $\delta r_\mathrm{h}^{\mathrm{PSF}}$ and delta e Subscript i Superscript normal upper P normal upper S normal upper F $\delta e_i^{\mathrm{PSF}}$ are offsets in PSF size and ellipticity between the training and test sets. As expected, PSF ellipticity errors influence the additive biases, while PSF size errors affect the multiplicative biases. Our results (Figure 4) show that the CNN meets the Euclid requirements provided StartAbsoluteValue delta e Subscript i Superscript normal upper P normal upper S normal upper F Baseline EndAbsoluteValue less than or equivalent to 10 Superscript negative 3 $|\delta e^{\mathrm{PSF}}_i|\lesssim10^{-3}$ and StartAbsoluteValue delta r Subscript normal h Superscript normal upper P normal upper S normal upper F Baseline EndAbsoluteValue divided by r Subscript normal h Superscript normal upper P normal upper S normal upper F Baseline less than or equivalent to 5 times 10 Superscript negative 3 $|\delta r^{\mathrm{PSF}}_{\mathrm{h}}|/r^{\mathrm{PSF}}_{\mathrm{h}}\lesssim5\times10^{-3}$ . These tolerances are an order of magnitude less stringent than the requirements on the PSF model size and shape set out in the Euclid Definition Study Report (Laureijs et al. Reference Laureijs2011).

Figure 4.

Impact of PSF misestimation on shear biases. Residual multiplicative normal upper Delta m Subscript i $\Delta m_i$ (top) and additive normal upper Delta c Subscript i $\Delta c_i$ (bottom) biases, relative to the fiducial case. Results are shown for offsets in PSF size (left) and ellipticity (middle/right), using both noise-free and noisy simulations. Points and shaded regions as in Figure 3. Figure from Voigt (Reference Voigt2024).

Graphs depict the impact of PSF misestimation on shear biases.

4. Conclusions

CNNs offer a promising approach for precision shear measurement in next-generation surveys. Using a shallow architecture with an appropriate loss function designed to mitigate noise bias, we achieve biases below Euclid requirements. Furthermore, the biases are relatively robust to PSF modeling uncertainties.

Acknowledgements

This work is based on research published in Voigt (Reference Voigt2024).

References

Amara, A., Réfrégier, A., 2008, MNRAS, 391, 228 CrossRefGoogle Scholar
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LSST Dark Energy Science Collaboration 2012, arXiv e-prints, p. arXiv:1211.0310 Google Scholar
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Figure 0

Figure 1. Schematic of the CNN used for shear estimation (from Voigt 2024). Each input is a 15×15$15\times15$ pixel stamp containing a PSF-convolved galaxy. The network predicts galaxy ellipticities e^i$\hat{e}_i$, which are then averaged across the sample to obtain shear estimates γ^i$\hat{\gamma}_i$.

Figure 1

Figure 2. Simulation workflow. Starting from analytic galaxy (1) and PSF (2) models, images are convolved on a large, fine grid (3), binned up to 0.1 arcsec pixels (4), cropped (5), and combined with noise (6) to produce the final 15×15$15\times15$ pixel stamps (7). Figure from Voigt (2024).

Figure 2

Figure 3. Dependence of multiplicative (mi$m_i$; top) and additive (ci$c_i$; bottom) biases on training set size (left), epochs (middle left), number of filters (middle right), and number of noisy realisations (right). Crosses (squares) denote the first (second) shear component. Dark shaded regions show Euclid requirements. Figure from Voigt (2024).

Figure 3

Figure 4. Impact of PSF misestimation on shear biases. Residual multiplicative Δmi$\Delta m_i$ (top) and additive Δci$\Delta c_i$ (bottom) biases, relative to the fiducial case. Results are shown for offsets in PSF size (left) and ellipticity (middle/right), using both noise-free and noisy simulations. Points and shaded regions as in Figure 3. Figure from Voigt (2024).