D$_4$CNN$\times$AnaCal: Physics-Informed Machine Learning for Accurate and Precise Weak Lensing Shear Estimation

Abstract

Traditional weak gravitational lensing shear estimators are carefully calibrated but struggle to fully capture realistic galaxy morphologies, point-spread-function (PSF) effects, blending, and noise in deep surveys, while blindly trained machine learning (ML) models can introduce significant calibration biases. Here we construct a fully D$_4$-equivariant deep neural network for galaxy shape measurement whose architecture enforces symmetry under 90$^{\circ}$ rotations and mirror transformations, and adopt the Analytical Calibration framework (AnaCal) to calibrate the model using its backpropagated gradients. For isolated galaxies in LSST-like single-band simulations, we demonstrate that our approach achieves $\sim$10% lower shape noise than the traditional moment-based Fourier Power Function Shapelets estimator in the high-noise regime, equivalent to a $\sim$20% gain in effective galaxy number density, while simultaneously achieving multiplicative biases consistent with zero across a wide range of noise levels, PSF sizes and ellipticities, and magnitude selection cuts, with all measurements satisfying $|m| {<} 10^{-3}$ (i.e., within the 0.2% LSST requirement) and most at the ${\sim}10^{-4}$ level. We demonstrate this framework on isolated single-band galaxy images with Gaussian noise and known PSF, establishing a rigorous, physics-informed foundation for future extensions of ML-based shear estimation to blended sources and multi-band observations in Stage-IV surveys. All codes and data products will be made publicly available upon acceptance.

Figures (8)

• Figure 1: Workflow of the analytical shear calibration for an ML–based shear estimator. Starting from a galaxy image, the ML model produces a raw shape estimate and corresponding model gradient via backpropagation. In parallel, the shear response of the resmoothed image is computed analytically at the pixel level using the expressions in \ref{['eq:pixel_rsp']}. The contraction of the model gradient with the pixel shear response yields the shear response matrix $R_{ij} = \partial e_i / \partial \gamma_j$, which is then used to perform a linear calibration of the ML-based shape estimator. The calibrated shear is finally used to evaluate the residual calibration biases, quantified by the multiplicative and additive bias parameters $m$ and $c$.
• Figure 2: Example architecture of the D4-equivariant shape measurement model ($\mathrm{D}_4$CNN). The input galaxy image is transformed over the full D4 orbit (four rotations and their mirrored counterparts), and we forward the CNN eight times on each of the eight transformed variants to produce a set of features. These features are then mapped back to the original reference frame and combined through a weighted average to construct a D4-equivariant feature representation. Finally, an unbiased odd MLP preserves the sign of this equivariant feature, ensuring the output shape transforms with the desired equivariance under D4 symmetry operations.
• Figure 3: Transformation of measured shape under rotation for our $\mathrm{D}_4$CNN. The same galaxy profile with fixed ellipticity ($e = 0.4$) but different orientations ($-90\degree$ to $+90\degree$ between the x-axis and the major axis) is fed into the model. The predicted $e_1$ and $e_2$ components (blue and orange points) are compared against theoretical sine/cosine functions (dashed and dash-dotted curves) scaled to match the measured amplitude. Vertical dotted lines mark key reference angles at $-45\degree$, $0\degree$, and $+45\degree$.
• Figure 4: Shear estimation results for galaxies with $m_i<25.3$ under variations in image noise level (top row), PSF ellipticity (middle row), and PSF full width at half maximum (bottom row). The left panels show the multiplicative bias $m_1$ (blue) and $m_2$ (red), while the right panels show the additive bias $c_1$ (blue) and $c_2$ (red). Data points are horizontally slightly shifted from the original value to avoid overlap. Points indicate the mean bias estimates, with error bars corresponding to $1\sigma$ (deep color) and $3\sigma$ (light color) statistical uncertainties with bootstrap resampling 20 times. The shaded gray region in the multiplicative-bias panels denotes the LSST requirement of $|m|<0.002$. Across all tested observing conditions, both multiplicative and additive biases remain consistent with zero within errors, demonstrating the accuracy of the calibrated ML-based shear estimator.
• Figure 5: Shear estimation results for galaxies with different i-band magnitude cuts. The shaded gray region in the multiplicative-bias panels denotes the LSST requirement of $|m|<0.002$. Left panels show the multiplicative bias $m_1$ (blue) and $m_2$ (red), while the right panels show the additive bias $c_1$ (blue) and $c_2$ (red). Points indicate the mean bias estimates, with error bars corresponding to $1\sigma$ (deep color) and $3\sigma$ (light color) statistical uncertainties with bootstrap resampling 20 times. Data points are horizontally slightly shifted from the original value to avoid overlap. With different magnitude cuts, both multiplicative and additive biases remain consistent with zero within errors, demonstrating the accuracy of the calibrated ML-based shear estimator.