MIU2Net: weak-lensing mass inversion using deep learning with nested U-structures

TL;DR

The paper tackles weak-lensing mass inversion by developing MIU2Net, a nested U2-Net–based framework that jointly optimizes pixelwise convergence and its frequency-domain power spectrum through a Radial-Averaged Power Spectrum (RAPS) loss. Trained on ray-traced simulations with realistic shape noise and masking, MIU2Net achieves about 4% accuracy in the convergence power spectrum up to $l \approx 500$, substantially outperforming traditional methods and previous DL approaches. Beyond two-point statistics, the method reliably reconstructs the convergence distribution, peak centroids, and amplitudes, and shows robustness to masks and reduced shear, while enabling partial lifting of the mass-sheet degeneracy via learned priors (with a correction factor that generalizes across cosmologies). The results suggest MIU2Net as a powerful tool for mapping dark matter and probing large-scale structure with next-generation space surveys like CSST and Euclid, with potential extensions to halo identification and cosmological parameter inference.

Abstract

One of the primary goals of next-generation gravitational lensing surveys is to measure the large-scale distribution of dark matter, which requires accurate mass inversion to convert weak-lensing shear maps into convergence (kappa) fields. This work develops a mass inversion method tailored for upcoming space missions such as CSST and Euclid, aiming to recover both the mass distribution and the convergence power spectrum with high fidelity. We introduce MIU2Net, a versatile deep-learning framework for kappa-map reconstruction based on the U2-Net architecture. A new loss function is constructed to jointly estimate the convergence field and its frequency-domain energy distribution, effectively balancing optimal mean squared error and optimal power-spectrum recovery. The method incorporates realistic observational effects into shear fields, including shape noise, reduced shear, and complex masks. Under noise levels anticipated for future space-based lensing surveys, MIU2Net recovers the convergence power spectrum with 4% uncertainties up to l approximately 500, significantly outperforming Wiener filtering and MCALens. Beyond two-point statistics, the method accurately reconstructs the convergence distribution, peak centroid, and peak amplitude. Compared to other learning-based approaches such as DeepMass, MIU2Net reduces the root-mean-square error by 5% without smoothing and by 38% with a 1-arcmin smoothing scale. MIU2Net represents a substantial advancement in mass inversion methodology, offering improved accuracy in both RMSE and power-spectrum reconstruction. It provides a promising tool for mapping dark matter environments and large-scale structures in the era of next-generation space lensing surveys.

Figures (18)

• Figure 1: Visual comparison between Truth, our method (MIU2Net), and established methods including Kaiser-Squires (KS), Wiener filtering (WF), MCALens (MCA), and SimpleUNet (UNet, based on DeepMass). Each panel covers a $1.75 \times 1.75$$\deg^2$ Euclidean patch of sky. The shear maps used for reconstruction have shape noise corresponding to galaxy number density $n_g = 20 \, \rm{arcmin}^{-2}$. We visualize in detail the scarlet dashed square region in each panel in Fig. \ref{['fig:profile3d']}.
• Figure 2: MIU2Net structure adapted from U2-Net. Figure is drawn based on Fig. $5$ in U2Net2020.
• Figure 3: Dynamic range for each of the $500$ convergence reconstructions from MIU2Net, KS, WF, MCA, and UNet.Left panel: minimum predicted convergence value $\min(\bm{\hat{\kappa}})$ against minimum true convergence value $\min(\bm\kappa)$ for each reconstruction. We also plot the best-fit lines, and the shadings indicate the $95\%$ confidence intervals for each method. The coloured dots follow the same labelling scheme as in the right panel. The black dashed line is the $y=x$ line denoting ideal reconstruction. Note that MIU2Net can best recover the lower bound, but it systematically overestimates it when $\min(\bm\kappa)$ is low. KS reconstructions cluster around the lower region far away from truth, because KS is symmetric around $0$ and does not provide a good estimate for the lower bound. Right panel: scatter plot and best-fit lines for maximum values for each true-prediction pair. The shadings indicate the $95\%$ confidence intervals for each method. Again, the black dashed line is the $y=x$ line tracing the ideal reconstruction. It is clear that MIU2Net has a slope closest to the ideal and can reliably predict the upper bound of individual convergence maps. Nevertheless, MIU2Net systematically underestimates the upper bound. This is expected because an MSE-based estimator is inclined to make blurry, middle-range predictions that smooth out intense spikes. Compared to UNet, which is also an MSE-based estimator, MIU2Net more reliably recovers the maximum amplitude. As seen by the larger slope, MIU2Net is more inclined to make bold, salient predictions than UNet for larger maximum peaks.
• Figure 4: Centroid and amplitude recovery from MIU2Net, KS, WF, MCA, and UNet for an individual convergence peak. The plotted regions have dimensions $8.2 \times 8.2 \, \rm arcmin^2$ and correspond to the scarlet dashed square regions in Fig. \ref{['fig:visual']}. The crosses indicate the highest peak in each of the recovered regions. The coordinates above the crosses are in the form $(\Delta x, \Delta y, \kappa_{\rm peak})$, where $\Delta x$ and $\Delta y$ are spatial coordinates of the predicted peak relative to that of the true peak, and $\kappa_{\rm peak}$ is the peak convergence value. The Full Width at Half Maximum (FWHM) for each peak is shown in the bottom left of each panel. MIU2Net best recovers the position, amplitude, and FWHM of the true peak without significant noise. However, MIU2Net fails to recover the highly non-Gaussian profile of the true peak, as the reconstruction still exhibits a relatively smoothed profile and a bell-like shape. Note that although KS recovers the central peak with decent accuracy, this is because the central peak has a large convergence value ($0.638$). For lower convergence values, noise can easily submerge the reconstructed peak.
• Figure 5: On the mass-sheet degeneracy. Left panel: for $500$ reconstructions, we plot the mean $\langle \bm{\hat{\kappa}} \rangle$ for each of the predicted convergence maps against the mean $\langle \bm\kappa \rangle$ for each of the true convergence maps. The black dashed line is the $y = x$ line denoting the ideal recovery of mean convergence. MIU2Net reconstructions clearly follow a linear relation whose slope is close to the ideal and has a negligible offset. The blue line and shading indicate the best-fit line for MIU2Net and the $95\%$ confidence interval. Both KS and WF assume $\langle \bm{\hat{\kappa}} \rangle = 0$, so they overlap on the $y = 0$ line. UNet also exhibits a positive slope, suggesting that deep learning MMSE estimators may utilize the prior information in convergence fields better than other methods.Middle panel: the same as left panel, except now we plot the corrected MIU2Net reconstructions with $\mu_{\rm ms} = 1.672$. The value of $\mu_{\rm ms}$ is chosen empirically based on the parameters of the best-fit line in the left panel. The best-fit line for corrected MIU2Net closely resembles the ideal linear relationship. Right panel: the corrected MIU2Net reconstructions with the same scaling factor $\mu_{\rm ms} = 1.672$ for reconstructions using the second cosmology. The corrected mean convergence is still very close to the ideal reconstruction despite the change in cosmology.