Euclid Quick Data Release (Q1). LEMON -- Lens Modelling with Neural networks. Automated and fast modelling of Euclid gravitational lenses with a singular isothermal ellipsoid mass profile

Technical Summary: LEMON – LEns MOdelling with Neural networks for Euclid Gravitational Lenses

Problem Statement
The Euclid mission is set to survey approximately 14,000 deg² of extragalactic sky, potentially yielding around $10^5$ gravitational lens images. While Convolutional Neural Networks (CNNs) have proven effective for the detection of lenses, the subsequent modelling of these systems to estimate mass and light parameters remains a bottleneck. Traditional modelling relies on computationally intensive, manual techniques such as maximum likelihood and Markov chain Monte Carlo (MCMC) methods. These approaches are too slow to handle the anticipated data volume and often require human intervention to set initial conditions, leading to potential failure rates and inefficiencies in the Euclid modelling pipeline.

Methodology
This paper presents an expansion of LEMON (LEns MOdelling with Neural networks), a Bayesian Neural Network (BNN) designed for the automated and rapid modelling of strong gravitational lenses.

• Architecture: LEMON utilizes a ResNet-50 architecture. It employs dropout layers during both training and inference to facilitate Bayesian uncertainty estimation via Monte Carlo sampling. This allows the network to distinguish between two types of uncertainty:
  • Aleatoric uncertainty: Intrinsic to the data (e.g., low signal-to-noise, image corruption).
  • Epistemic uncertainty: Arising from the model's lack of knowledge in specific regions of the parameter space.
• Training Data: The network was trained and tested on 100,000 mock Euclid lenses generated by the Euclid Consortium's Strong Lensing Science Working Group. These simulations include contaminants (foreground/background objects) and cover a wide range of environmental scenarios. The lenses are modelled with a Singular Isothermal Ellipsoid (SIE) mass profile plus external shear, and the light profile is described by a single Sérsic model.
• Target Parameters: LEMON predicts eight key parameters:
  1. Einstein radius ($R_{Ein}$)
  2. Mass ellipticity components ($\epsilon_{x,m}, \epsilon_{y,m}$)
  3. Light ellipticity components ($\epsilon_{x,l}, \epsilon_{y,l}$)
  4. Lens effective radius ($R_{e, lens}$)
  5. Lens Sérsic index ($n_{lens}$)
  6. Lens magnitude ($m_{lens}$)
  7. Unlensed source x and y positions ($x_{src}, y_{src}$)
• Calibration: To address systematic over- or under-estimation of confidence intervals common in BNNs, the authors applied Platt-scaling to calibrate the total uncertainty predictions using a validation set of simulated lenses.
• Hybrid Pipeline: A secondary application of LEMON involves using its predictions as initial starting points for the standard Euclid modelling pipeline (PyAutoLens). This replaces the standard nested-sampling initialisation with a gradient-ascent optimiser, leveraging the BNN's fast inference to accelerate the convergence of traditional methods.

Key Results
The performance of LEMON was validated across three distinct datasets:

1. Mock Euclid Test Set:

   • LEMON demonstrated high accuracy in recovering parameters for simulated lenses.
   • Einstein Radius: Well recovered with a coefficient of determination ($R^2$) of 0.91 and low bias.
   • Ellipticities: Light ellipticity was recovered with exceptional precision ($R^2 \approx 0.96$), while mass ellipticity showed slightly lower but robust performance ($R^2 \approx 0.85-0.86$).
   • Sérsic Index: This was the most challenging parameter, showing the largest scatter ($R^2 = 0.77$) and non-linear trends at high values, consistent with difficulties in recovering this parameter via photometry alone.
   • Uncertainties: The calibrated uncertainties were reliable, with epistemic uncertainties generally lower than aleatoric uncertainties.

2. Euclidised HST Lenses (Real Data, Degraded to Euclid Depth):

   • Applied to real lenses from SLACS, EEL, COSMOS, and ACS surveys converted to Euclid resolution.
   • LEMON matched classical modelling results for Einstein radius ($R^2 = 0.53$) and ellipticities ($R^2 = 0.87-0.88$).
   • Effective radius predictions showed systematic underestimation for values $>1''$, likely due to distributional shifts between the training simulations and real data, and differences in modelling assumptions (e.g., fixed vs. variable Sérsic indices in literature).
   • Uncertainties were higher than in the mock set, attributed to the network encountering parameter space regions outside its training distribution.

3. Real Euclid Data (Perseus ERO and Q1):

   • Perseus ERO: Applied to five confirmed lenses, LEMON recovered Einstein radii within $1\sigma$ of classical modelling (Pronto software), even in the presence of bright contaminants, though with increased uncertainty.
   • Q1 Data: Applied to 354 strong lens candidates from the Euclid Quick Data Release 1.
     • Einstein radius recovery matched PyAutoLens well ($R^2 = 0.71$, bias $= 0.01''$).
     • Light ellipticity and magnitude matched classical results well ($R^2 \approx 0.74-0.90$).
     • Mass ellipticity and Sérsic index showed poor correlation with PyAutoLens, likely due to differences in modelling assumptions (e.g., PyAutoLens imposing a maximum $n=5$).

Speed-Up and Pipeline Integration
A critical finding is the potential for LEMON to accelerate the Euclid modelling pipeline. When LEMON predictions were used as initial guesses for a gradient-ascent optimiser in PyAutoLens:

• The modelling time was reduced by a factor of 26 compared to the standard nested-sampling approach (from ~7 hours to ~0.27 hours per lens on a single CPU).
• Random initialisations failed to recover correct models for any of the test lenses, demonstrating that the BNN's accurate initial guesses are essential for the gradient-ascent method to function effectively.
• The hybrid approach also reduced failure rates by complementing the limitations of the standard pipeline; models that failed with standard methods were sometimes recovered by the hybrid approach.

Significance and Claims
The authors position this work as a significant step toward the efficient, automated modelling required for the Euclid mission's large data volumes. They claim that LEMON:

• Provides a robust, fast method for estimating lens mass and light parameters with quantified uncertainties.
• Can serve as a cross-validation tool to reduce the failure rate of the traditional modelling pipeline.
• Enables a speed-up of up to 26 times for individual lens modelling when integrated with classical optimisers, a feat impossible with random initialisation.
• Is capable of handling the expected ~100,000 lenses from Euclid, facilitating the statistical analysis of dark matter fractions, mass density slopes, and initial mass functions across unprecedented sample sizes.

The paper remains modest regarding the Sérsic index and mass ellipticity recovery on real data, acknowledging that these parameters are more difficult to constrain and that distributional shifts between simulations and real data affect performance. Future work is proposed to expand the training set to cover broader parameter spaces and to implement deblending techniques to improve parameter prediction.