Description

Strong gravitational lensing is a powerful probe of dark matter substructure and cosmological structure formation. Current simulation pipelines (e.g., Lenstronomy-based ray tracing) require solving the lens equation repeatedly for different mass distributions and source configurations. While accurate, these simulations are computationally expensive and limit large-scale parameter sweeps, uncertainty quantification, and real-time inference. This project proposes the development of a Neural Operator framework to learn the functional mapping between mass distributions and lensed images. Unlike traditional neural networks that operate on finite-dimensional vectors, neural operators learn mappings between infinite-dimensional function spaces. This makes them particularly well-suited for approximating solutions of physical systems governed by partial differential equations. The goal is to train a neural operator that directly maps:

• Lens mass distribution → Lensed image or
• (Mass distribution, source light profile) → Observed lensed image

Such a model would serve as a fast surrogate simulator capable of producing high-fidelity lensing outputs at a fraction of the computational cost of traditional ray-tracing solvers. This would represent the first exploration of neural operators within the ML4SCI DeepLense ecosystem.

Duration

Total project length: 175/350 hours.

Difficulty level

Advanced

Task ideas

1. Literature Study
   1. Study Fourier Neural Operators (FNO)
   2. Study DeepONets
   3. Review operator learning in physical systems
2. Neural Operator Implementation
   1. Implement Fourier Neural Operator for 2D fields
   2. Compare with DeepONet-style architectures
   3. Explore spectral vs spatial operator parameterizations
3. Evaluation
   1. Compare speed vs traditional solver
   2. Measure pixel-wise reconstruction error
   3. Evaluate preservation of physical invariants
   4. Test generalization across:
      1. Different mass profiles
      2. Different redshifts
      3. Different source morphologies
4. Extensions (if time permits)
   1. Conditional neural operators
   2. Uncertainty-aware operator learning
   3. Physics-informed operator constraints
   4. Hybrid operator + diffusion refinement

Expected results

• A neural operator capable of approximating the lensing simulation operator.
• Significant speedup compared to ray-tracing pipelines.
• Demonstration of generalization across varying astrophysical conditions.
• A benchmark study comparing neural operators with CNN-based surrogate models.

Requirements

Python, PyTorch, experience with machine learning and deep learning. Partial understanding of:

• Representation learning
• Spectral methods
• Partial differential equations (basic familiarity)
• Scientific machine learning concepts Familiarity with gravitational lensing is preferred but not mandatory.

Mentors

• Michael Toomey (Massachusetts Institute of Technology)
• Sergei Gleyzer (University of Alabama)
• Pranath Reddy (University of Florida)
• Ashutosh Ojha (Indian Institute of Technology (Indian School of Mines), Dhanbad)

Please DO NOT contact mentors directly by email. Instead, please email ml4-sci@cern.ch with Project Title and include your CV and test results. The relevant mentors will then get in touch with you.

Links

• Paper 1
• Paper 2
• Paper 3

Corresponding Project

• DEEPLENSE

Participating Organizations

• Alabama
• MIT

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