# Translation and Rotation Equivariant Normalizing Flow (TRENF) for Optimal Cosmological Analysis

arXiv: https://arxiv.org/abs/2202.05282 · HTML: https://mlsanity.com/paper/2202.05282/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 0 of its 8 references that are in the map.
- First posted on arXiv 10 Feb 2022.

## Built on by (most cited first, 11 of 11)

- [Beyond Euclid: An Illustrated Guide to Modern Machine Learning with Geometric, Topological, and Algebraic Structures](https://mlsanity.com/paper/2407.09468/) (2024)
- [Field Level Neural Network Emulator for Cosmological N-body Simulations](https://mlsanity.com/paper/2206.04594/) (2022)
- [Multiscale Flow for Robust and Optimal Cosmological Analysis](https://mlsanity.com/paper/2306.04689/) (2023)
- [Machine Learning and Cosmology](https://mlsanity.com/paper/2203.08056/) (2022)
- [Is machine learning good or bad for the natural sciences?](https://mlsanity.com/paper/2405.18095/) (2024)
- [Wavelet Flow For Extragalactic Foreground Simulations](https://mlsanity.com/paper/2505.21220/) (2025)
- [FAIR Universe Weak Lensing ML Uncertainty Challenge: Handling Uncertainties and Distribution Shifts for Precision Cosmology](https://mlsanity.com/paper/2604.14451/) (2026)
- [Reconstructing the Universe with Variational self-Boosted Sampling](https://mlsanity.com/paper/2206.15433/) (2022)
- [Parallelized Acquisition for Active Learning using Monte Carlo Sampling](https://mlsanity.com/paper/2305.19267/) (2023)
- [Velocityformer: Broken-Symmetry-Matched Equivariant Graph Transformers for Cosmological Velocity Reconstruction](https://mlsanity.com/paper/2605.21483/) (2026)
- [Emulating Cosmic Structure Formation with a Lagrangian Neural Cellular Automaton](https://mlsanity.com/paper/2607.27320/) (2026)

## Abstract (arXiv)

Our universe is homogeneous and isotropic, and its perturbations obey translation and rotation symmetry. In this work we develop Translation and Rotation Equivariant Normalizing Flow (TRENF), a generative Normalizing Flow (NF) model which explicitly incorporates these symmetries, defining the data likelihood via a sequence of Fourier space-based convolutions and pixel-wise nonlinear transforms. TRENF gives direct access to the high dimensional data likelihood p(x|y) as a function of the labels y, such as cosmological parameters. In contrast to traditional analyses based on summary statistics, the NF approach has no loss of information since it preserves the full dimensionality of the data. On Gaussian random fields, the TRENF likelihood agrees well with the analytical expression and saturates the Fisher information content in the labels y. On nonlinear cosmological overdensity fields from N-body simulations, TRENF leads to significant improvements in constraining power over the standard power spectrum summary statistic. TRENF is also a generative model of the data, and we show that TRENF samples agree well with the N-body simulations it trained on, and that the inverse mapping of the data agrees well with a Gaussian white noise both visually and on various summary statistics: when this is perfectly achieved the resulting p(x|y) likelihood analysis becomes optimal. Finally, we develop a generalization of this model that can handle effects that break the symmetry of the data, such as the survey mask, which enables likelihood analysis on data without periodic boundaries.