# Multiscale Flow for Robust and Optimal Cosmological Analysis

arXiv: https://arxiv.org/abs/2306.04689 · HTML: https://mlsanity.com/paper/2306.04689/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 94 references that are in the map.
- First posted on arXiv 7 Jun 2023.

## Builds on (most cited first, 19 of 19)

- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [Masked Autoregressive Flow for Density Estimation](https://mlsanity.com/paper/1705.07057/) (2017)
- [The frontier of simulation-based inference](https://mlsanity.com/paper/1911.01429/) (2019)
- [A note on the evaluation of generative models](https://mlsanity.com/paper/1511.01844/) (2015)
- [How to Train Your Energy-Based Models](https://mlsanity.com/paper/2101.03288/) (2021)
- [Are Generative Classifiers More Robust to Adversarial Attacks?](https://mlsanity.com/paper/1802.06552/) (2018)
- [Wavelet Flow: Fast Training of High Resolution Normalizing Flows](https://mlsanity.com/paper/2010.13821/) (2020)
- [Generative and Discriminative Text Classification with Recurrent Neural Networks](https://mlsanity.com/paper/1703.01898/) (2017)
- [Revisiting Discriminative vs. Generative Classifiers: Theory and Implications](https://mlsanity.com/paper/2302.02334/) (2023)
- [Representational aspects of depth and conditioning in normalizing flows](https://mlsanity.com/paper/2010.01155/) (2020)
- [Hybrid Discriminative-Generative Training via Contrastive Learning](https://mlsanity.com/paper/2007.09070/) (2020)
- [Realistic galaxy image simulation via score-based generative models](https://mlsanity.com/paper/2111.01713/) (2021)
- [Learning cosmology and clustering with cosmic graphs](https://mlsanity.com/paper/2204.13713/) (2022)
- [Multifield Cosmology with Artificial Intelligence](https://mlsanity.com/paper/2109.09747/) (2021)
- [Non-Gaussian information from weak lensing data via deep learning](https://mlsanity.com/paper/1802.01212/) (2018)
- [Translation and Rotation Equivariant Normalizing Flow (TRENF) for Optimal Cosmological Analysis](https://mlsanity.com/paper/2202.05282/) (2022)
- [Can denoising diffusion probabilistic models generate realistic astrophysical fields?](https://mlsanity.com/paper/2211.12444/) (2022)

## Built on by (most cited first, 10 of 10)

- [Field-level simulation-based inference with galaxy catalogs: the impact of systematic effects](https://mlsanity.com/paper/2310.15234/) (2023)
- [Cosmological Field Emulation and Parameter Inference with Diffusion Models](https://mlsanity.com/paper/2312.07534/) (2023)
- [Diffusion-HMC: Parameter Inference with Diffusion-model-driven Hamiltonian Monte Carlo](https://mlsanity.com/paper/2405.05255/) (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)
- [Generative Models and Statistical Validation](https://mlsanity.com/paper/2605.30453/) (2026)
- [Factorizable Normalizing Flows for parameter-dependent density morphing](https://mlsanity.com/paper/2606.30489/) (2026)
- [Hybrid summary statistics: neural weak lensing inference beyond the power spectrum](https://mlsanity.com/paper/2407.18909/) (2024)
- [Learning the Universe: Posterior Reliability of Neural Generative Models in High-Dimensional Field-Level Inference of Cosmic Initial Conditions](https://mlsanity.com/paper/2606.10023/) (2026)
- [Emulating Cosmic Structure Formation with a Lagrangian Neural Cellular Automaton](https://mlsanity.com/paper/2607.27320/) (2026)

## Abstract (arXiv)

We propose Multiscale Flow, a generative Normalizing Flow that creates samples and models the field-level likelihood of two-dimensional cosmological data such as weak lensing. Multiscale Flow uses hierarchical decomposition of cosmological fields via a wavelet basis, and then models different wavelet components separately as Normalizing Flows. The log-likelihood of the original cosmological field can be recovered by summing over the log-likelihood of each wavelet term. This decomposition allows us to separate the information from different scales and identify distribution shifts in the data such as unknown scale-dependent systematics. The resulting likelihood analysis can not only identify these types of systematics, but can also be made optimal, in the sense that the Multiscale Flow can learn the full likelihood at the field without any dimensionality reduction. We apply Multiscale Flow to weak lensing mock datasets for cosmological inference, and show that it significantly outperforms traditional summary statistics such as power spectrum and peak counts, as well as novel Machine Learning based summary statistics such as scattering transform and convolutional neural networks. We further show that Multiscale Flow is able to identify distribution shifts not in the training data such as baryonic effects. Finally, we demonstrate that Multiscale Flow can be used to generate realistic samples of weak lensing data.