# SCORENF: Score-based Normalizing Flows for Sampling Unnormalized distributions

arXiv: https://arxiv.org/abs/2510.21330 · HTML: https://mlsanity.com/paper/2510.21330/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 26 references that are in the map.
- First posted on arXiv 24 Oct 2025.

## Builds on (most cited first, 15 of 15)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [An Introduction to Variational Autoencoders](https://mlsanity.com/paper/1906.02691/) (2019)
- [WaveGrad: Estimating Gradients for Waveform Generation](https://mlsanity.com/paper/2009.00713/) (2020)
- [A Review on Generative Adversarial Networks: Algorithms, Theory, and Applications](https://mlsanity.com/paper/2001.06937/) (2020)
- [SE(3)-Stochastic Flow Matching for Protein Backbone Generation](https://mlsanity.com/paper/2310.02391/) (2023)
- [Flow-based generative models for Markov chain Monte Carlo in lattice field theory](https://mlsanity.com/paper/1904.12072/) (2019)
- [Flow Annealed Importance Sampling Bootstrap](https://mlsanity.com/paper/2208.01893/) (2022)
- [Iterated Denoising Energy Matching for Sampling from Boltzmann Densities](https://mlsanity.com/paper/2402.06121/) (2024)
- [Neural Network Renormalization Group](https://mlsanity.com/paper/1802.02840/) (2018)
- [Full-band General Audio Synthesis with Score-based Diffusion](https://mlsanity.com/paper/2210.14661/) (2022)
- [AdvNF: Reducing Mode Collapse in Conditional Normalising Flows using Adversarial Learning](https://mlsanity.com/paper/2401.15948/) (2024)

## Built on by (most cited first, 2 of 2)

- [SIMANF: Sample Free Learning of Unnormalized Distributions via Simulated Annealing in Normalizing Flows](https://mlsanity.com/paper/2609.32279/) (2026)
- [FUND: Density Flow for Sampling Unnormalised Distributions](https://mlsanity.com/paper/2609.32296/) (2026)

## Abstract (arXiv)

Unnormalized probability distributions are central to modeling complex physical systems across various scientific domains. Traditional sampling methods, such as Markov Chain Monte Carlo (MCMC), often suffer from slow convergence, critical slowing down, poor mode mixing, and high autocorrelation. In contrast, likelihood-based and adversarial machine learning models, though effective, are heavily data-driven, requiring large datasets and often encountering mode covering and mode collapse. In this work, we propose ScoreNF, a score-based learning framework built on the Normalizing Flow (NF) architecture, integrated with an Independent Metropolis-Hastings (IMH) module, enabling efficient and unbiased sampling from unnormalized target distributions. We show that ScoreNF maintains high performance even with small training ensembles, thereby reducing reliance on computationally expensive MCMC-generated training data. We also present a method for assessing mode-covering and mode-collapse behaviours. We validate our method on synthetic 2D distributions (MOG-4 and MOG-8) and the high-dimensional φ4 lattice field theory distribution, demonstrating its effectiveness for sampling tasks.