# SESaMo: Symmetry-Enforcing Stochastic Modulation for Normalizing Flows

arXiv: https://arxiv.org/abs/2505.19619 · HTML: https://mlsanity.com/paper/2505.19619/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 50 of its 76 references that are in the map.
- First posted on arXiv 26 May 2025.

## Builds on (most cited first, 30 of 50)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (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)
- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Diffusion Models: A Comprehensive Survey of Methods and Applications](https://mlsanity.com/paper/2209.00796/) (2022)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [E(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [Conditional Image Generation with PixelCNN Decoders](https://mlsanity.com/paper/1606.05328/) (2016)
- [Pixel Recurrent Neural Networks](https://mlsanity.com/paper/1601.06759/) (2016)
- [Equivariant Diffusion for Molecule Generation in 3D](https://mlsanity.com/paper/2203.17003/) (2022)
- [Neural Spline Flows](https://mlsanity.com/paper/1906.04032/) (2019)
- [Spherical CNNs](https://mlsanity.com/paper/1801.10130/) (2018)
- [Equivariant message passing for the prediction of tensorial properties and molecular spectra](https://mlsanity.com/paper/2102.03150/) (2021)
- [Invariant and Equivariant Graph Networks](https://mlsanity.com/paper/1812.09902/) (2018)
- [Gauge Equivariant Convolutional Networks and the Icosahedral CNN](https://mlsanity.com/paper/1902.04615/) (2019)
- [Deep Generative Modelling: A Comparative Review of VAEs, GANs, Normalizing Flows, Energy-Based and Autoregressive Models](https://mlsanity.com/paper/2103.04922/) (2021)
- [Equivariant Flows: Exact Likelihood Generative Learning for Symmetric Densities](https://mlsanity.com/paper/2006.02425/) (2020)
- [E(n) Equivariant Normalizing Flows](https://mlsanity.com/paper/2105.09016/) (2021)
- [On the Universality of Invariant Networks](https://mlsanity.com/paper/1901.09342/) (2019)
- [Flow-based generative models for Markov chain Monte Carlo in lattice field theory](https://mlsanity.com/paper/1904.12072/) (2019)
- [Equivariant flow matching](https://mlsanity.com/paper/2306.15030/) (2023)
- [Gradient Estimation Using Stochastic Computation Graphs](https://mlsanity.com/paper/1506.05254/) (2015)
- [Lorentz Group Equivariant Neural Network for Particle Physics](https://mlsanity.com/paper/2006.04780/) (2020)
- [Equivariant flow-based sampling for lattice gauge theory](https://mlsanity.com/paper/2003.06413/) (2020)
- [Solving Statistical Mechanics Using Variational Autoregressive Networks](https://mlsanity.com/paper/1809.10606/) (2018)

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

- [Scaling flow-based approaches for topology sampling in SU(3) gauge theory](https://mlsanity.com/paper/2510.25704/) (2025)
- [Toward Scalable Normalizing Flows for the Hubbard Model](https://mlsanity.com/paper/2601.18273/) (2026)

## Abstract (arXiv)

Deep generative models have recently garnered significant attention across various fields, from physics to chemistry, where sampling from unnormalized Boltzmann-like distributions represents a fundamental challenge. In particular, autoregressive models and normalizing flows have become prominent due to their appealing ability to yield closed-form probability densities. Moreover, it is well-established that incorporating prior knowledge - such as symmetries - into deep neural networks can substantially improve training performances. In this context, recent advances have focused on developing symmetry-equivariant generative models, achieving remarkable results. Building upon these foundations, this paper introduces Symmetry-Enforcing Stochastic Modulation (SESaMo). Similar to equivariant normalizing flows, SESaMo enables the incorporation of inductive biases (e.g., symmetries) into normalizing flows through a novel technique called stochastic modulation. This approach enhances the flexibility of the generative model, allowing to effectively learn a variety of exact and broken symmetries. Our numerical experiments benchmark SESaMo in different scenarios, including an 8-Gaussian mixture model and physically relevant field theories, such as the φ4 theory and the Hubbard model.