# Equivariant Flow Matching for Symmetry-Breaking Bifurcation Problems

arXiv: https://arxiv.org/abs/2509.03340 · HTML: https://mlsanity.com/paper/2509.03340/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 24 of its 44 references that are in the map.
- First posted on arXiv 3 Sep 2025.

## Placed in

- Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/)

## Builds on (most cited first, 24 of 24)

- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [E(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [Learning Mesh-Based Simulation with Graph Networks](https://mlsanity.com/paper/2010.03409/) (2020)
- [Autoencoding beyond pixels using a learned similarity metric](https://mlsanity.com/paper/1512.09300/) (2015)
- [Machine Learning for Fluid Mechanics](https://mlsanity.com/paper/1905.11075/) (2019)
- [Generating Images with Perceptual Similarity Metrics based on Deep Networks](https://mlsanity.com/paper/1602.02644/) (2016)
- [Equivariant flow matching](https://mlsanity.com/paper/2306.15030/) (2023)
- [Artificial Intelligence for Science in Quantum, Atomistic, and Continuum Systems](https://mlsanity.com/paper/2307.08423/) (2023)
- [Equivariance with Learned Canonicalization Functions](https://mlsanity.com/paper/2211.06489/) (2022)
- [Equivariant Flow Matching with Hybrid Probability Transport](https://mlsanity.com/paper/2312.07168/) (2023)
- [From optimal transport to generative modeling: the VEGAN cookbook](https://mlsanity.com/paper/1705.07642/) (2017)
- [Can Push-forward Generative Models Fit Multimodal Distributions?](https://mlsanity.com/paper/2206.14476/) (2022)
- [Symmetry Breaking and Equivariant Neural Networks](https://mlsanity.com/paper/2312.09016/) (2023)
- [Improving Equivariant Networks with Probabilistic Symmetry Breaking](https://mlsanity.com/paper/2503.21985/) (2025)
- [Learning and discovering multiple solutions using physics-informed neural networks with random initialization and deep ensemble](https://mlsanity.com/paper/2503.06320/) (2025)
- [Enabling Local Neural Operators to perform Equation-Free System-Level Analysis](https://mlsanity.com/paper/2505.02308/) (2025)
- [Equivariant Symmetry Breaking Sets](https://mlsanity.com/paper/2402.02681/) (2024)
- [STFlow: Data-Coupled Flow Matching for Geometric Trajectory Simulation](https://mlsanity.com/paper/2505.18647/) (2025)
- [Equivariant Neural Simulators for Stochastic Spatiotemporal Dynamics](https://mlsanity.com/paper/2305.14286/) (2023)
- [Understanding complex crowd dynamics with generative neural simulators](https://mlsanity.com/paper/2412.01491/) (2024)
- [Similarity Equivariant Graph Neural Networks for Homogenization of Metamaterials](https://mlsanity.com/paper/2404.17365/) (2024)

## Built on by (most cited first, 4 of 4)

- [Multimodal Scientific Learning Beyond Diffusions and Flows](https://mlsanity.com/paper/2602.00960/) (2026)
- [BubbleSH: A Dataset of Rising Bubbles with Deformable Interfaces](https://mlsanity.com/paper/2607.07275/) (2026)
- [Multi-Attractor GNNs: Set-Valued Expressivity Beyond Unique Equilibria](https://mlsanity.com/paper/2609.35274/) (2026)
- [Bi-FORK: Generative Modeling of High-Dimensional Bifurcating Systems](https://mlsanity.com/paper/2610.12449/) (2026)

## Abstract (arXiv)

Bifurcation phenomena in nonlinear dynamical systems often lead to multiple coexisting stable solutions, particularly in the presence of symmetry breaking. Deterministic machine learning models are unable to capture this multiplicity, averaging over solutions and failing to represent lower-symmetry outcomes. In this work, we formalize the use of generative AI, specifically flow matching, as a principled way to model the full probability distribution over bifurcation outcomes. Our approach builds on existing techniques by combining flow matching with equivariant architectures and an optimal-transport-based coupling mechanism. We generalize equivariant flow matching to a symmetric coupling strategy that aligns predicted and target outputs under group actions, allowing accurate learning in equivariant settings. We validate our approach on a range of systems, from simple conceptual systems to physical problems such as buckling beams and the Allen--Cahn equation. The results demonstrate that the approach accurately captures multimodal distributions and symmetry-breaking bifurcations. Moreover, our results demonstrate that flow matching significantly outperforms non-probabilistic and variational methods. This offers a principled and scalable solution for modeling multistability in high-dimensional systems.