# Switched Flow Matching: Eliminating Singularities via Switching ODEs

arXiv: https://arxiv.org/abs/2405.11605 · HTML: https://mlsanity.com/paper/2405.11605/

## 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 35 of its 69 references that are in the map.
- First posted on arXiv 19 May 2024.

## 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, 30 of 35)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Photorealistic Text-to-Image Diffusion Models with Deep Language Understanding](https://mlsanity.com/paper/2205.11487/) (2022)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [GLIDE: Towards Photorealistic Image Generation and Editing with Text-Guided Diffusion Models](https://mlsanity.com/paper/2112.10741/) (2021)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [WaveNet: A Generative Model for Raw Audio](https://mlsanity.com/paper/1609.03499/) (2016)
- [Cascaded Diffusion Models for High Fidelity Image Generation](https://mlsanity.com/paper/2106.15282/) (2021)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [Conditional Image Generation with PixelCNN Decoders](https://mlsanity.com/paper/1606.05328/) (2016)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [WaveGrad: Estimating Gradients for Waveform Generation](https://mlsanity.com/paper/2009.00713/) (2020)
- [FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models](https://mlsanity.com/paper/1810.01367/) (2018)
- [Conditional Image Synthesis With Auxiliary Classifier GANs](https://mlsanity.com/paper/1610.09585/) (2016)
- [Towards Principled Methods for Training Generative Adversarial Networks](https://mlsanity.com/paper/1701.04862/) (2017)
- [Augmented Neural ODEs](https://mlsanity.com/paper/1904.01681/) (2019)
- [MADE: Masked Autoencoder for Distribution Estimation](https://mlsanity.com/paper/1502.03509/) (2015)
- [Multisample Flow Matching: Straightening Flows with Minibatch Couplings](https://mlsanity.com/paper/2304.14772/) (2023)
- [How to Train Your Energy-Based Models](https://mlsanity.com/paper/2101.03288/) (2021)
- [Dissecting Neural ODEs](https://mlsanity.com/paper/2002.08071/) (2020)

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

- [Governing equation discovery of a complex system from snapshots](https://mlsanity.com/paper/2410.16694/) (2024)
- [Rectified Schr"odinger Bridge Matching for Few-Step Visual Navigation](https://mlsanity.com/paper/2604.05673/) (2026)
- [TracingFlow: A Simulation-Free Trajectory Inference Framework Based on Second-Order Dynamics](https://mlsanity.com/paper/2608.21070/) (2026)
- [MODE: Learning compositional representations of complex systems with Mixtures Of Dynamical Experts](https://mlsanity.com/paper/2510.09594/) (2025)

## Abstract (arXiv)

Continuous-time generative models, such as Flow Matching (FM), construct probability paths to transport between one distribution and another through the simulation-free learning of the neural ordinary differential equations (ODEs). During inference, however, the learned model often requires multiple neural network evaluations to accurately integrate the flow, resulting in a slow sampling speed. We attribute the reason to the inherent (joint) heterogeneity of source and/or target distributions, namely the singularity problem, which poses challenges for training the neural ODEs effectively. To address this issue, we propose a more general framework, termed Switched FM (SFM), that eliminates singularities via switching ODEs, as opposed to using a uniform ODE in FM. Importantly, we theoretically show that FM cannot transport between two simple distributions due to the existence and uniqueness of initial value problems of ODEs, while these limitations can be well tackled by SFM. From an orthogonal perspective, our framework can seamlessly integrate with the existing advanced techniques, such as minibatch optimal transport, to further enhance the straightness of the flow, yielding a more efficient sampling process with reduced costs. We demonstrate the effectiveness of the newly proposed SFM through several numerical examples.