# Ergodic Generative Flows

arXiv: https://arxiv.org/abs/2505.03561 · HTML: https://mlsanity.com/paper/2505.03561/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion and flow policies for robotics, one of 14 directions in Diffusion models.
- Builds on 35 of its 59 references that are in the map.
- First posted on arXiv 6 May 2025.

## Placed in

- Diffusion models / [Diffusion and flow policies for robotics](https://mlsanity.com/diffusion/diffusion-and-flow-policies-for-robotics/)
- Offline RL and imitation learning from data / [Diffusion and flow policies for offline RL](https://mlsanity.com/offlinerl/diffusion-and-flow-policies-for-offline-rl/)

## Builds on (most cited first, 30 of 35)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [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)
- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Scaling Instruction-Finetuned Language Models](https://mlsanity.com/paper/2210.11416/) (2022)
- [DPM-Solver: A Fast ODE Solver for Diffusion Probabilistic Model Sampling in Around 10 Steps](https://mlsanity.com/paper/2206.00927/) (2022)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Exact solutions to the nonlinear dynamics of learning in deep linear neural networks](https://mlsanity.com/paper/1312.6120/) (2013)
- [Deep Equilibrium Models](https://mlsanity.com/paper/1909.01377/) (2019)
- [FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models](https://mlsanity.com/paper/1810.01367/) (2018)
- [On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transport](https://mlsanity.com/paper/1805.09545/) (2018)
- [Flow Network based Generative Models for Non-Iterative Diverse Candidate Generation](https://mlsanity.com/paper/2106.04399/) (2021)
- [GFlowNet Foundations](https://mlsanity.com/paper/2111.09266/) (2021)
- [Flow Matching on General Geometries](https://mlsanity.com/paper/2302.03660/) (2023)
- [Trajectory balance: Improved credit assignment in GFlowNets](https://mlsanity.com/paper/2201.13259/) (2022)
- [Stochastic Normalizing Flows](https://mlsanity.com/paper/2002.06707/) (2020)
- [Multiscale Deep Equilibrium Models](https://mlsanity.com/paper/2006.08656/) (2020)
- [Riemannian Continuous Normalizing Flows](https://mlsanity.com/paper/2006.10605/) (2020)
- [Normalizing Flows on Tori and Spheres](https://mlsanity.com/paper/2002.02428/) (2020)
- [Generative Flow Networks for Discrete Probabilistic Modeling](https://mlsanity.com/paper/2202.01361/) (2022)
- [A theory of continuous generative flow networks](https://mlsanity.com/paper/2301.12594/) (2023)
- [Diffusion Normalizing Flow](https://mlsanity.com/paper/2110.07579/) (2021)
- [Provable Benefit of Orthogonal Initialization in Optimizing Deep Linear Networks](https://mlsanity.com/paper/2001.05992/) (2020)
- [Normalizing Flows on Riemannian Manifolds](https://mlsanity.com/paper/1611.02304/) (2016)
- [Moser Flow: Divergence-based Generative Modeling on Manifolds](https://mlsanity.com/paper/2108.08052/) (2021)
- [Improved off-policy training of diffusion samplers](https://mlsanity.com/paper/2402.05098/) (2024)
- [Accelerating Diffusion Models with Parallel Sampling: Inference at Sub-Linear Time Complexity](https://mlsanity.com/paper/2405.15986/) (2024)
- [Unifying Generative Models with GFlowNets and Beyond](https://mlsanity.com/paper/2209.02606/) (2022)
- [CFlowNets: Continuous Control with Generative Flow Networks](https://mlsanity.com/paper/2303.02430/) (2023)

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

- [Beyond the Proxy: Trajectory-Distilled Guidance for Offline GFlowNet Training](https://mlsanity.com/paper/2505.20110/) (2025)
- [Your GFlowNet Secretly Learns an Optimal Transport Plan](https://mlsanity.com/paper/2606.06272/) (2026)

## Abstract (arXiv)

Generative Flow Networks (GFNs) were initially introduced on directed acyclic graphs to sample from an unnormalized distribution density. Recent works have extended the theoretical framework for generative methods allowing more flexibility and enhancing application range. However, many challenges remain in training GFNs in continuous settings and for imitation learning (IL), including intractability of flow-matching loss, limited tests of non-acyclic training, and the need for a separate reward model in imitation learning. The present work proposes a family of generative flows called Ergodic Generative Flows (EGFs) which are used to address the aforementioned issues. First, we leverage ergodicity to build simple generative flows with finitely many globally defined transformations (diffeomorphisms) with universality guarantees and tractable flow-matching loss (FM loss). Second, we introduce a new loss involving cross-entropy coupled to weak flow-matching control, coined KL-weakFM loss. It is designed for IL training without a separate reward model. We evaluate IL-EGFs on toy 2D tasks and real-world datasets from NASA on the sphere, using the KL-weakFM loss. Additionally, we conduct toy 2D reinforcement learning experiments with a target reward, using the FM loss.