# Co-Evolving Paths and Flows via Path-Flow Alignment

arXiv: https://arxiv.org/abs/2610.08717 · HTML: https://mlsanity.com/paper/2610.08717/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 19 of its 20 references that are in the map.
- First posted on arXiv 6 Oct 2026.

## 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, 19 of 19)

- [Classifier-Free Diffusion Guidance](https://mlsanity.com/paper/2207.12598/) (2022)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Improved Techniques for Training GANs](https://mlsanity.com/paper/1606.03498/) (2016)
- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [SiT: Exploring Flow and Diffusion-based Generative Models with Scalable Interpolant Transformers](https://mlsanity.com/paper/2401.08740/) (2024)
- [Multisample Flow Matching: Straightening Flows with Minibatch Couplings](https://mlsanity.com/paper/2304.14772/) (2023)
- [GANs Trained by a Two Time-Scale Update Rule Converge to a Local Nash Equilibrium](https://mlsanity.com/paper/1706.08500/) (2017)
- [Metric Flow Matching for Smooth Interpolations on the Data Manifold](https://mlsanity.com/paper/2405.14780/) (2024)
- [Neural Flow Diffusion Models: Learnable Forward Process for Improved Diffusion Modelling](https://mlsanity.com/paper/2404.12940/) (2024)
- [Diffusion Models without Classifier-free Guidance](https://mlsanity.com/paper/2502.12154/) (2025)
- [Curly Flow Matching for Learning Non-gradient Field Dynamics](https://mlsanity.com/paper/2510.26645/) (2025)
- [FlowDCN: Exploring DCN-like Architectures for Fast Image Generation with Arbitrary Resolution](https://mlsanity.com/paper/2410.22655/) (2024)
- [Beyond Optimal Transport: Model-Aligned Coupling for Flow Matching](https://mlsanity.com/paper/2505.23346/) (2025)
- [Weighted Conditional Flow Matching](https://mlsanity.com/paper/2507.22270/) (2025)
- [Multi-Marginal Flow Matching with Adversarially Learnt Interpolants](https://mlsanity.com/paper/2510.01159/) (2025)
- [Learning Straight Flows by Learning Curved Interpolants](https://mlsanity.com/paper/2503.20719/) (2025)

## Abstract (arXiv)

We study path-flow alignment as a unified training objective for flow matching. Instead of fixing the interpolation path and learning only the velocity field, we jointly train an endpoint-preserving path network and a flow network using the same alignment loss: the flow learns to match the path velocity, and the path learns to align its velocity to the current flow. Although every fixed learned path defines a valid flow-matching objective, the alignment loss alone is not a reliable criterion for path learning. We identify path overfitting, a failure mode in which the alignment loss decreases while sample quality worsens. We find that this failure is associated with low-entropy bottlenecks in the induced probability path, where the learned path routes samples through overly concentrated intermediate marginals. Motivated by this diagnosis, we introduce a stochastic path regularizer that hides part of the source information from the path network while preserving exact endpoints. The resulting regularization gives an explicit entropy floor for the stochastic training-path marginals and empirically suppresses the bottleneck in the learned sampler, making joint path-flow training effective. On ImageNet-256x256 with SiT backbones, our method consistently improves FID across model scales, extends to model-guidance training, and leaves the inference-time architecture and sampler unchanged. Code is available at https://github.com/lizeyu090312/traj_opt_paper