# Spatial Transport of Integration Error in Generative ODEs

arXiv: https://arxiv.org/abs/2607.16361 · HTML: https://mlsanity.com/paper/2607.16361/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 18 of its 21 references that are in the map.
- First posted on arXiv 17 Jul 2026.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Builds on (most cited first, 18 of 18)

- [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)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Scalable Diffusion Models with Transformers](https://mlsanity.com/paper/2212.09748/) (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)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [Consistency Models](https://mlsanity.com/paper/2303.01469/) (2023)
- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [SiT: Exploring Flow and Diffusion-based Generative Models with Scalable Interpolant Transformers](https://mlsanity.com/paper/2401.08740/) (2024)
- [Minimizing Trajectory Curvature of ODE-based Generative Models](https://mlsanity.com/paper/2301.12003/) (2023)
- [Region-Adaptive Sampling for Diffusion Transformers](https://mlsanity.com/paper/2502.10389/) (2025)
- [Training-free Mixed-Resolution Latent Upsampling for Spatially Accelerated Diffusion Transformers](https://mlsanity.com/paper/2507.08422/) (2025)
- [CurveFlow: Curvature-Guided Flow Matching for Image Generation](https://mlsanity.com/paper/2508.15093/) (2025)
- [Foveated Diffusion: Efficient Spatially Adaptive Image and Video Generation](https://mlsanity.com/paper/2603.23491/) (2026)
- [Isokinetic Flow Matching for Pathwise Straightening of Generative Flows](https://mlsanity.com/paper/2604.04491/) (2026)
- [SDiT: Semantic Region-Adaptive for Diffusion Transformers](https://mlsanity.com/paper/2601.12283/) (2026)

## Built on by (most cited first, 1 of 1)

- [Cross-Time Directional Selection in Diffusion Sampling](https://mlsanity.com/paper/2610.05199/) (2026)

## Abstract (arXiv)

A trained flow or diffusion model is usually run with only a handful of solver steps, and the integration error this leaves behind is unevenly distributed across the image. We ask where that error is injected and how it reaches the endpoint, and answer with a signed source-and-transport accounting of few-step integration error, tested to first order. A perturbation experiment on five models at 256^2 resolution shows the learned dynamics spread local disturbances widely: near the start of sampling, under 10% of the summed endpoint response remains at the source. Signed one-step truncation residuals, propagated through the model's own linearized dynamics, reconstruct much of the endpoint error's direction and regional structure (cosine 0.81-0.87), and a region's error owes more to what arrives from elsewhere than to its own injection. Structure-destroying nulls, with protocols frozen before evaluation, locate what carries the account: randomizing contribution signs halves it, and reassigning which region receives each contribution, with content, norms, and signs intact, destroys it entirely. Where the injections land is readable from the model itself. The variation of its velocity or prediction field along the trajectory, a structure that emerges during training, predicts the final per-region gap (within-image rho of 0.57-0.70 on fine trajectories, weaker from the cheap solve alone). The prediction is partial because endpoint error depends not only on injected magnitude but on its sign, timing, and transport through the learned dynamics. A training penalty on the injected variation lowers few-step error, so the structure is one a model can be trained to change.