# Probing the Geometry of Diffusion Models with the String Method

arXiv: https://arxiv.org/abs/2602.22122 · HTML: https://mlsanity.com/paper/2602.22122/

## 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 12 of its 32 references that are in the map.
- First posted on arXiv 25 Feb 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, 12 of 12)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [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)
- [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)
- [AlphaFold Meets Flow Matching for Generating Protein Ensembles](https://mlsanity.com/paper/2402.04845/) (2024)
- [DiffMorpher: Unleashing the Capability of Diffusion Models for Image Morphing](https://mlsanity.com/paper/2312.07409/) (2023)
- [Protein Conformation Generation via Force-Guided SE(3) Diffusion Models](https://mlsanity.com/paper/2403.14088/) (2024)
- [Interpolating between Images with Diffusion Models](https://mlsanity.com/paper/2307.12560/) (2023)
- [Diffusion Models as Cartoonists: The Curious Case of High Density Regions](https://mlsanity.com/paper/2411.01293/) (2024)
- [Learning normalized image densities via dual score matching](https://mlsanity.com/paper/2506.05310/) (2025)

## Abstract (arXiv)

Understanding the geometry of learned distributions is fundamental to improving and interpreting diffusion models, yet systematic tools for exploring their landscape remain limited. Standard latent-space interpolations fail to respect the structure of the learned distribution, often traversing low-density regions. We introduce a framework based on the string method that computes continuous paths between samples by evolving curves under the learned score function. Operating on pretrained models without retraining, our approach interpolates between three regimes: pure generative transport, which yields continuous sample paths; gradient-dominated dynamics, which recover minimum energy paths (MEPs); and finite-temperature string dynamics, which compute principal curves -- self-consistent paths that balance energy and entropy. We demonstrate that the choice of regime matters in practice. For image diffusion models, MEPs contain high-likelihood but unrealistic ''cartoon'' images, confirming prior observations that likelihood maxima appear unrealistic; principal curves instead yield realistic morphing sequences despite lower likelihood. For protein structure prediction, our method computes transition pathways between metastable conformers directly from models trained on static structures, yielding paths with physically plausible intermediates. Together, these results establish the string method as a principled tool for probing the modal structure of diffusion models -- identifying modes, characterizing barriers, and mapping connectivity in complex learned distributions.