# Optimal and Diffusion Transports in Machine Learning

arXiv: https://arxiv.org/abs/2512.06797 · HTML: https://mlsanity.com/paper/2512.06797/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 50 of its 95 references that are in the map.
- First posted on arXiv 7 Dec 2025.

## Builds on (most cited first, 30 of 50)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Scaling Laws for Neural Language Models](https://mlsanity.com/paper/2001.08361/) (2020)
- [Hierarchical Text-Conditional Image Generation with CLIP Latents](https://mlsanity.com/paper/2204.06125/) (2022)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Neural Machine Translation of Rare Words with Subword Units](https://mlsanity.com/paper/1508.07909/) (2015)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Video Diffusion Models](https://mlsanity.com/paper/2204.03458/) (2022)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Optimization Methods for Large-Scale Machine Learning](https://mlsanity.com/paper/1606.04838/) (2016)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [An Explanation of In-context Learning as Implicit Bayesian Inference](https://mlsanity.com/paper/2111.02080/) (2021)
- [Exact solutions to the nonlinear dynamics of learning in deep linear neural networks](https://mlsanity.com/paper/1312.6120/) (2013)
- [Wide Neural Networks of Any Depth Evolve as Linear Models Under Gradient Descent](https://mlsanity.com/paper/1902.06720/) (2019)
- [What Can Transformers Learn In-Context? A Case Study of Simple Function Classes](https://mlsanity.com/paper/2208.01066/) (2022)
- [On Lazy Training in Differentiable Programming](https://mlsanity.com/paper/1812.07956/) (2018)
- [On Exact Computation with an Infinitely Wide Neural Net](https://mlsanity.com/paper/1904.11955/) (2019)
- [Linear Convergence of Gradient and Proximal-Gradient Methods Under the Polyak-Lojasiewicz Condition](https://mlsanity.com/paper/1608.04636/) (2016)
- [A Mean Field View of the Landscape of Two-Layers Neural Networks](https://mlsanity.com/paper/1804.06561/) (2018)
- [On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transport](https://mlsanity.com/paper/1805.09545/) (2018)
- [A Survey of Transformers](https://mlsanity.com/paper/2106.04554/) (2021)
- [Stable Architectures for Deep Neural Networks](https://mlsanity.com/paper/1705.03341/) (2017)
- [Implicit Regularization in Matrix Factorization](https://mlsanity.com/paper/1705.09280/) (2017)

## Built on by (most cited first, 6 of 6)

- [On The Hidden Biases of Flow Matching Samplers](https://mlsanity.com/paper/2512.16768/) (2025)
- [The physics of AI weather models](https://mlsanity.com/paper/2605.23778/) (2026)
- [Reachability and asymptotics of Gaussian Transformer dynamics](https://mlsanity.com/paper/2606.07600/) (2026)
- [Generative Modeling by Value-Driven Transport](https://mlsanity.com/paper/2605.22507/) (2026)
- [Diffusion enabled Optimal Transport distances for graph matching](https://mlsanity.com/paper/2607.06646/) (2026)
- [Euclidean Distance-Based Flow Matching](https://mlsanity.com/paper/2610.18534/) (2026)

## Abstract (arXiv)

Several problems in machine learning are naturally expressed as the design and analysis of time-evolving probability distributions. This includes sampling via diffusion methods, optimizing the weights of neural networks, and analyzing the evolution of token distributions across layers of large language models. While the targeted applications differ (samples, weights, tokens), their mathematical descriptions share a common structure. A key idea is to switch from the Eulerian representation of densities to their Lagrangian counterpart through vector fields that advect particles. This dual view introduces challenges, notably the non-uniqueness of Lagrangian vector fields, but also opportunities to craft density evolutions and flows with favorable properties in terms of regularity, stability, and computational tractability. This survey presents an overview of these methods, with emphasis on two complementary approaches: diffusion methods, which rely on stochastic interpolation processes and underpin modern generative AI, and optimal transport, which defines interpolation by minimizing displacement cost. We illustrate how both approaches appear in applications ranging from sampling, neural network optimization, to modeling the dynamics of transformers for large language models.