# Dynamic Conditional Optimal Transport through Simulation-Free Flows

arXiv: https://arxiv.org/abs/2404.04240 · HTML: https://mlsanity.com/paper/2404.04240/

## Key facts

- Cited by 25 papers in the map (data through 14 Oct 2026).
- Builds on 61 of its 92 references that are in the map.
- First posted on arXiv 5 Apr 2024.

## Builds on (most cited first, 30 of 61)

- [Learning Transferable Visual Models From Natural Language Supervision](https://mlsanity.com/paper/2103.00020/) (2021)
- [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)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Image-to-Image Translation with Conditional Adversarial Networks](https://mlsanity.com/paper/1611.07004/) (2016)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Conditional Generative Adversarial Nets](https://mlsanity.com/paper/1411.1784/) (2014)
- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [StarGAN v2: Diverse Image Synthesis for Multiple Domains](https://mlsanity.com/paper/1912.01865/) (2019)
- [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)
- [Unsupervised Image-to-Image Translation Networks](https://mlsanity.com/paper/1703.00848/) (2017)
- [Self-Normalizing Neural Networks](https://mlsanity.com/paper/1706.02515/) (2017)
- [The frontier of simulation-based inference](https://mlsanity.com/paper/1911.01429/) (2019)
- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [Multisample Flow Matching: Straightening Flows with Minibatch Couplings](https://mlsanity.com/paper/2304.14772/) (2023)
- [How to train your neural ODE: the world of Jacobian and kinetic regularization](https://mlsanity.com/paper/2002.02798/) (2020)
- [Sequential Neural Likelihood: Fast Likelihood-free Inference with Autoregressive Flows](https://mlsanity.com/paper/1805.07226/) (2018)
- [Optimal transport mapping via input convex neural networks](https://mlsanity.com/paper/1908.10962/) (2019)
- [EnhanceNet: Single Image Super-Resolution Through Automated Texture Synthesis](https://mlsanity.com/paper/1612.07919/) (2016)
- [TrajectoryNet: A Dynamic Optimal Transport Network for Modeling Cellular Dynamics](https://mlsanity.com/paper/2002.04461/) (2020)
- [Flow Matching in Latent Space](https://mlsanity.com/paper/2307.08698/) (2023)
- [OT-Flow: Fast and Accurate Continuous Normalizing Flows via Optimal Transport](https://mlsanity.com/paper/2006.00104/) (2020)
- [Neural Optimal Transport](https://mlsanity.com/paper/2201.12220/) (2022)
- [Image-to-Image Translation: Methods and Applications](https://mlsanity.com/paper/2101.08629/) (2021)

## Built on by (most cited first, 25 of 25)

- [Guided Diffusion Sampling on Function Spaces with Applications to PDEs](https://mlsanity.com/paper/2505.17004/) (2025)
- [Flow Matching: Markov Kernels, Stochastic Processes and Transport Plans](https://mlsanity.com/paper/2501.16839/) (2025)
- [Stochastic Process Learning via Operator Flow Matching](https://mlsanity.com/paper/2501.04126/) (2025)
- [Flow Matching with Semidiscrete Couplings](https://mlsanity.com/paper/2509.25519/) (2025)
- [Conditional simulation via entropic optimal transport: Toward non-parametric estimation of conditional Brenier maps](https://mlsanity.com/paper/2411.07154/) (2024)
- [Fast Flow-based Visuomotor Policies via Conditional Optimal Transport Couplings](https://mlsanity.com/paper/2505.01179/) (2025)
- [Composite Flow Matching for Reinforcement Learning with Shifted-Dynamics Data](https://mlsanity.com/paper/2505.23062/) (2025)
- [Source Separation by Flow Matching](https://mlsanity.com/paper/2505.16119/) (2025)
- [Weighted Conditional Flow Matching](https://mlsanity.com/paper/2507.22270/) (2025)
- [Conditional Flow Matching for Bayesian Posterior Inference](https://mlsanity.com/paper/2510.09534/) (2025)
- [Modeling Stochastic Conditional Dynamics from Sparse Observations via Kernel-Stabilized Flow Matching](https://mlsanity.com/paper/2411.08314/) (2024)
- [A Geometric Approach to Optimal Experimental Design](https://mlsanity.com/paper/2510.14848/) (2025)
- [GRASP: Guided Residual Adapters with Sample-wise Partitioning](https://mlsanity.com/paper/2512.01675/) (2025)
- [Pairwise Optimal Transports for Training All-to-All Flow-Based Condition Transfer Model](https://mlsanity.com/paper/2504.03188/) (2025)
- [Bridging Simulators with Conditional Optimal Transport](https://mlsanity.com/paper/2510.24631/) (2025)
- [Conditional Normalizing Flows for Forward and Backward Joint State and Parameter Estimation](https://mlsanity.com/paper/2601.07013/) (2026)
- [Conditional Unbalanced Optimal Transport Maps: An Outlier-Robust Framework for Conditional Generative Modeling](https://mlsanity.com/paper/2603.06972/) (2026)
- [Prior-Fitted Functional Flow: In-Context Generative Models for Pharmacokinetics](https://mlsanity.com/paper/2604.17670/) (2026)
- [Intrinsic Flow Matching on Quantum Pure-State Manifolds with Phase-Aligned Transport](https://mlsanity.com/paper/2606.21256/) (2026)
- [Optimal Transport Q-Learning for Flow Policy Steering and Acceleration](https://mlsanity.com/paper/2607.06262/) (2026)
- [Restoration Flow Matching-Based Channel Refinement and Equalization Correction for MIMO Semantic Communications](https://mlsanity.com/paper/2607.23615/) (2026)
- [Approximating Measures on Function Spaces: Transport and Truncation](https://mlsanity.com/paper/2609.17802/) (2026)
- [One-Step Generative Surrogate Models via Block-Triangular Joint Drifting](https://mlsanity.com/paper/2609.26435/) (2026)
- [DRIFT: Disentangled Responsive-Invariant Flow Transport for Single-Cell Perturbation Prediction](https://mlsanity.com/paper/2609.35106/) (2026)
- [Deterministic Prediction as Global Velocity: Flow Matching for Probabilistic Time-Series Forecast](https://mlsanity.com/paper/2610.18621/) (2026)

## Abstract (arXiv)

We study the geometry of conditional optimal transport (COT) and prove a dynamical formulation which generalizes the Benamou-Brenier Theorem. Equipped with these tools, we propose a simulation-free flow-based method for conditional generative modeling. Our method couples an arbitrary source distribution to a specified target distribution through a triangular COT plan, and a conditional generative model is obtained by approximating the geodesic path of measures induced by this COT plan. Our theory and methods are applicable in infinite-dimensional settings, making them well suited for a wide class of Bayesian inverse problems. Empirically, we demonstrate that our method is competitive on several challenging conditional generation tasks, including an infinite-dimensional inverse problem.