# Extended Flow Matching: a Method of Conditional Generation with Generalized Continuity Equation arXiv: https://arxiv.org/abs/2402.18839 ยท HTML: https://mlsanity.com/paper/2402.18839/ ## Key facts - Cited by 11 papers in the map (data through 14 Oct 2026). - Placed in Flow matching, time series and science, one of 14 directions in Diffusion models. - Builds on 32 of its 49 references that are in the map. - First posted on arXiv 29 Feb 2024. ## 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, 30 of 32) - [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020) - [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020) - [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015) - [Classifier-Free Diffusion Guidance](https://mlsanity.com/paper/2207.12598/) (2022) - [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022) - [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022) - [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013) - [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022) - [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023) - [Equivariant Diffusion for Molecule Generation in 3D](https://mlsanity.com/paper/2203.17003/) (2022) - [Junction Tree Variational Autoencoder for Molecular Graph Generation](https://mlsanity.com/paper/1802.04364/) (2018) - [Wasserstein Auto-Encoders](https://mlsanity.com/paper/1711.01558/) (2017) - [Multisample Flow Matching: Straightening Flows with Minibatch Couplings](https://mlsanity.com/paper/2304.14772/) (2023) - [Flow Matching in Latent Space](https://mlsanity.com/paper/2307.08698/) (2023) - [SE(3)-Stochastic Flow Matching for Protein Backbone Generation](https://mlsanity.com/paper/2310.02391/) (2023) - [Equivariant flow matching](https://mlsanity.com/paper/2306.15030/) (2023) - [Guided Flows for Generative Modeling and Decision Making](https://mlsanity.com/paper/2311.13443/) (2023) - [Stochastic interpolants with data-dependent couplings](https://mlsanity.com/paper/2310.03725/) (2023) - [Simulation-free Schr"odinger bridges via score and flow matching](https://mlsanity.com/paper/2307.03672/) (2023) - [Equivariant Energy-Guided SDE for Inverse Molecular Design](https://mlsanity.com/paper/2209.15408/) (2022) - [Functional Flow Matching](https://mlsanity.com/paper/2305.17209/) (2023) - [Efficient Video Prediction via Sparsely Conditioned Flow Matching](https://mlsanity.com/paper/2211.14575/) (2022) - [Improved motif-scaffolding with SE(3) flow matching](https://mlsanity.com/paper/2401.04082/) (2024) - [Linear Optimal Transport Embedding: Provable Wasserstein classification for certain rigid transformations and perturbations](https://mlsanity.com/paper/2008.09165/) (2020) - [Latent Space Editing in Transformer-Based Flow Matching](https://mlsanity.com/paper/2312.10825/) (2023) - [Adaptive Guidance: Training-free Acceleration of Conditional Diffusion Models](https://mlsanity.com/paper/2312.12487/) (2023) - [On the Complexity of Approximating Multimarginal Optimal Transport](https://mlsanity.com/paper/1910.00152/) (2019) - [MagDiff: Multi-Alignment Diffusion for High-Fidelity Video Generation and Editing](https://mlsanity.com/paper/2311.17338/) (2023) - [Generating Synthetic Datasets by Interpolating along Generalized Geodesics](https://mlsanity.com/paper/2306.06866/) (2023) - [Wasserstein Geodesic Generator for Conditional Distributions](https://mlsanity.com/paper/2308.10145/) (2023) ## Built on by (most cited first, 11 of 11) - [Meta Flow Matching: Integrating Vector Fields on the Wasserstein Manifold](https://mlsanity.com/paper/2408.14608/) (2024) - [Conditional Wasserstein Distances with Applications in Bayesian OT Flow Matching](https://mlsanity.com/paper/2403.18705/) (2024) - [Flow matching achieves almost minimax optimal convergence](https://mlsanity.com/paper/2405.20879/) (2024) - [Dynamic Conditional Optimal Transport through Simulation-Free Flows](https://mlsanity.com/paper/2404.04240/) (2024) - [DiverseFlow: Sample-Efficient Diverse Mode Coverage in Flows](https://mlsanity.com/paper/2504.07894/) (2025) - [Flow Matching Meets Biology and Life Science: A Survey](https://mlsanity.com/paper/2507.17731/) (2025) - [Designing a Conditional Prior Distribution for Flow-Based Generative Models](https://mlsanity.com/paper/2502.09611/) (2025) - [Theoretical Guarantees for High Order Trajectory Refinement in Generative Flows](https://mlsanity.com/paper/2503.09069/) (2025) - [Modeling Stochastic Conditional Dynamics from Sparse Observations via Kernel-Stabilized Flow Matching](https://mlsanity.com/paper/2411.08314/) (2024) - [Pairwise Optimal Transports for Training All-to-All Flow-Based Condition Transfer Model](https://mlsanity.com/paper/2504.03188/) (2025) - [Structured Coupling for Flow Matching](https://mlsanity.com/paper/2605.07676/) (2026) ## Abstract (arXiv) The task of conditional generation is one of the most important applications of generative models, and numerous methods have been developed to date based on the celebrated flow-based models. However, many flow-based models in use today are not built to allow one to introduce an explicit inductive bias to how the conditional distribution to be generated changes with respect to conditions. This can result in unexpected behavior in the task of style transfer, for example. In this research, we introduce extended flow matching (EFM), a direct extension of flow matching that learns a "matrix field" corresponding to the continuous map from the space of conditions to the space of distributions. We show that we can introduce inductive bias to the conditional generation through the matrix field and demonstrate this fact with MMOT-EFM, a version of EFM that aims to minimize the Dirichlet energy or the sensitivity of the distribution with respect to conditions. We will present our theory along with experimental results that support the competitiveness of EFM in conditional generation.