mlsanityData to 14 Oct 2026

A Computational Framework for Solving Wasserstein Lagrangian Flows

arXiv 2310.10649 ↗First posted 16 Oct 2023cs.LG math.OC stat.MLcited by 46

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Abstract (arXiv)

The dynamical formulation of the optimal transport can be extended through various choices of the underlying geometry (kinetic energy), and the regularization of density paths (potential energy). These combinations yield different variational problems (Lagrangians), encompassing many variations of the optimal transport problem such as the Schr"odinger bridge, unbalanced optimal transport, and optimal transport with physical constraints, among others. In general, the optimal density path is unknown, and solving these variational problems can be computationally challenging. We propose a novel deep learning based framework approaching all of these problems from a unified perspective. Leveraging the dual formulation of the Lagrangians, our method does not require simulating or backpropagating through the trajectories of the learned dynamics, and does not need access to optimal couplings. We showcase the versatility of the proposed framework by outperforming previous approaches for the single-cell trajectory inference, where incorporating prior knowledge into the dynamics is crucial for correct predictions.

Key facts. Cited by 46 papers in the map (data through 14 Oct 2026). Builds on 44 of its 86 references that are in the map. First posted on arXiv 16 Oct 2023.

Builds on

44 of 86 references.

References · 44

  1. 2015U-Net: Convolutional Networks for Biomedical Image Segmentationcited by 19,713
  2. 2020Score-Based Generative Modeling through Stochastic Differential Equationscited by 9,122
  3. 2022Flow Matching for Generative Modelingcited by 5,702
  4. 2022Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flowcited by 3,700
  5. 2022Building Normalizing Flows with Stochastic Interpolantscited by 995
  6. 2023Stochastic Interpolants: A Unifying Framework for Flows and Diffusionscited by 939
  7. 2023Improving and generalizing flow-based generative models with minibatch optimal transportcited by 879
  8. 2021Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modelingcited by 602
  9. 2016Input Convex Neural Networkscited by 414
  10. 2013A survey of the Schr"odinger problem and some of its connections with optimal transportcited by 292
  11. 2022Rectified Flow: A Marginal Preserving Approach to Optimal Transportcited by 283
  12. 2023Multisample Flow Matching: Straightening Flows with Minibatch Couplingscited by 278
32 more
  1. 2020Scalable Gradients for Stochastic Differential Equationscited by 271
  2. 2021Likelihood Training of Schr"odinger Bridge using Forward-Backward SDEs Theorycited by 219
  3. 2019Optimal transport mapping via input convex neural networkscited by 206
  4. 2019Ab-Initio Solution of the Many-Electron Schr"odinger Equation with Deep Neural Networkscited by 200
  5. 2020TrajectoryNet: A Dynamic Optimal Transport Network for Modeling Cellular Dynamicscited by 160
  6. 2021Solving Schr"odinger Bridges via Maximum Likelihoodcited by 141
  7. 2015Optimal Entropy-Transport problems and a new Hellinger-Kantorovich distance between positive measurescited by 139
  8. 2014On the relation between optimal transport and Schr"odinger bridges: A stochastic control viewpointcited by 99
  9. 2022Action Matching: Learning Stochastic Dynamics from Samplescited by 98
  10. 2021Do Neural Optimal Transport Solvers Work? A Continuous Wasserstein-2 Benchmarkcited by 96
  11. 2021Proximal Optimal Transport Modeling of Population Dynamicscited by 93
  12. 2023Simulation-free Schr"odinger bridges via score and flow matchingcited by 83
  13. 2023Diffusion Bridge Mixture Transports, Schr"odinger Bridge Problems and Generative Modelingcited by 80
  14. 2021Large-Scale Wasserstein Gradient Flowscited by 77
  15. 2022Supervised Training of Conditional Monge Mapscited by 61
  16. 2021Optimizing Functionals on the Space of Probabilities with Input Convex Neural Networkscited by 53
  17. 2021Variational Wasserstein gradient flowcited by 53
  18. 2019Accelerating Langevin Sampling with Birth-deathcited by 45
  19. 2023Deep Momentum Multi-Marginal Schr"odinger Bridgecited by 42
  20. 2023Generalized Schr"odinger Bridge Matchingcited by 40
  21. 2019Accelerated Information Gradient flowcited by 39
  22. 2023The Monge Gap: A Regularizer to Learn All Transport Mapscited by 32
  23. 2021Towards a mathematical theory of trajectory inferencecited by 31
  24. 2020Fast and Smooth Interpolation on Wasserstein Spacecited by 22
  25. 2022Riemannian Metric Learning via Optimal Transportcited by 21
  26. 2018Second order models for optimal transport and cubic splines on the Wasserstein spacecited by 20
  27. 2018Measure-valued spline curves: an optimal transport viewpointcited by 17
  28. 2019Accelerated Flow for Probability Distributionscited by 15
  29. 2021Learning High Dimensional Wasserstein Geodesicscited by 10
  30. 2023Unbalanced Diffusion Schr"odinger Bridgecited by 9
  31. 2019Hopf-Cole transformation via generalized Schr"odinger bridge problemcited by 8
  32. 2022Wasserstein-Fisher-Rao Splinescited by 6

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2024: 14; 2025: 17; 2026: 15.

46 papers.

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  1. 2024Metric Flow Matching for Smooth Interpolations on the Data Manifoldcited by 77
  2. 2024Schr"odinger Bridge Flow for Unpaired Data Translationcited by 47
  3. 2024Meta Flow Matching: Integrating Vector Fields on the Wasserstein Manifoldcited by 43
  4. 2024Learning stochastic dynamics from snapshots through regularized unbalanced optimal transportcited by 33
  5. 2024Doob's Lagrangian: A Sample-Efficient Variational Approach to Transition Path Samplingcited by 27
  6. 2024Dynamic Conditional Optimal Transport through Simulation-Free Flowscited by 25
  7. 2025Joint Velocity-Growth Flow Matching for Single-Cell Dynamics Modelingcited by 20
  8. 2025Curly Flow Matching for Learning Non-gradient Field Dynamicscited by 20
  9. 2024Geometry-Aware Generative Autoencoders for Warped Riemannian Metric Learning and Generative Modeling on Data Manifoldscited by 13
  10. 2025Modeling Cell Dynamics and Interactions with Unbalanced Mean Field Schr"odinger Bridgecited by 12
  11. 2025Variational Regularized Unbalanced Optimal Transport: Single Network, Least Actioncited by 11
  12. 2024Efficient Trajectory Inference in Wasserstein Space Using Consecutive Averagingcited by 10
34 more
  1. 2025Integrating Dynamical Systems Modeling with Spatiotemporal scRNA-seq Data Analysiscited by 9
  2. 2025Learning of Population Dynamics: Inverse Optimization Meets JKO Schemecited by 9
  3. 2024Score-based Neural Ordinary Differential Equations for Computing Mean Field Control Problemscited by 7
  4. 2024Improving Neural Optimal Transport via Displacement Interpolationcited by 7
  5. 2026WFR-FM: Simulation-Free Dynamic Unbalanced Optimal Transportcited by 6
  6. 2024Scalable Simulation-free Entropic Unbalanced Optimal Transportcited by 5
  7. 2025Branched Schr"odinger Bridge Matchingcited by 5
  8. 2026A Call to Lagrangian Action: Learning Population Mechanics from Temporal Snapshotscited by 5
  9. 2024Conditional Lagrangian Wasserstein Flow for Time Series Imputationcited by 4
  10. 2024Generalized Flow Matching for Transition Dynamics Modelingcited by 4
  11. 2024Symplectic Neural Flows for Modeling and Discoverycited by 4
  12. 2025Multi-Marginal Flow Matching with Adversarially Learnt Interpolantscited by 4
  13. 2026Beyond Continuity: Simulation-free Reconstruction of Discrete Branching Dynamics from Single-cell Snapshotscited by 2
  14. 2026Lagrangian Flow Matching: A Least-Action Framework for Principled Path Designcited by 2
  15. 2025Minimum-Excess-Work Guidancecited by 1
  16. 2025Flows and Diffusions on the Neural Manifoldcited by 1
  17. 2025HOTA: Hamiltonian framework for Optimal Transport Advectioncited by 1
  18. 2025Learning Explicit Single-Cell Dynamics Using ODE Representationscited by 1
  19. 2026WFR-MFM: One-Step Inference for Dynamic Unbalanced Optimal Transportcited by 1
  20. 2026Wasserstein Residuals: Learning Gradient Flows from Population Dynamicscited by 1
  21. 2026Twisted Schr"odinger Bridge Matchingcited by 1
  22. 2026Trajectory inference via Acceleration Matchingcited by 1
  23. 2026TracingFlow: A Simulation-Free Trajectory Inference Framework Based on Second-Order Dynamicscited by 1
  24. 2025Follow the MEP: Scalable Neural Representations for Minimum-Energy Path Discovery in Molecular Systemscited by 0
  25. 2025Machine-Learned Sampling of Conditioned Path Measurescited by 0
  26. 2025Energy Guided Geometric Flow Matchingcited by 0
  27. 2025Multi-marginal temporal Schr"odinger Bridge Matching from unpaired datacited by 0
  28. 2025Multi-Marginal Schr"odinger Bridge Matchingcited by 0
  29. 2026Generative Modeling with Flux Matchingcited by 0
  30. 2026Multimarginal flow matching with optimal transport potentialscited by 0
  31. 2026A Variational Optimal Transport Operator on Incompressible Flowcited by 0
  32. 2026Simulation-Free Learning of Population Dynamics with Wasserstein Lagrangian Residualscited by 0
  33. 2026Discrete Action Matching: Learning Stochastic Dynamics from Samples via State Graphscited by 0
  34. 2026Neural Wasserstein Splines: Representation-Invariant Second-Order Priors for Trajectory Inferencecited by 0