A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem

2000/01/01 by Jean-David Benamou, Jean‐David Benamou, Yann Brenier · 276 citations
Mathematics · #Geometry and complex manifolds #Geometric Analysis and Curvature Flows #Navier-Stokes equation solutions

paper · doi:10.1007/s002110050002

Citations

• The geometry of optimal transportation
• Polar factorization and monotone rearrangement of vector‐valued functions
• Boundary Regularity of Maps with Convex Potentials--II

Cited by

• Stochastic Control Liaisons: Richard Sinkhorn Meets Gaspard Monge on a Schrödinger Bridge
• Neural ODE Control for Classification, Approximation, and Transport
• Mixed Finite Element Methods for the Fully Nonlinear Monge–Ampère Equation Based on the Vanishing Moment Method
• Local-Global Geometric Insights for Graph Neural Networks via Entropic Curvature
• A convergence result of a continuous model of deep learning via a Łojasiewicz--Simon inequality
• Arnold--Nielsen Geometry for Complexity-Deformed Noncommutative Transport
• Reflected Schrodinger Bridge Problem over Sub-Riemannian Manifold
• A Dynamical Approximation Scheme on the Stiefel manifold for Wasserstein Gradient Flows
• An Efficient Augmented Lagrangian Framework for Dynamic Optimal Transport on Surfaces Based on Second-Order Cone Programming Reformulation
• Twisted Schrödinger Bridge Matching
• A Unified Variational Framework for Optimal Transport with Lagrangian Costs
• On Conformal Flexibility of Completeness and Minimizing Geodesics on Hilbert Manifolds
• Conditioning Gaussian Processes on Almost Anything
• Muon Dynamics as a Spectral Wasserstein Flow
• Quantum Optimal Transport for Tensor Field Processing
• Regularity of the Schrödinger cost
• The data-driven Schroedinger bridge
• Geometry of diffeomorphism groups, complete integrability and optimal transport
• Large-Scale Optimal Transport and Mapping Estimation
• Optimal Transportation of Vector-Valued Measures
• W_1,+-interpolation of probability measures on graphs
• Geometric decomposition of information flow for overdamped Langevin systems and optimal transport in subsystems
• Nonlinear diffusion equations with degenerate fast-decay mobility by coordinate transformation
• Uniqueness of the solution to the Vlasov-Poisson system with bounded density
• Generative Modeling by Value-Driven Transport
• Stochastic control, entropic interpolation and gradient flows on Wasserstein product spaces
• Multilevel Optimal Transport: a Fast Approximation of Wasserstein-1 distances
• Geodesics for a class of distances in the space of probability measures
• A Dacorogna-Moser construction of transport maps on ℝ^d with application to geodesics on the space of couplings
• Fused Gromov-Wasserstein distance for structured objects: theoretical foundations and mathematical properties
• Latent Confounded Causal Discovery via Lie Bracket Geometry
• Marginal flows of non-entropic weak Schrödinger bridges
• Diffusion Approximations to Schrödinger Bridges on Manifolds
• A Benamou-Brenier formulation for the multi-marginal optimal transport problem with infimal convolution cost
• The tangent space to the Wasserstein space: parallel transport and other applications
• A Benamou-Brenier Proximal Splitting Method for Constrained Unbalanced Optimal Transport
• Optimal and Diffusion Transports in Machine Learning
• Deep Neural Network-Based Aerial Transport in the Presence of Cooperative and Uncooperative UAS
• Entropy and Fisher information in non-convex domains: one chain to rule them all
• Wasserstein Evolution : Evolutionary Optimization as Phase Transition
• The swap transpose on couplings translates to Petz' recovery map on quantum channels
• Diffusion-Augmented Markov Decision Processes for Maximum Entropy Reinforcement Learning
• High-dimensional Mean-Field Games by Particle-based Flow Matching
• Wasserstein Style Transfer
• Quantitative Stability and Error Estimates for Optimal Transport Plans
• Optimal transport for vector Gaussian mixture models
• TrajectoryNet: A Dynamic Optimal Transport Network for Modeling Cellular\n Dynamics
• Dynamic characterization of barycentric optimal transport problems and their martingale relaxation
• EnfoPath: Energy-Informed Analysis of Generative Trajectories in Flow Matching
• Deterministic Mean Field Games on Networks and Related Optimal Control Problems
• VeCoR -- Velocity Contrastive Regularization for Flow Matching
• Domain Decomposition Parabolic Monge-Ampère Approach for Fast Generation of Adaptive Moving Meshes
• Covariance-Modulated Optimal Transport and Gradient Flows
• A PDE-constrained Optimization Approach to Optimal Trajectory Planning under Uncertainty via Reflected Schrödinger Bridges
• Equivariant Flows: Exact Likelihood Generative Learning for Symmetric\n Densities
• Intrinsic Wasserstein Correlation Analysis
• Optimization of continuous-flow over traffic networks with fundamental diagram constraints
• CellStream: Dynamical Optimal Transport Informed Embeddings for Reconstructing Cellular Trajectories from Snapshots Data
• FMMI: Flow Matching Mutual Information Estimation
• Controlling conservation laws I: entropy-entropy flux
• Contact Wasserstein Geodesics for Non-Conservative Schrödinger Bridges
• Thermodynamic interpretation of Wasserstein distance
• A kernel method for the learning of Wasserstein geometric flows
• Knothe-Rosenblatt maps via soft-constrained optimal transport
• Numerical methods for matching for teams and Wasserstein barycenters
• Convergent finite difference solvers for viscosity solutions of the\n elliptic Monge-Amp `ere equation in dimensions two and higher
• Optimal transport with a density-dependent cost function
• Discretization and convergence of the ballistic Benamou-Brenier formulation of the porous medium and Burgers equations
• Generalized Finite-time Optimal Control Framework in Stochastic Thermodynamics
• Optimization of continuous-flow over traffic networks with fundamental diagram constraints
• A Primal-dual Forward-backward Splitting Method for Cross-diffusion Gradient Flows with General Mobility Matrices
• Bi-martingale optimal transport and its applications
• Gradient Flows as Optimal Controlled Evolutions: From Rn to Wasserstein product spaces
• Relaxed Schroedinger bridges and robust network routing
• A primal-dual approach for a total variation Wasserstein flow
• Euclidean, Metric, and Wasserstein Gradient Flows: an overview
• Maximum Mean Discrepancy Gradient Flow
• Scalable Dynamic Optimal Transport via Distributed Linearized ADMM
• Variational time-fractional Mean Field Games
• Equilibration of aggregation-diffusion equations with weak interaction forces
• Optimal Mass Transport: Signal processing and machine-learning applications
• Well-posedness of a parabolic moving-boundary problem in the setting of\n Wasserstein gradient flows
• Regularity theory and geometry of unbalanced optimal transport
• Matricial Wasserstein-1 Distance
• Optimal transport mapping via input convex neural networks
• On the gradient flow structure of the isotropic Landau equation
• Primal dual methods for Wasserstein gradient flows
• A fitness-driven cross-diffusion system from polulation dynamics as a\n gradient flow
• Natural gradient via optimal transport
• Approximation of Optimal Transport problems with marginal moments\n constraints
• Computing the time-continuous Optimal Mass Transport Problem without Lagrangian techniques
• Continuous-Flow Graph Transportation Distances
• A convergent Lagrangian discretization for p-Wasserstein and flux-limited diffusion equations
• Maximal power output of a stochastic thermodynamic engine
• On the relation between optimal transport and Schrödinger bridges: A stochastic control viewpoint
• Analysis and Application of Optimal Transport For Challenging Seismic Inverse Problems
• Quantum Wasserstein isometries of the n-qubit state space: a Wigner-type result
• A monotonic method for solving nonlinear optimal control problems
• Ricci curvature for parametric statistics via optimal transport
• On properties of the Generalized Wasserstein distance
• Cellular flow control design for mixing based on the least action principle
• Stochastic control liaisons: Richard Sinkhorn meets Gaspard Monge on a Schroedinger bridge
• Reconstruction of the early Universe as a convex optimization problem
• Variational Modelling: Energies, gradient flows, and large deviations
• Computing Optimal Trajectories for Optimal Transport in Nonuniform Environments
• Finding geodesics with the Deep Ritz method
• Some notes on the Hellinger distance and various Fisher-Rao distances
• Martingale Optimal Transport and Martingale Schrödinger Bridges for Calibration of Stochastic Volatility Models
• Universal vector and matrix optimal transport
• Counterfactual Identifiability via Dynamic Optimal Transport
• A generalized alternating NGMRES method for PDE-constrained optimization problems governed by transport equations
• Frictional martingale optimal transport and robust hedging
• Field-Theoretic Simulation of Dean-Kawasaki Dynamics for Interacting Particles
• Non-conservative optimal transport
• Learning Explicit Single-Cell Dynamics Using ODE Representations
• Neural Hamilton--Jacobi Characteristic Flows for Optimal Transport
• Flow Matching with Semidiscrete Couplings
• OAT-FM: Optimal Acceleration Transport for Improved Flow Matching
• Stochastic Interpolants via Conditional Dependent Coupling
• Thermodynamic geometry of friction on graphs: Resistance, commute times, and optimal transport
• Transport Based Mean Flows for Generative Modeling
• An energy stable and positivity-preserving scheme for the Maxwell-Stefan diffusion system
• A dynamical formulation of multi-marginal optimal transport
• Federated Flow Matching
• Rates of Estimation of Optimal Transport Maps using Plug-in Estimators via Barycentric Projections
• Fast Unbalanced Optimal Transport on a Tree
• GlymphVIS: Visualizing Glymphatic Transport Pathways Using Regularized\n Optimal Transport
• Optimal mass transport and kernel density estimation for state-dependent networked dynamic systems
• Connecting GANs, MFGs, and OT
• A Transport for imaging process
• The Vanishing Moment Method for Fully Nonlinear Second Order Partial Differential Equations: Formulation, Theory, and Numerical Analysis
• Geometric methods for estimation of structured covariances
• A viscosity solution approach to the Monge-Ampere formulation of the Optimal Transportation Problem
• Numerical solution of the Optimal Transportation problem using the\n Monge-Ampere equation
• Discrete geodesic calculus in the space of viscous fluidic objects
• Perspective Functions: Properties, Constructions, and Examples
• Uniform Sampling from the Reachable Set Using Optimal Transport
• Computation and implementation of an optimal mean field control for smart charging
• Variable-preconditioned transformed primal-dual method for generalized Wasserstein Gradient Flows
• Risk-Sensitive Mean-Field-Type Games with Lp-norm Drifts
• Uniqueness and non-uniqueness of steady states of aggregation-diffusion equations
• W-entropy formulas on super Ricci flows and Langevin deformation on Wasserstein space over Riemannian manifolds
• W-entropy formulas and Langevin deformation of flows on Wasserstein space over Riemannian manifolds
• Scalable Computation of Dynamic Flow Problems via Multimarginal Graph-Structured Optimal Transport
• Optimizing optimal transport: Role of final distributions in finite-time thermodynamics
• Enhancing 3D Medical Image Understanding with Pretraining Aided by 2D Multimodal Large Language Models
• Strongly continuous fields of operators over varying Hilbert spaces
• On the geometry of Stein variational gradient descent
• Optimal transport for causal discovery
• On the quadratic barycentric transport problem
• Characterization of radially symmetric finite time blowup in\n multidimensional aggregation equations,
• Traversing the Schroedinger Bridge strait: Robert Fortet's marvelous\n proof redux
• State tracking of linear ensembles via optimal mass transport
• Structure preserving primal dual methods for gradient flows with nonlinear mobility transport distances
• Open problems in information geometry: a discussion at FDIG 2025
• Preconditioned Regularized Wasserstein Proximal Sampling
• Geodesic convexity of the relative entropy in reversible Markov chains
• Smooth p-Wasserstein Distance: Structure, Empirical Approximation, and Statistical Applications
• Optimizing Functionals on the Space of Probabilities with Input Convex\n Neural Networks
• A Shooting Formulation of Deep Learning
• Entropic estimation of optimal transport maps
• Finite Horizon Density Control for Static State Feedback Linearizable Systems
• On a class of modified Wasserstein distances induced by concave mobility functions defined on bounded intervals
• Computational methods for nonlocal mean field games with applications
• Computations of optimal transport distance with Fisher information regularization
• Geodesic convexity and strengthened functional inequalities in submanifolds of Wasserstein space
• On the mean field limit of the Random Batch Method for interacting particle systems
• A new approach to posterior contraction rates via Wasserstein dynamics
• Kernelized Wasserstein Natural Gradient
• Efficient preconditioners for solving dynamical optimal transport via interior point methods
• Multi-Marginal Stochastic Flow Matching for High-Dimensional Snapshot Data at Irregular Time Points
• Wasserstein Distances on Quantum Structures: an Overview
• Flow Matching for Probabilistic Learning of Dynamical Systems from Missing or Noisy Data
• WeightFlow: Learning Stochastic Dynamics via Evolving Weight of Neural Network
• A positivity-preserving and energy stable scheme for a quantum diffusion equation
• Scaling Algorithms for Unbalanced Transport Problems
• Discrete approximation of the minimizing movement scheme for evolution equations of Wasserstein gradient flow type with nonlinear mobility
• Benamou-Brenier and Kantorovich on sub-Riemannian manifolds with no abnormal geodesics
• Fundamental diagram constrained dynamic optimal transport via proximal splitting methods
• A Noncommutative Transport Metric and Symmetric Quantum Markov Semigroups as Gradient Flows of the Entropy
• A fully non-linear version of the Euler incompressible equations: the semi-geostrophic system
• Stochastic Optimal Control via Measure Relaxations
• Joint Inference of Trajectory and Obstacle in Mean-Field Games via Bilevel Optimization
• Large Deviations of Irreversible Processes
• Optimal Transportation in the presence of a prescribed pressure field
• Transport-based analysis, modeling, and learning from signal and data\n distributions
• Learning Implicit Generative Models with Theoretical Guarantees
• Variational time discretization of geodesic calculus
• HOTA: Hamiltonian framework for Optimal Transport Advection
• GradNetOT: Learning Optimal Transport Maps with GradNets
• Momentum Multi-Marginal Schrödinger Bridge Matching
• Convergence of drift-diffusion PDEs arising as Wasserstein gradient flows of convex functions
• Spline element method for the Monge-Ampere equation
• Enforcing Latent Euclidean Geometry in Single-Cell VAEs for Manifold Interpolation
• A data-driven linear-programming methodology for optimal transport
• From Kinetic Theory to AI: a Rediscovery of High-Dimensional Divergences and Their Properties
• The Camassa-Holm equation as an incompressible Euler equation: a\n geometric point of view
• Around the entropic Talagrand inequality
• Numerical analysis of strongly nonlinear PDEs
• Pseudo transient continuation and time marching methods for Monge-Ampere\n type equations
• Application of an unbalanced optimal transport distance and a mixed L1/Wasserstein distance to full waveform inversion
• Flow matching for reaction pathway generation
• A geometric perspective on regularized optimal transport
• An Analog of the 2-Wasserstein Metric in Non-commutative Probability\n under which the Fermionic Fokker-Planck Equation is Gradient Flow for the\n Entropy
• Discovery and visualization of structural biomarkers from MRI using\n transport-based morphometry
• FRAME Revisited: An Interpretation View Based on Particle Evolution
• Convergence of a fourth order singular perturbation of the n-dimensional radially symmetric Monge-Ampère equation
• DICE: Discrete inverse continuity equation for learning population dynamics
• q-Moment Estimates for the Singular p-Laplace Equation and Applications
• An efficient linear programming method for Optimal Transportation
• A Mixed Formulation of the Monge-Kantorovich Equations
• An Optimal Transport View On Schroedinger's Equation
• Parallel-in-Time Preconditioning for Time-Dependent Variational Mean Field Games
• QC-OT: Optimal Transport with Quasiconformal Mapping
• Minimally dissipative multi-bit logical operations
• Perspective: Time irreversibility in systems observed at coarse resolution
• Thermodynamic Bounds on Generalized Transport: From Single-Molecule to Bulk Observables
• Optimal transport natural gradient for statistical manifolds with continuous sample space
• Mean Field Game GAN
• Approximating the order 2 quantum Wasserstein distance using the moment-SOS hierarchy
• Neural Monge Map estimation and its applications
• Transporting a Dirac mass in a mean field planning problem
• Dataset Dynamics via Gradient Flows in Probability Space
• Mean Field Games with state constraints: from mild to pointwise\n solutions of the PDE system
• On the attainment of the Wasserstein--Cramer--Rao lower bound
• The LQR-Schrödinger Bridge
• On Fitting Flow Models with Large Sinkhorn Couplings
• p-Euler equations and p-Navier-Stokes equations
• Interpolation of Density Matrices and Matrix-Valued Measures: The Unbalanced Case
• Entropic turnpike estimates for the kinetic Schrödinger problem
• Transport-Based Pattern Theory: A Signal Transformation Approach
• Entropy dissipation semi-discretization schemes for Fokker-Planck equations
• Adversarial Optimal Transport Through The Convolution Of Kernels With Evolving Measures
• Geometric Visual Servo Via Optimal Transport
• Optimal mass transport over bridges
• Hopf-Cole transformation via generalized Schrödinger bridge problem
• Machine-Learned Sampling of Conditioned Path Measures
• Learning of Population Dynamics: Inverse Optimization Meets JKO Scheme
• W-entropy formulas and Langevin deformation on the L^q-Wasserstein space over Riemannian manifolds
• Learning High Dimensional Wasserstein Geodesics
• Weak Solutions for Potential Mean Field Games of Controls
• Inexact JKO and proximal-gradient algorithms in the Wasserstein space
• A positivity-preserving hybrid DDG method for Poisson--Nernst--Planck systems
• 3/4-discrete optimal transport
• On the Relation between Rectified Flows and Optimal Transport
• An optimal transport model for imaging in atmospheric turbulence
• Non-Asymptotic Mean-Field Games
• PDPO: Parametric Density Path Optimization
• Generative Distribution Embeddings: Lifting autoencoders to the space of distributions for multiscale representation learning
• A continuation multiple shooting method for Wasserstein geodesic equation
• On the Connection between Dynamical Optimal Transport and Functional\n Lifting
• Model-Based Localization and Tracking Using Bluetooth Low-Energy Beacons. [europepmc]
• Characterizing Cancer Drug Response and Biological Correlates: A Geometric Network Approach. [europepmc]
• Computing diffusivities from particle models out of equilibrium. [europepmc]
• Optimal Mass Transport with Lagrangian Workflow Reveals Advective and Diffusion Driven Solute Transport in the Glymphatic System. [europepmc]
• The glymphatic system and its role in cerebral homeostasis. [europepmc]
• Generalizations of Talagrand Inequality for Sinkhorn Distance Using Entropy Power Inequality. [europepmc]
• Some Recent Advances in Energetic Variational Approaches. [europepmc]
• Spatial and Texture Analysis of Root System distribution with Earth mover's Distance (STARSEED). [europepmc]
• Homogenisation of dynamical optimal transport on periodic graphs. [europepmc]
• Continuous positive airway pressure increases CSF flow and glymphatic transport. [europepmc]
• Unbalanced regularized optimal mass transport with applications to fluid flows in the brain. [europepmc]
• Reconstructing growth and dynamic trajectories from single-cell transcriptomics data. [europepmc]
• Optimal Control of Underdamped Systems: An Analytic Approach. [europepmc]
• Integrating Dynamical Systems Modeling with Spatiotemporal scRNA-Seq Data Analysis. [europepmc]
• Analysis of mean-field models arising from self-attention dynamics in transformer architectures with layer normalization. [europepmc]
• SYNCHRONIZED OPTIMAL TRANSPORT FOR JOINT MODELING OF DYNAMICS ACROSS MULTIPLE SPACES. [europepmc]
• FatePredictor: Cell fate decision-making prediction with an ensemble deep learning model. [europepmc]
• Experimentally achieving minimal dissipation via thermodynamically optimal transport. [europepmc]
• Variational structures for the Fokker-Planck equation with general Dirichlet boundary conditions. [europepmc]
• Beyond metaphor: quantitative reconstruction of Waddington landscape and exploration of cellular behavior. [europepmc]
• VIST: variational inference for single cell time series. [europepmc]
• Learning collective multicellular dynamics with an interacting mean field neural SDE model. [europepmc]
• Generative models of cell dynamics: from Neural ODEs to flow matching. [europepmc]
• Minimum-Excess-Work Guidance: Score-Based Sampling with Experimental Data or Sparse Restraints. [europepmc]
• Reconstructing Waddington's Landscape from Data [europepmc]