mlsanityData to 7 Oct 2026

Monge-Amp`ere Flow for Generative Modeling

arXiv 1809.10188 ↗First posted 26 Sep 2018cs.LG cond-mat.stat-mech math.DScited by 33

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Abstract (arXiv)

We present a deep generative model, named Monge-Amp`ere flow, which builds on continuous-time gradient flow arising from the Monge-Amp`ere equation in optimal transport theory. The generative map from the latent space to the data space follows a dynamical system, where a learnable potential function guides a compressible fluid to flow towards the target density distribution. Training of the model amounts to solving an optimal control problem. The Monge-Amp`ere flow has tractable likelihoods and supports efficient sampling and inference. One can easily impose symmetry constraints in the generative model by designing suitable scalar potential functions. We apply the approach to unsupervised density estimation of the MNIST dataset and variational calculation of the two-dimensional Ising model at the critical point. This approach brings insights and techniques from Monge-Amp`ere equation, optimal transport, and fluid dynamics into reversible flow-based generative models.

Key facts. Cited by 33 papers in the map (data through 7 Oct 2026). First posted on arXiv 26 Sep 2018.

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Citing papers per year: 2021: 8; 2022: 8; 2023: 4; 2024: 3; 2025: 5; 2026: 5.

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  1. 2023Stochastic Interpolants: A Unifying Framework for Flows and Diffusionscited by 897
  2. 2021Deep Generative Modelling: A Comparative Review of VAEs, GANs, Normalizing Flows, Energy-Based and Autoregressive Modelscited by 251
  3. 2021An Introduction to Deep Generative Modelingcited by 104
  4. 2024How Discrete and Continuous Diffusion Meet: Comprehensive Analysis of Discrete Diffusion Models via a Stochastic Integral Frameworkcited by 48
  5. 2025Fast Solvers for Discrete Diffusion Models: Theory and Applications of High-Order Algorithmscited by 47
  6. 2024Accelerating Diffusion Models with Parallel Sampling: Inference at Sub-Linear Time Complexitycited by 42
  7. 2022Normalizing flow neural networks by JKO schemecited by 39
  8. 2021Adaptive deep density approximation for Fokker-Planck equationscited by 28
  9. 2022Aspects of scaling and scalability for flow-based sampling of lattice QCDcited by 26
  10. 2025DriftLite: Lightweight Drift Control for Inference-Time Scaling of Diffusion Modelscited by 21
  11. 2023Efficient Neural Network Approaches for Conditional Optimal Transport with Applications in Bayesian Inferencecited by 19
  12. 2025Solving Inverse Problems via Diffusion-Based Priors: An Approximation-Free Ensemble Sampling Approachcited by 18
21 more
  1. 2022Taming Hyperparameter Tuning in Continuous Normalizing Flows Using the JKO Schemecited by 17
  2. 2023Gaussian Interpolation Flowscited by 17
  3. 2021Ab-initio study of interacting fermions at finite temperature with neural canonical transformationcited by 16
  4. 2021Jacobian Determinant of Normalizing Flowscited by 8
  5. 2022A Mathematical Framework for Learning Probability Distributionscited by 7
  6. 2022Deep Variational Free Energy Approach to Dense Hydrogencited by 6
  7. 2022Turning Normalizing Flows into Monge Maps with Geodesic Gaussian Preserving Flowscited by 6
  8. 2021Augmented KRnet for density estimation and approximationcited by 4
  9. 2022High-dimensional density estimation with tensorizing flowcited by 4
  10. 2021EBMs Trained with Maximum Likelihood are Generator Models Trained with a Self-adverserial Losscited by 2
  11. 2026Simple Approximation and Derivative Free Inference-Time Scaling for Diffusion Models via Sequential Monte Carlo on Path Measurescited by 2
  12. 2021Characteristic Neural Ordinary Differential Equationscited by 1
  13. 2023PINF: Continuous Normalizing Flows for Physics-Constrained Deep Learningcited by 1
  14. 2026FluxLite: Inference-Time Proposal Control for Discrete Diffusion Modelscited by 1
  15. 2022Multisymplectic Formulation of Deep Learning Using Mean--Field Type Control and Nonlinear Stability of Training Algorithmcited by 0
  16. 2024Variational Optimization for Quantum Problems using Deep Generative Networkscited by 0
  17. 2025Accurate and thermodynamically consistent hydrogen equation of state for planetary modeling with flow matchingcited by 0
  18. 2025Estimation of Stochastic Optimal Transport Mapscited by 0
  19. 2026Complex normalizing flows can almost be information K"ahler-Ricci flowscited by 0
  20. 2026Quantum Dynamics via Score Matching on Bohmian Trajectoriescited by 0
  21. 2026FES-FM: Free Energy Surface Sampling via Reduced Flow Matchingcited by 0