# Learning Brenier Potentials with Convex Generative Adversarial Neural Networks

arXiv: https://arxiv.org/abs/2504.19779 · HTML: https://mlsanity.com/paper/2504.19779/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 42 references that are in the map.
- First posted on arXiv 28 Apr 2025.

## Builds on (most cited first, 24 of 24)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [ESRGAN: Enhanced Super-Resolution Generative Adversarial Networks](https://mlsanity.com/paper/1809.00219/) (2018)
- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [f-GAN: Training Generative Neural Samplers using Variational Divergence Minimization](https://mlsanity.com/paper/1606.00709/) (2016)
- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [Deep Generative Modelling: A Comparative Review of VAEs, GANs, Normalizing Flows, Energy-Based and Autoregressive Models](https://mlsanity.com/paper/2103.04922/) (2021)
- [Convex Potential Flows: Universal Probability Distributions with Optimal Transport and Convex Optimization](https://mlsanity.com/paper/2012.05942/) (2020)
- [Optimal Control Via Neural Networks: A Convex Approach](https://mlsanity.com/paper/1805.11835/) (2018)
- [Coupling-based Invertible Neural Networks Are Universal Diffeomorphism Approximators](https://mlsanity.com/paper/2006.11469/) (2020)
- [Simultaneous approximation of a smooth function and its derivatives by deep neural networks with piecewise-polynomial activations](https://mlsanity.com/paper/2206.09527/) (2022)
- [Some Theoretical Properties of GANs](https://mlsanity.com/paper/1803.07819/) (2018)
- [Regularity as Regularization: Smooth and Strongly Convex Brenier Potentials in Optimal Transport](https://mlsanity.com/paper/1905.10812/) (2019)
- [Input Convex Gradient Networks](https://mlsanity.com/paper/2111.12187/) (2021)
- [Rates of convergence for density estimation with generative adversarial networks](https://mlsanity.com/paper/2102.00199/) (2021)
- [A Convenient Infinite Dimensional Framework for Generative Adversarial Learning](https://mlsanity.com/paper/2011.12087/) (2020)
- [LU-Net: Invertible Neural Networks Based on Matrix Factorization](https://mlsanity.com/paper/2302.10524/) (2023)

## Built on by (most cited first, 3 of 3)

- [When do World Models Successfully Learn Dynamical Systems?](https://mlsanity.com/paper/2507.04898/) (2025)
- [Consistency of Learned Sparse Grid Quadrature Rules using NeuralODEs](https://mlsanity.com/paper/2507.01533/) (2025)
- [Regularity of Solutions to Beckmann's Parametric Optimal Transport](https://mlsanity.com/paper/2603.19755/) (2026)

## Abstract (arXiv)

Brenier proved that under certain conditions on a source and a target probability measure there exists a strictly convex function such that its gradient is a transport map from the source to the target distribution. This function is called the Brenier potential. Furthermore, detailed information on the H"older regularity of the Brenier potential is available. In this work we develop the statistical learning theory of generative adversarial neural networks that learn the Brenier potential. As by the transformation of densities formula, the density of the generated measure depends on the second derivative of the Brenier potential, we develop the universal approximation theory of ReCU networks with cubic activation mathttReCU(x)=max0,x3 that combines the favorable approximation properties of H"older functions with a Lipschitz continuous density. In order to assure the convexity of such general networks, we introduce an adversarial training procedure for a potential function represented by the ReCU networks that combines the classical discriminator cross entropy loss with a penalty term that enforces (strict) convexity. We give a detailed decomposition of learning errors and show that for a suitable high penalty parameter all networks chosen in the adversarial min-max optimization problem are strictly convex. This is further exploited to prove the consistency of the learning procedure for (slowly) expanding network capacity. We also implement the described learning algorithm and apply it to a number of standard test cases from Gaussian mixture to image data as target distributions. As predicted in theory, we observe that the convexity loss becomes inactive during the training process and the potentials represented by the neural networks have learned convexity.