# Physics Informed Convex Artificial Neural Networks (PICANNs) for Optimal Transport based Density Estimation

arXiv: https://arxiv.org/abs/2104.01194 · HTML: https://mlsanity.com/paper/2104.01194/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 50 references that are in the map.
- First posted on arXiv 2 Apr 2021.

## Builds on (most cited first, 19 of 19)

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Improved Training of Wasserstein GANs](https://mlsanity.com/paper/1704.00028/) (2017)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [Masked Autoregressive Flow for Density Estimation](https://mlsanity.com/paper/1705.07057/) (2017)
- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [Invertible Residual Networks](https://mlsanity.com/paper/1811.00995/) (2018)
- [Flow++: Improving Flow-Based Generative Models with Variational Dequantization and Architecture Design](https://mlsanity.com/paper/1902.00275/) (2019)
- [Self-Adaptive Physics-Informed Neural Networks using a Soft Attention Mechanism](https://mlsanity.com/paper/2009.04544/) (2020)
- [Optimal transport mapping via input convex neural networks](https://mlsanity.com/paper/1908.10962/) (2019)
- [Large-Scale Optimal Transport and Mapping Estimation](https://mlsanity.com/paper/1711.02283/) (2017)
- [Wasserstein-2 Generative Networks](https://mlsanity.com/paper/1909.13082/) (2019)
- [Learning in Modal Space: Solving Time-Dependent Stochastic PDEs Using Physics-Informed Neural Networks](https://mlsanity.com/paper/1905.01205/) (2019)
- [2-Wasserstein Approximation via Restricted Convex Potentials with Application to Improved Training for GANs](https://mlsanity.com/paper/1902.07197/) (2019)
- [On the convergence of physics informed neural networks for linear second-order elliptic and parabolic type PDEs](https://mlsanity.com/paper/2004.01806/) (2020)
- [A fast approach to optimal transport: The back-and-forth method](https://mlsanity.com/paper/1905.12154/) (2019)
- [(q,p)-Wasserstein GANs: Comparing Ground Metrics for Wasserstein GANs](https://mlsanity.com/paper/1902.03642/) (2019)
- [Diffeomorphic Learning](https://mlsanity.com/paper/1806.01240/) (2018)

## Built on by (most cited first, 2 of 2)

- [A Natural Primal-Dual Hybrid Gradient Method for Adversarial Neural Network Training on Solving Partial Differential Equations](https://mlsanity.com/paper/2411.06278/) (2024)
- [Convex Physics Informed Neural Networks for the Monge-Amp`ere Optimal Transport Problem](https://mlsanity.com/paper/2501.10162/) (2025)

## Abstract (arXiv)

Optimal Mass Transport (OMT) is a well studied problem with a variety of applications in a diverse set of fields ranging from Physics to Computer Vision and in particular Statistics and Data Science. Since the original formulation of Monge in 1781 significant theoretical progress been made on the existence, uniqueness and properties of the optimal transport maps. The actual numerical computation of the transport maps, particularly in high dimensions, remains a challenging problem. By Brenier's theorem, the continuous OMT problem can be reduced to that of solving a non-linear PDE of Monge-Ampere type whose solution is a convex function. In this paper, building on recent developments of input convex neural networks and physics informed neural networks for solving PDE's, we propose a Deep Learning approach to solve the continuous OMT problem. To demonstrate the versatility of our framework we focus on the ubiquitous density estimation and generative modeling tasks in statistics and machine learning. Finally as an example we show how our framework can be incorporated with an autoencoder to estimate an effective probabilistic generative model.