# Deep Learning for Mean Field Optimal Transport

arXiv: https://arxiv.org/abs/2302.14739 · HTML: https://mlsanity.com/paper/2302.14739/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 60 references that are in the map.
- First posted on arXiv 28 Feb 2023.

## Builds on (most cited first, 25 of 25)

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations](https://mlsanity.com/paper/1706.04702/) (2017)
- [On the relation between optimal transport and Schr"odinger bridges: A stochastic control viewpoint](https://mlsanity.com/paper/1412.4430/) (2014)
- [A Machine Learning Framework for Solving High-Dimensional Mean Field Game and Mean Field Control Problems](https://mlsanity.com/paper/1912.01825/) (2019)
- [Deep Generalized Schr"odinger Bridge](https://mlsanity.com/paper/2209.09893/) (2022)
- [Convergence Analysis of Machine Learning Algorithms for the Numerical Solution of Mean Field Control and Games: II -- The Finite Horizon Case](https://mlsanity.com/paper/1908.01613/) (2019)
- [Neural networks-based algorithms for stochastic control and PDEs in finance](https://mlsanity.com/paper/2101.08068/) (2021)
- [Deep Learning Methods for Mean Field Control Problems with Delay](https://mlsanity.com/paper/1905.00358/) (2019)
- [Optimal steering of a linear stochastic system to a final probability distribution](https://mlsanity.com/paper/1408.2222/) (2014)
- [Numerical Methods for Mean Field Games and Mean Field Type Control](https://mlsanity.com/paper/2106.06231/) (2021)
- [Deep Learning for Mean Field Games and Mean Field Control with Applications to Finance](https://mlsanity.com/paper/2107.04568/) (2021)
- [Numerical resolution of McKean-Vlasov FBSDEs using neural networks](https://mlsanity.com/paper/1909.12678/) (2019)
- [Optimal steering of a linear stochastic system to a final probability distribution, part II](https://mlsanity.com/paper/1410.3447/) (2014)
- [Learning High-Dimensional McKean-Vlasov Forward-Backward Stochastic Differential Equations with General Distribution Dependence](https://mlsanity.com/paper/2204.11924/) (2022)
- [Optimal steering of a linear stochastic system to a final probability distribution, Part III](https://mlsanity.com/paper/1608.03622/) (2016)
- [An entropy minimization approach to second-order variational mean-field games](https://mlsanity.com/paper/1807.09078/) (2018)
- [Connecting GANs, MFGs, and OT](https://mlsanity.com/paper/2002.04112/) (2020)
- [Optimal transport: discretization and algorithms](https://mlsanity.com/paper/2003.00855/) (2020)
- [Mean Field Type Control with Congestion (II): An Augmented Lagrangian Method](https://mlsanity.com/paper/1611.02023/) (2016)
- [The Master Equation in Mean Field Theory](https://mlsanity.com/paper/1404.4150/) (2014)
- [On the system of partial differential equations arising in mean field type control](https://mlsanity.com/paper/1503.05044/) (2015)
- [Mean field games systems of first order](https://mlsanity.com/paper/1401.1789/) (2014)
- [The planning problem in Mean Field Games as regularized mass transport](https://mlsanity.com/paper/1811.02706/) (2018)

## Built on by (most cited first, 1 of 1)

- [Convergence analysis of controlled particle systems arising in deep learning: from finite to infinite sample size](https://mlsanity.com/paper/2404.05185/) (2024)

## Abstract (arXiv)

Mean field control (MFC) problems have been introduced to study social optima in very large populations of strategic agents. The main idea is to consider an infinite population and to simplify the analysis by using a mean field approximation. These problems can also be viewed as optimal control problems for McKean-Vlasov dynamics. They have found applications in a wide range of fields, from economics and finance to social sciences and engineering. Usually, the goal for the agents is to minimize a total cost which consists in the integral of a running cost plus a terminal cost. In this work, we consider MFC problems in which there is no terminal cost but, instead, the terminal distribution is prescribed. We call such problems mean field optimal transport problems since they can be viewed as a generalization of classical optimal transport problems when mean field interactions occur in the dynamics or the running cost function. We propose three numerical methods based on neural networks. The first one is based on directly learning an optimal control. The second one amounts to solve a forward-backward PDE system characterizing the solution. The third one relies on a primal-dual approach. We illustrate these methods with numerical experiments conducted on two families of examples.