# Mean-field neural networks: learning mappings on Wasserstein space

arXiv: https://arxiv.org/abs/2210.15179 · HTML: https://mlsanity.com/paper/2210.15179/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 26 references that are in the map.
- First posted on arXiv 27 Oct 2022.

## Builds on (most cited first, 12 of 12)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (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)
- [A Machine Learning Framework for Solving High-Dimensional Mean Field Game and Mean Field Control Problems](https://mlsanity.com/paper/1912.01825/) (2019)
- [An overview on deep learning-based approximation methods for partial differential equations](https://mlsanity.com/paper/2012.12348/) (2020)
- [Alternating the Population and Control Neural Networks to Solve High-Dimensional Stochastic Mean-Field Games](https://mlsanity.com/paper/2002.10113/) (2020)
- [Machine Learning for semi linear PDEs](https://mlsanity.com/paper/1809.07609/) (2018)
- [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)
- [Ito's formula for flows of measures on semimartingales](https://mlsanity.com/paper/2010.05288/) (2020)
- [Reservoir optimization and Machine Learning methods](https://mlsanity.com/paper/2106.08097/) (2021)

## Built on by (most cited first, 16 of 16)

- [Mean-field neural networks-based algorithms for McKean-Vlasov control problems *](https://mlsanity.com/paper/2212.11518/) (2022)
- [Actor critic learning algorithms for mean-field control with moment neural networks](https://mlsanity.com/paper/2309.04317/) (2023)
- [Universal Approximation of Mean-Field Models via Transformers](https://mlsanity.com/paper/2410.16295/) (2024)
- [An Overview on Machine Learning Methods for Partial Differential Equations: from Physics Informed Neural Networks to Deep Operator Learning](https://mlsanity.com/paper/2408.13222/) (2024)
- [Neural McKean-Vlasov Processes: Distributional Dependence in Diffusion Processes](https://mlsanity.com/paper/2404.09402/) (2024)
- [Continuous-time q-learning for mean-field control with common noise, part-II: q-learning algorithms](https://mlsanity.com/paper/2604.27378/) (2026)
- [Policy Gradient for Continuous-Time Mean-Field Control](https://mlsanity.com/paper/2605.20718/) (2026)
- [Learning operators on labelled conditional distributions with applications to mean field control of non exchangeable systems](https://mlsanity.com/paper/2603.21683/) (2026)
- [Deep Hilbert--Galerkin Methods for Infinite-Dimensional PDEs and Optimal Control](https://mlsanity.com/paper/2603.19463/) (2026)
- [Algorithmically Designed Artificial Neural Networks (ADANNs): Higher order deep operator learning for parametric partial differential equations](https://mlsanity.com/paper/2302.03286/) (2023)
- [Quantile and moment neural networks for learning functionals of distributions](https://mlsanity.com/paper/2303.11060/) (2023)
- [MVNN: A Measure-Valued Neural Network for Learning McKean-Vlasov Dynamics from Particle Data](https://mlsanity.com/paper/2604.00333/) (2026)
- [Mean-Field PhiBE: Continuous-Time Mean-Field Reinforcement Learning from Discrete-Time Data](https://mlsanity.com/paper/2606.26498/) (2026)
- [Universal Approximation of Nonlinear Operators and Their Derivatives](https://mlsanity.com/paper/2605.15285/) (2026)
- [A cylindrical neural approximation theorem for conditional laws of McKean-Vlasov equations with common noise](https://mlsanity.com/paper/2608.08040/) (2026)
- [From Distributions to Stochastic Processes: Neural Approximation of Measure-Valued Maps](https://mlsanity.com/paper/2609.33649/) (2026)

## Abstract (arXiv)

We study the machine learning task for models with operators mapping between the Wasserstein space of probability measures and a space of functions, like e.g. in mean-field games/control problems. Two classes of neural networks, based on bin density and on cylindrical approximation, are proposed to learn these so-called mean-field functions, and are theoretically supported by universal approximation theorems. We perform several numerical experiments for training these two mean-field neural networks, and show their accuracy and efficiency in the generalization error with various test distributions. Finally, we present different algorithms relying on mean-field neural networks for solving time-dependent mean-field problems, and illustrate our results with numerical tests for the example of a semi-linear partial differential equation in the Wasserstein space of probability measures.