# Unsupervised Solution Operator Learning for Mean-Field Games via Sampling-Invariant Parametrizations

arXiv: https://arxiv.org/abs/2401.15482 · HTML: https://mlsanity.com/paper/2401.15482/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 60 references that are in the map.
- First posted on arXiv 27 Jan 2024.

## Builds on (most cited first, 28 of 28)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Layer Normalization](https://mlsanity.com/paper/1607.06450/) (2016)
- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [SGDR: Stochastic Gradient Descent with Warm Restarts](https://mlsanity.com/paper/1608.03983/) (2016)
- [PointNet++: Deep Hierarchical Feature Learning on Point Sets in a Metric Space](https://mlsanity.com/paper/1706.02413/) (2017)
- [Gaussian Error Linear Units (GELUs)](https://mlsanity.com/paper/1606.08415/) (2016)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [HyperNetworks](https://mlsanity.com/paper/1609.09106/) (2016)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [Optimal transport mapping via input convex neural networks](https://mlsanity.com/paper/1908.10962/) (2019)
- [Training generative neural networks via Maximum Mean Discrepancy optimization](https://mlsanity.com/paper/1505.03906/) (2015)
- [In-Context Operator Learning with Data Prompts for Differential Equation Problems](https://mlsanity.com/paper/2304.07993/) (2023)
- [Wasserstein-2 Generative Networks](https://mlsanity.com/paper/1909.13082/) (2019)
- [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)
- [Accurate Point Cloud Registration with Robust Optimal Transport](https://mlsanity.com/paper/2111.00648/) (2021)
- [Alternating the Population and Control Neural Networks to Solve High-Dimensional Stochastic Mean-Field Games](https://mlsanity.com/paper/2002.10113/) (2020)
- [Chart Auto-Encoders for Manifold Structured Data](https://mlsanity.com/paper/1912.10094/) (2019)
- [Generalization in Mean Field Games by Learning Master Policies](https://mlsanity.com/paper/2109.09717/) (2021)
- [Meta Optimal Transport](https://mlsanity.com/paper/2206.05262/) (2022)
- [Bridging Mean-Field Games and Normalizing Flows with Trajectory Regularization](https://mlsanity.com/paper/2206.14990/) (2022)
- [Solving Large-Scale Optimization Problems with a Convergence Rate Independent of Grid Size](https://mlsanity.com/paper/1805.09453/) (2018)
- [A Fast Proximal Gradient Method and Convergence Analysis for Dynamic Mean Field Planning](https://mlsanity.com/paper/2102.13260/) (2021)
- [Computational Mean-field Games on Manifolds](https://mlsanity.com/paper/2206.01622/) (2022)

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

- [Neural Operators for Multi-Task Control and Adaptation](https://mlsanity.com/paper/2604.03449/) (2026)
- [PIONM: A Generalized Approach to Solving Density-Constrained Mean-Field Games Equilibrium under Modified Boundary Conditions](https://mlsanity.com/paper/2504.03209/) (2025)
- [All in One: Generative Modeling as Mean-Field Game Design](https://mlsanity.com/paper/2607.23026/) (2026)

## Abstract (arXiv)

Recent advances in deep learning has witnessed many innovative frameworks that solve high dimensional mean-field games (MFG) accurately and efficiently. These methods, however, are restricted to solving single-instance MFG and demands extensive computational time per instance, limiting practicality. To overcome this, we develop a novel framework to learn the MFG solution operator. Our model takes a MFG instances as input and output their solutions with one forward pass. To ensure the proposed parametrization is well-suited for operator learning, we introduce and prove the notion of sampling invariance for our model, establishing its convergence to a continuous operator in the sampling limit. Our method features two key advantages. First, it is discretization-free, making it particularly suitable for learning operators of high-dimensional MFGs. Secondly, it can be trained without the need for access to supervised labels, significantly reducing the computational overhead associated with creating training datasets in existing operator learning methods. We test our framework on synthetic and realistic datasets with varying complexity and dimensionality to substantiate its robustness.