# Learning Surrogate Potential Mean Field Games via Gaussian Processes: A Data-Driven Approach to Ill-Posed Inverse Problems

arXiv: https://arxiv.org/abs/2502.11506 · HTML: https://mlsanity.com/paper/2502.11506/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 35 of its 72 references that are in the map.
- First posted on arXiv 17 Feb 2025.

## Builds on (most cited first, 30 of 35)

- [Gaussian Processes and Kernel Methods: A Review on Connections and Equivalences](https://mlsanity.com/paper/1807.02582/) (2018)
- [Machine Learning of Linear Differential Equations using Gaussian Processes](https://mlsanity.com/paper/1701.02440/) (2017)
- [Solving and Learning Nonlinear PDEs with Gaussian Processes](https://mlsanity.com/paper/2103.12959/) (2021)
- [In-Context Operator Learning with Data Prompts for Differential Equation Problems](https://mlsanity.com/paper/2304.07993/) (2023)
- [A Machine Learning Framework for Solving High-Dimensional Mean Field Game and Mean Field Control Problems](https://mlsanity.com/paper/1912.01825/) (2019)
- [Numerical Gaussian Processes for Time-dependent and Non-linear Partial Differential Equations](https://mlsanity.com/paper/1703.10230/) (2017)
- [Alternating the Population and Control Neural Networks to Solve High-Dimensional Stochastic Mean-Field Games](https://mlsanity.com/paper/2002.10113/) (2020)
- [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)
- [Sparse Cholesky Factorization for Solving Nonlinear PDEs via Gaussian Processes](https://mlsanity.com/paper/2304.01294/) (2023)
- [A mean-field games laboratory for generative modeling](https://mlsanity.com/paper/2304.13534/) (2023)
- [Mean Field Game of Controls and An Application To Trade Crowding](https://mlsanity.com/paper/1610.09904/) (2016)
- [Mean Field Games and Applications: Numerical Aspects](https://mlsanity.com/paper/2003.04444/) (2020)
- [Numerical Methods for Mean Field Games and Mean Field Type Control](https://mlsanity.com/paper/2106.06231/) (2021)
- [Learning Dynamical Systems from Data: A Simple Cross-Validation Perspective, Part V: Sparse Kernel Flows for 132 Chaotic Dynamical Systems](https://mlsanity.com/paper/2301.10321/) (2023)
- [Learning dynamical systems from data: A simple cross-validation perspective, part III: Irregularly-Sampled Time Series](https://mlsanity.com/paper/2111.13037/) (2021)
- [Proximal methods for stationary Mean Field Games with local couplings](https://mlsanity.com/paper/1608.07701/) (2016)
- [Individual-Level Inverse Reinforcement Learning for Mean Field Games](https://mlsanity.com/paper/2202.06401/) (2022)
- [On Imitation in Mean-field Games](https://mlsanity.com/paper/2306.14799/) (2023)
- [On the implementation of a primal-dual algorithm for second order time-dependent mean field games with local couplings](https://mlsanity.com/paper/1802.07902/) (2018)
- [A mean field game inverse problem](https://mlsanity.com/paper/2007.11551/) (2020)
- [Numerical methods for Mean field Games based on Gaussian Processes and Fourier Features](https://mlsanity.com/paper/2112.05414/) (2021)
- [Convergence Analysis of Machine Learning Algorithms for the Numerical Solution of Mean Field Control and Games: I -- The Ergodic Case](https://mlsanity.com/paper/1907.05980/) (2019)
- [Splitting methods for a class of non-potential mean field games](https://mlsanity.com/paper/2007.00099/) (2020)
- [Mean field control problems for vaccine distribution](https://mlsanity.com/paper/2104.11887/) (2021)
- [Inverse problems for mean field games](https://mlsanity.com/paper/2205.11350/) (2022)
- [A Policy Iteration Method for Inverse Mean Field Games](https://mlsanity.com/paper/2409.06184/) (2024)
- [Mean field games systems of first order](https://mlsanity.com/paper/1401.1789/) (2014)
- [Long Time Behavior of First Order Mean Field Games on Euclidean Space](https://mlsanity.com/paper/1809.09057/) (2018)
- [Simultaneously recovering running cost and Hamiltonian in Mean Field Games system](https://mlsanity.com/paper/2303.13096/) (2023)
- [Decoding Mean Field Games from Population and Environment Observations By Gaussian Processes](https://mlsanity.com/paper/2312.06625/) (2023)

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

- [Gaussian process policy iteration with additive Schwarz acceleration for forward and inverse HJB and mean field game problems](https://mlsanity.com/paper/2505.00909/) (2025)
- [Inverse Learning of the Altruism and Cost Level in Mixed-Individual Mean Field Games](https://mlsanity.com/paper/2609.13469/) (2026)

## Abstract (arXiv)

Mean field games (MFGs) describe the collective behavior of large populations of interacting agents. In this work, we tackle ill-posed inverse problems in potential MFGs, aiming to recover the agents' population, momentum, and environmental setup from limited, noisy measurements and partial observations. These problems are ill-posed because multiple MFG configurations can explain the same data, or different parameters can yield nearly identical observations. Nonetheless, they remain crucial in practice for real-world scenarios where data are inherently sparse or noisy, or where the MFG structure is not fully determined. Our focus is on finding surrogate MFGs that accurately reproduce the observed data despite these challenges. We propose two Gaussian process (GP)-based frameworks: an inf-sup formulation and a bilevel approach. The choice between them depends on whether the unknown parameters introduce concavity in the objective. In the inf-sup framework, we use the linearity of GPs and their parameterization structure to maintain convex-concave properties, allowing us to apply standard convex optimization algorithms. In the bilevel framework, we employ a gradient-descent-based algorithm and introduce two methods for computing the outer gradient. The first method leverages an existing solver for the inner potential MFG and applies automatic differentiation, while the second adopts an adjoint-based strategy that computes the outer gradient independently of the inner solver. Our numerical experiments show that when sufficient prior information is available, the unknown parameters can be accurately recovered. Otherwise, if prior information is limited, the inverse problem is ill-posed, but our frameworks can still produce surrogate MFG models that closely match observed data.