# Numerical methods for Mean field Games based on Gaussian Processes and Fourier Features

arXiv: https://arxiv.org/abs/2112.05414 · HTML: https://mlsanity.com/paper/2112.05414/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Dec 2021.

## Built on by (most cited first, 7 of 7)

- [Error Analysis of Kernel/GP Methods for Nonlinear and Parametric PDEs](https://mlsanity.com/paper/2305.04962/) (2023)
- [A Kernel Approach for PDE Discovery and Operator Learning](https://mlsanity.com/paper/2210.08140/) (2022)
- [Decoding Mean Field Games from Population and Environment Observations By Gaussian Processes](https://mlsanity.com/paper/2312.06625/) (2023)
- [Scaling up Probabilistic PDE Simulators with Structured Volumetric Information](https://mlsanity.com/paper/2406.05020/) (2024)
- [Learning Surrogate Potential Mean Field Games via Gaussian Processes: A Data-Driven Approach to Ill-Posed Inverse Problems](https://mlsanity.com/paper/2502.11506/) (2025)
- [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)
- [Solving Functional PDEs with Gaussian Processes and Applications to Functional Renormalization Group Equations](https://mlsanity.com/paper/2512.20956/) (2025)

## Abstract (arXiv)

In this article, we propose two numerical methods, the Gaussian Process (GP) method and the Fourier Features (FF) algorithm, to solve mean field games (MFGs). The GP algorithm approximates the solution of a MFG with maximum a posteriori probability estimators of GPs conditioned on the partial differential equation (PDE) system of the MFG at a finite number of sample points. The main bottleneck of the GP method is to compute the inverse of a square gram matrix, whose size is proportional to the number of sample points. To improve the performance, we introduce the FF method, whose insight comes from the recent trend of approximating positive definite kernels with random Fourier features. The FF algorithm seeks approximated solutions in the space generated by sampled Fourier features. In the FF method, the size of the matrix to be inverted depends only on the number of Fourier features selected, which is much less than the size of sample points. Hence, the FF method reduces the precomputation time, saves the memory, and achieves comparable accuracy to the GP method. We give the existence and the convergence proofs for both algorithms. The convergence argument of the GP method does not depend on any monotonicity condition, which suggests the potential applications of the GP method to solve MFGs with non-monotone couplings in future work. We show the efficacy of our algorithms through experiments on a stationary MFG with a non-local coupling and on a time-dependent planning problem. We believe that the FF method can also serve as an alternative algorithm to solve general PDEs.