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

TL;DR

This work tackles ill-posed inverse problems in potential MFGs, aiming to recover the agents' population, momentum, and environmental setup from limited, noisy measurements and partial observations, and proposes two Gaussian process (GP)-based frameworks: an inf-sup formulation and a bilevel approach.

Abstract

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.

Figures (14)

• Figure 1: Numerical results for solving the inverse problem of the MFG system in \ref{['eq:MFG_system']} using the inf-sup framework: (a) the sample (grid) points and observation points of $m$; (b) the observation points of $V$; (c) the discretized $L^2$ error $\mathcal{E}(m^k, m^*)$ versus the iteration number $k$; (d) the exact solution $m^*$; (e) the recovered $m$; (f) the pointwise error between the recovered $m$ and the exact $m^*$; (g) the ground truth $V$; (h) the recovered $V$; (i) the pointwise error between the recovered values and the exact solution of $V$.
• Figure 2: Numerical results for solving the inverse problem of the MFG system in \ref{['eq:MFG_system']} using the inf-sup framework: (a) log-log plot of $L^2$ errors for $m$ versus $m^*$ as the number of observation points increases. (b) log-log plot of $L^2$ errors for $V$ versus its exact values as the number of observation points increases.
• Figure 3: Reference results of $m$ and $V$ for the MFG \ref{['General']}
• Figure 4: Numerical results for solving the inverse problem of the MFG system in \ref{['General']} using a power function to approximate the coupling function $F$: (a) sample (grid) points and observation points of $m$; (b) observation points of $V$; (c) discretized $L^2$ error $\mathcal{E}(m^k, m^*)$ versus iteration number $k$; (d) recovered $F$ vs. reference $F$; (e) recovered $m$; (f) pointwise error between the recovered $m$ and the exact $m^*$; (g) recovered $V$; (h) pointwise error between the recovered $V$ and its exact solution.
• Figure 5: Numerical results for solving the inverse problem of the MFG system in \ref{['General']} using a power function approximation for the coupling function $F$: (a) log-log plot of $L^2$ errors for $m$ versus $m^*$ as the number of observation points increases. (b) log-log plot of $L^2$ errors for $V$ versus its exact values as the number of observation points increases.