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

2025-02-17

AI Summary

This work addresses an ill-posed inverse problem in potential-type mean-field games (MFGs), where sparse and noisy observations hinder joint recovery of the population distribution, momentum field, and environmental parameters. To overcome limitations of conventional methods—namely, reliance on expensive inner solvers and poor generalization—we propose a dual framework grounded in Gaussian process priors: (i) a convex–concave optimization formulation guided by inf-sup stability, and (ii) an adjoint-gradient method that eliminates inner solvers and avoids automatic differentiation through nested optimization. Theoretically and empirically, we demonstrate that with sufficient prior information, the framework achieves high-fidelity reconstruction of unknown parameters; under weak priors, it yields surrogate MFG models with strong data fidelity, significantly enhancing robustness and practicality for ill-conditioned inverse problems.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Game Theory and Economic Paradigms: Adversarial Learning
• Search and Optimization: Non-convex Optimization

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

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.

Problem

• Recovering MFG configurations from noisy data
• Solving ill-posed inverse problems in potential MFGs
• Developing GP-based frameworks for surrogate MFGs

Innovation

• Gaussian Processes for MFGs
• Inf-sup and bilevel frameworks
• Adjoint-based gradient computation