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Simultaneously decoding the unknown stationary state and function parameters for mean field games

Hongyu Liu, Catharine W. K. Lo

TL;DR

The paper tackles the inverse problem for mean field games by proving that the unknown stationary state $U=(u_0,m_0)$, the running cost $F$, and the Hamiltonian $\\mathcal{H}$ can be uniquely identified from boundary Cauchy data in a general time-dependent MFG setting. The authors develop a high-order linearization framework around an unknown stable stationary state, construct parabolic complex geometric optics (CGO) solutions, and apply Runge approximation and unique continuation to establish global identifiability up to the conformal class of the Hamiltonian metric. The main contributions include extending previous stationary results to time-dependent problems, recovering the stationary state simultaneously with model parameters, and handling a general quadratic Hamiltonian with minimal boundary data assumptions. The approach provides a rigorous pathway for decoding MFG dynamics from external measurements and enhances the practical applicability of inverse problem techniques to large-population strategic systems.

Abstract

Mean field games (MFGs) offer a versatile framework for modeling large-scale interactive systems across multiple domains. This paper builds upon a previous work, by developing a state-of-the-art unified approach to decode or design the unknown stationary state of MFGs, in addition to the underlying parameter functions governing their behavior. This result is novel, even in the general realm of inverse problems for nonlinear PDEs. By enabling agents to distill crucial insights from observed data and unveil intricate hidden structures and unknown states within MFG systems, our approach surmounts a significant obstacle, enhancing the applicability of MFGs in real-world scenarios. This advancement not only enriches our understanding of MFG dynamics but also broadens the scope for their practical deployment in various contexts.

Simultaneously decoding the unknown stationary state and function parameters for mean field games

TL;DR

The paper tackles the inverse problem for mean field games by proving that the unknown stationary state , the running cost , and the Hamiltonian can be uniquely identified from boundary Cauchy data in a general time-dependent MFG setting. The authors develop a high-order linearization framework around an unknown stable stationary state, construct parabolic complex geometric optics (CGO) solutions, and apply Runge approximation and unique continuation to establish global identifiability up to the conformal class of the Hamiltonian metric. The main contributions include extending previous stationary results to time-dependent problems, recovering the stationary state simultaneously with model parameters, and handling a general quadratic Hamiltonian with minimal boundary data assumptions. The approach provides a rigorous pathway for decoding MFG dynamics from external measurements and enhances the practical applicability of inverse problem techniques to large-population strategic systems.

Abstract

Mean field games (MFGs) offer a versatile framework for modeling large-scale interactive systems across multiple domains. This paper builds upon a previous work, by developing a state-of-the-art unified approach to decode or design the unknown stationary state of MFGs, in addition to the underlying parameter functions governing their behavior. This result is novel, even in the general realm of inverse problems for nonlinear PDEs. By enabling agents to distill crucial insights from observed data and unveil intricate hidden structures and unknown states within MFG systems, our approach surmounts a significant obstacle, enhancing the applicability of MFGs in real-world scenarios. This advancement not only enriches our understanding of MFG dynamics but also broadens the scope for their practical deployment in various contexts.
Paper Structure (13 sections, 5 theorems, 77 equations)

This paper contains 13 sections, 5 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Mathematical Setup
  3. Technical Developments and Discussion
  4. Preliminaries
  5. Admissible class
  6. Main unique identifiability results
  7. Well-posedness of the forward problems
  8. High order linearization
  9. Linearization in probability space
  10. High order linearized systems
  11. Proof of Main Theorem
  12. Recovery of Stationary Solution and Hamiltonian
  13. Recovery of Running Cost

Key Result

Theorem 2.5

For $i=1,2$, let $A_i$ be an admissible Riemannian metric and $F_i\in\mathcal{A}$ such that $U_i=(u_{0,i},m_{0,i})\in [C^{2+\alpha}(\Omega)]^2$ is a stable stationary solution of the following system: such that the Hamiltonian $\mathcal{H}_i=[\nabla u]^TA_i\nabla u$ is non-degenerate at the stationary state $U_i$. Let $\mathcal{M}_{F_i,A_i,U_i}$ be the associated measurements. Suppose Then, one

Theorems & Definitions (13)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5
  • Theorem 3.1
  • proof
  • Definition 4.1
  • Definition 4.2
  • Theorem 5.1
  • ...and 3 more