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Determining state space anomalies in mean field games

Hongyu Liu, Catharine W. K. Lo

TL;DR

This work addresses the inverse boundary problem for stationary mean field games with state-space anomalies, aiming to uniquely identify both the anomaly \(\omega\) and its discontinuous parameter configurations in the Hamiltonian and running cost from boundary data. It introduces a higher-order linearisation framework combined with microlocal analysis of corner singularities to recover \(\omega\) and either the Hamiltonian jump \(H\) or the running cost jump \(F\), under admissibility conditions that permit discontinuities and relax analytic assumptions on \(F\). The authors enforce positivity of the population density \(m\) via selective boundary data and show that an infinite number of measurements is required due to nonlinearity. The results extend inverse problems for nonlinear coupled PDEs and have potential applications in traffic flow, economics, and epidemic modeling by enabling detection of obstructions or anomalies in the state space from boundary observations.

Abstract

In this paper, we are concerned with the inverse problem of determining anomalies in the state space associated with the stationary mean field game (MFG) system. We establish novel unique identifiability results for the intrinsic structure of these anomalies in mean field games systems, including their topological structure and parameter configurations, in several general scenarios of practical interest, including traffic flow, market economics and epidemics. To the best of our knowledge, this is the first work that considers anomalies in the state space for the nonlinear coupled MFG system.

Determining state space anomalies in mean field games

TL;DR

This work addresses the inverse boundary problem for stationary mean field games with state-space anomalies, aiming to uniquely identify both the anomaly and its discontinuous parameter configurations in the Hamiltonian and running cost from boundary data. It introduces a higher-order linearisation framework combined with microlocal analysis of corner singularities to recover and either the Hamiltonian jump or the running cost jump , under admissibility conditions that permit discontinuities and relax analytic assumptions on . The authors enforce positivity of the population density via selective boundary data and show that an infinite number of measurements is required due to nonlinearity. The results extend inverse problems for nonlinear coupled PDEs and have potential applications in traffic flow, economics, and epidemic modeling by enabling detection of obstructions or anomalies in the state space from boundary observations.

Abstract

In this paper, we are concerned with the inverse problem of determining anomalies in the state space associated with the stationary mean field game (MFG) system. We establish novel unique identifiability results for the intrinsic structure of these anomalies in mean field games systems, including their topological structure and parameter configurations, in several general scenarios of practical interest, including traffic flow, market economics and epidemics. To the best of our knowledge, this is the first work that considers anomalies in the state space for the nonlinear coupled MFG system.
Paper Structure (11 sections, 7 theorems, 82 equations)

This paper contains 11 sections, 7 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. Problem Setup and Background
  3. Technical Developments and Discussion
  4. Preliminaries
  5. Notations and Basic Setting
  6. Geometrical Setup
  7. Main Results
  8. Higher-Order Linearisation
  9. Proof of Main Results
  10. A Corner Singularity due to Different Hamiltonians
  11. A Corner Singularity due to Different Running Costs $F$

Key Result

Theorem 1.1

Consider a certain general scenario where there exists an anomalous polyhedral inclusion $\omega\Subset\Omega$, such that $H$, $\lambda$ and $F$ has a jump discontinuity given by their adherence to certain admissibilility properties. Then $\omega$ and its parameter configurations $H$ and $F$ are uni

Theorems & Definitions (15)

  • Theorem 1.1
  • Theorem 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.6
  • Definition 2.7
  • Theorem 2.8
  • ...and 5 more