Travel-time tomography from mean field game dynamics

TL;DR

This paper introduces a novel mean field game framework for travel-time tomography, modeling active population dynamics to recover environmental features from observed data.

Contribution

It unifies propagation, observation, and inversion into a PDE-constrained model and proposes a two-stage inversion method with demonstrated stability.

Findings

Stable recovery of environment features under noise

Effective two-stage inversion pipeline combining diffusion and MFG refinement

Framework applicable to biological, vascular, and groundwater systems

Abstract

Travel-time tomography seeks to recover a hidden environment from external measurements generated by propagation through an anomalous region. Standard formulations treat propagation as passive, so the environment influences observations mainly by bending paths or changing travel times. Many collective systems do not operate in that regime: observed arrivals are shaped by strategic motion, congestion, and environmental costs. We formulate this active setting through mean field games, in which the unknown environment enters the running cost through a spatial cost field and observations are read from the resulting population dynamics. This yields three contributions. First, it places propagation, observation generation, and inversion within one PDE-constrained model. Second, it clarifies why the inverse problem differs structurally from passive tomography: kinetic, congestion, and environmental effects are coupled endogenously and appear through space- and time-dependent local weights rather than externally chosen global coefficients. Third, it introduces a two-stage inversion pipeline that combines diffusion-based initialization with full mean field game refinement, and numerical experiments show stable recovery under noise and across multichannel scene families. Taken together, these ingredients establish a foundational framework for a class of inverse problems in which data are generated by optimizing and interacting populations rather than by passive signals. The framework identifies the forward MFG model, admissible observation channels, structural mechanisms, and computational recovery route needed to study active tomography under collective dynamics. Such problems arise, among other settings, in biological transport, vascular flow, and subsurface groundwater dynamics.