A Globally Convergent Flow for Time-Dependent Mean Field Games and a Solver-Agnostic Framework for Inverse Problems

Mean field games (MFGs) model the limit of large populations of strategically interacting agents, yet both forward and inverse problems remain challenging. For the forward problem, a difficulty is to design numerical methods with global convergence guarantees whose convergence does not depend on careful initialization. For the inverse problem, a difficulty is to decouple parameter optimization from the forward solver, so that parameter updates do not depend on implementation details, and the inverse method does not need to be reformulated when the forward solver is changed. We address both issues as follows. For the forward problem, we propose a monotone Hessian-Riemannian flow for time-dependent MFGs on the feasible manifold of densities. The flow preserves the positivity of densities and is proved to be globally convergent. For the inverse problem, we cast parameter estimation as an outer optimization problem over the unknown coefficients, with the MFG system solved in an inner step for each parameter value. For solving this problem, we consider an adjoint-based gradient method together with a Gauss-Newton acceleration. This leads to a solver-agnostic framework, in which parameter updates are computed by implicitly differentiating the discrete MFG equations satisfied by the converged MFG solution, rather than by differentiating through a particular forward solver. We demonstrate the approach on several stationary and time-dependent MFG examples, where the Gauss-Newton method consistently requires fewer outer iterations than gradient descent.

💡 Research Summary

This paper presents a unified methodological framework addressing two central challenges in the numerical analysis of Mean Field Games (MFGs): the forward problem of computing solutions reliably, and the inverse problem of estimating model parameters from data.

For the forward problem, the key difficulty lies in designing numerical methods with global convergence guarantees that do not depend on careful initialization. The authors propose a novel monotone Hessian-Riemannian flow (HRF) for time-dependent MFGs. Previous monotone flows evolved in the full ambient space and did not preserve the intrinsic constraints of the MFG system, notably the positivity of the density. While an HRF was developed for stationary MFGs to preserve positivity via a Riemannian metric derived from an entropy functional, extending it to time-dependent problems faced the obstacle of handling mixed initial-terminal conditions globally in time. The authors’ crucial innovation is a “discretize-then-flow” strategy. They first discretize the MFG system in time, which reduces the endpoint conditions to simple coordinate constraints on the first and last time slices. On this finite-dimensional system, they construct an HRF that intrinsically evolves on the manifold of positive densities with unit mass. This flow preserves positivity by design (as it can be rewritten as an evolution for ln(m)) and is proven to be globally convergent to the unique solution of the discretized MFG system under standard convexity and monotonicity assumptions, regardless of the initial guess.

For the inverse problem, the core challenge is to decouple parameter optimization from the specific forward solver used, enabling a “plug-and-play” framework where the inverse method does not need reformulation if the forward solver changes. The authors formulate parameter estimation (e.g., for the spatial cost V) as a bilevel optimization problem: an outer loop optimizes the parameters, and an inner loop solves the MFG equations for the current parameter value. The major contribution is a solver-agnostic gradient computation technique. Instead of differentiating through the iterations of a specific solver (like policy iteration), gradients for the outer loop are computed by implicitly differentiating the discrete MFG equations satisfied by the converged solution. This treats the forward solve as a black box that simply provides an accurate solution; the internal algorithm is irrelevant. This approach liberates the inverse problem from the implementation details of the forward solver. Within this framework, the authors implement both an adjoint-based first-order method and a Gauss-Newton acceleration for the outer optimization, demonstrating that the latter consistently requires fewer iterations.

In summary, the paper makes two significant contributions: 1) A globally convergent, positivity-preserving flow method for time-dependent MFGs that robustly computes solutions without sensitivity to initialization. 2) A flexible, solver-agnostic framework for MFG inverse problems that allows any accurate forward solver to be plugged in, using implicit differentiation and Gauss-Newton acceleration for efficient parameter recovery. The methods are demonstrated on both stationary and time-dependent examples, showcasing their effectiveness and practicality.