Fourier approximation methods for first-order nonlocal mean-field games

Abstract

In this note, we develop Fourier approximation methods for the solutions of first-order nonlocal mean-field games (MFG) systems. Using Fourier expansion techniques, we approximate a given MFG system by a simpler one that is equivalent to a convex optimization problem over a finite-dimensional subspace of continuous curves. Furthermore, we perform a time-discretization for this optimization problem and arrive at a finite-dimensional saddle point problem. Finally, we solve this saddle-point problem by a variant of a primal dual hybrid gradient method.

A machine learning framework for solving high-dimensional mean field game and mean field control problems

“…However, a crucial computational challenge in applying these techniques in MFG stems from the density estimation, which is critical, e.g., to compute the congestion-cost incurred by an individual agent. In [73], the authors overcome this difficulty for non-local interactions by passing to Fourier coordinates in the congestion term and thus avoiding the density estimation. Our neural network parameterization aims to reduce the computational effort and memory footprint of the methods in [73] and provides a tractable way to estimate the population density.…”

“…In this paper, we tackle the curse of dimensionality in two steps. First, extending the approach in [73], we solve the continuity equation and compute all other terms involved in the MFG using Lagrangian coordinates. In practice, this requires computation of characteristic curves and the Jacobian determinant of the induced transformation; both terms can be inferred from the potential.…”

Splitting methods for a class of non-potential mean field games

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We extend the methods from [39,37] to a class of non-potential mean-field game (MFG) systems with mixed couplings. Up to now, splitting methods have been applied to potential MFG systems that can be cast as convex-concave saddle-point problems.

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“…Hence, this procedure is prone to high memory and computational costs, especially on a fine grid. In [39,37], the authors remedy this issue by passing to Fourier coordinates in the nonlocal terms. Relying on approximations of K, S in a suitable basis, they approximate nonlocal terms with a relatively small number of parameters independent of the grid-size.…”

“…Systems considered in [39,37] have only nonlocal interactions and are potential. Here, we extend these results to systems of the form (1) that are non-potential in general and contain both local and nonlocal interactions.…”