Simultaneously Solving Infinitely Many LQ Mean Field Games In Hilbert Spaces: The Power of Neural Operators

TL;DR

This paper introduces neural operators trained to efficiently approximate solutions for a wide class of infinite-dimensional linear-quadratic mean field games, providing a scalable and reliable approach for solving numerous related problems.

Contribution

It develops a neural operator framework with theoretical guarantees for solving infinite-dimensional LQ MFGs, enabling generalization across problem variants with controlled complexity.

Findings

Neural operators can reliably approximate solutions to unseen LQ MFGs.

The method provides statistical guarantees even in infinite-dimensional settings.

Sample complexity bounds are established for neural operators in this context.

Abstract

Traditional mean-field game (MFG) solvers operate on an instance-by-instance basis, which becomes infeasible when many related problems must be solved (e.g., for seeking a robust description of the solution under perturbations of the dynamics or utilities, or in settings involving continuum-parameterized agents.). We overcome this by training neural operators (NOs) to learn the rules-to-equilibrium map from the problem data (``rules'': dynamics and cost functionals) of LQ MFGs defined on separable Hilbert spaces to the corresponding equilibrium strategy. Our main result is a statistical guarantee: an NO trained on a small number of randomly sampled rules reliably solves unseen LQ MFG variants, even in infinite-dimensional settings. The number of NO parameters needed remains controlled under appropriate rule sampling during training. Our guarantee follows from three results: (i) local-Lipschitz estimates for the highly nonlinear rules-to-equilibrium map; (ii) a universal approximation theorem using NOs with a prespecified Lipschitz regularity (unlike traditional NO results where the NO's Lipschitz constant can diverge as the approximation error vanishes); and (iii) new sample-complexity bounds for $L$-Lipschitz learners in infinite dimensions, directly applicable as the Lipschitz constants of our approximating NOs are controlled in (ii).