Spectral Latent Methods for High-Dimensional Mean-Field Games

We develop a spectral Latent method for high-dimensional mean-field games (MFGs). Our starting point is the observation that the MFG system — the coupled Hamilton-Jacobi-Bellman and Fokker-Planck equations — is the Euler-Lagrange system of the grade-1 Latent truncation of the underlying \(N\)-player game (Nagy, 2026b). This interpretation yields three results. First, we prove that if the Hamiltonian \(H\) and coupling \(F\) are \(\rho\)-analytic, the value function \(v(t,x)\) and population distribution \(m(t,x)\) each have Latent representations with \(N^ = O(\log(1/\varepsilon)/\log\rho)\) modes, independent of the state space dimension \(d\). The coupled PDE system reduces to an ODE system of size \(2N^\), solvable in \(O(N^{*3})\) time per timestep. Second, we derive Latent finite-player corrections: grade-2 corrections capture the leading \(O(1/N)\) effects that the mean-field limit discards, giving the first systematic beyond-mean-field expansion with explicit convergence rates. Third, we benchmark the Latent method against tensor-train decomposition, particle-based flow matching, and random Fourier features on three test problems — a linear-quadratic MFG (exact solution available), a congestion game in \(d = 50\) dimensions, and a systemic risk model with heterogeneous banks. The Latent method achieves the target accuracy with \(5\)–\(50\times\) fewer degrees of freedom than tensor-train and provides convergence guarantees absent from neural methods.