High Order Tensor-Train-Based Schemes for High-Dimensional Mean Field Games

Abstract. We introduce a fully discrete scheme to solve a class of high-dimensional mean field games systems. Our approach couples semi-Lagrangian (SL) time discretizations with Tensor-Train (TT) decompositions to tame the curse of dimensionality. By reformulating the classical Hamilton–Jacobi–Bellman and Fokker–Planck equations as a sequence of advection–diffusion–reaction subproblems within a smoothed policy iteration, we construct both first and second order in time SL schemes. The TT format and appropriate quadrature rules reduce storage and computational cost from exponential to polynomial in the dimension. Numerical experiments demonstrate that our TT-accelerated SL methods achieve their theoretical convergence rates, exhibit modest growth in memory usage and runtime with dimension, and significantly outperform grid-based SL in accuracy per CPU-second.