A Neural Score-Based Probability Flows Approach for Second-Order Mean-Field Games

Second-order mean-field games (MFGs) model strategic interactions among large populations with stochastic dynamics, but their coupled forward-backward PDE systems are difficult to solve in high dimensions. We develop a unified probability-flow framework for general and potential second-order MFGs using score-based neural ODEs. Building on the flow-velocity formulation based on the Nelson's approach, we rewrite the Fokker-Planck equation as a continuity equation whose velocity depends on the density score. Under suitable regularity conditions, the resulting deterministic probability flow and the controlled diffusion have identical time marginals, yielding a grid-free Lagrangian formulation that avoids spatial discretization. We study an analytical interbank benchmark, synthetic transport including a 1000-dimensional case, nonlinear 12D quadcopter control, and unpaired CelebA image translation task at latent space. The experiments recover analytical solutions in benchmark settings and demonstrate the framework's feasibility for high-dimensional transport and nonlinear control, without claiming that finite training necessarily converges to an MFG equilibrium.