Neural Solvers for Mean-Field Games with Equilibrium Certificates

Large-scale multi-agent systems can be modeled by mean-field games (MFGs), in which individual decision-making and population evolution are coupled through an equilibrium between individual optimality, where the representative agent's policy is a best response to the population distribution, and population consistency, where the distribution is generated by the resulting policy. Solving such an equilibrium typically requires repeated forward–backward iterations to a fixed point for each new environment, objective, or population state. Neural solvers can reduce per-instance computation by learn a mapping from instances to solutions, but their predicted policies and populations need not form an equilibrium. We introduce Certified Equilibrium Neural Operator (CENO), a value-only neural MFG solver that couples solution construction with equilibrium certification. CENO predicts only candidate value functions. A Gibbs policy is derived from the predicted values and the population is then propagated through the policy-induced dynamics, ensuring population consistency by construction. To assess individual optimality, we derive an equilibrium certificate from the Bellman defect of the predicted values. The certificate upper-bounds the exploitability of the resulting policy-population pair, without computing a reference equilibrium or an additional best-response solve. The certificate also provides a label-free training objective, enabling the neural operator to learn across MFG instances without equilibrium solution labels. Under monotone population interactions, the certificate further bounds the error of the induced population trajectory relative to equilibrium, extending the guarantee from individual optimality to population-level accuracy. Experiments on a game-theoretic benchmark and three multi-agent population tasks evaluate solution quality and illustrate population-level coordination behaviors.