Unsupervised Solution Operator Learning for Mean-Field Games via Sampling-Invariant Parametrizations

MFGs arise as infinite-population limits of symmetric noncooperative games. Their equilibria are commonly expressed through coupled Hamilton–Jacobi–Bellman and Fokker–Planck equations. Finite-difference, finite-element, and grid solvers become infeasible in high dimensions. Neural approaches, including value-function networks, normalizing flows, adversarial training, and Schrödinger-Bridge methods, remove spatial grids but usually optimize a separate model for each instance. Neural operators such as Fourier neural operators learn PDE solution maps, yet typically assume fixed discretizations and supervised solutions.

The target is G*(P0,P1), mapping initial and terminal distributions to optimal trajectories F*. Distributions are infinite-dimensional; samples have arbitrary order and potentially variable cardinality; and supervised trajectories are expensive because each label requires solving an MFG. Hypernetworks avoid labels but may need to predict millions of trajectory-network parameters, making accurate, scalable prediction difficult. A practical method must therefore be grid-free, sample-based, nonlocal, label-free, and reusable.

• �� Directly optimize the operator G rather than predicting parameters for a separate solver.
• �� Represent P0 and P1 by iid point clouds, avoiding fixed spatial grids.
• �� Use an unbiased MMD estimator for terminal distribution matching from samples alone.
• �� Combine pointwise MLPs with multi-head attention to model individual and collective behavior.
• �� Define and prove sampling invariance, linking finite samples to a continuous operator.
• �� When I≡0, learn a time-independent Monge map T, which automatically satisfies the initial condition.

• �� Trajectory formulation: F(x,t) tracks an agent beginning at x; the population at time t is F(·,t)*P0.
• �� Energy: minimize λL∫||∂tF||²p0, plus λI∫I(F*P0) and λMM(F(·,T)*P0).
• �� Input: draw n iid samples from each distribution and stack them as X0,X1∈R^(n×d).
• �� Terminal loss: use unbiased MMD² with Gaussian or Laplacian kernels; its estimator converges at O((M+N)^−1/2).
• �� Architecture: multi-head attention computes softmax(QKᵀ/√dk)V, followed by layer normalization and a pointwise MLP.
• �� Optimization: minimize expected energy over an instance distribution μ; inference requires one forward pass and no weight update.

The evaluation spans synthetic and realistic distributions, changing dimensionality, complexity, and sampling representations. MNIST digits 0, 6, 5, and 4 are used as distributions through 1053 samples from pixel-value densities. The principal baseline is the single-instance neural MFG solver, which independently optimizes for each configuration. Evaluation considers terminal distribution agreement, trajectory or transport cost, qualitative solution structure, and inference time. The excerpt indicates architecture and sampling studies, but does not provide a complete table of hyperparameters or every ablation value.

The proposed learner produces accurate and visually intuitive solutions across the reported synthetic and realistic tasks. Its main quantitative claim is a reduction of more than five orders of magnitude in the time required to solve a new MFG compared with single-instance neural solvers, without compromising computed-solution quality. It accepts varying finite sample representations and has a continuous-limit justification through sampling invariance. MMD is attractive for high-dimensional samples because its stated convergence rate is dimension-independent, although the excerpt does not include exact error bars for each dataset.

Drone and robot formation planners can input sampled starting and destination positions, together with obstacle or interaction costs, and receive coordinated trajectories immediately. Traffic-flow, logistics, economic-migration, and industrial-scheduling systems can reuse the same operator for many scenarios. Deployment requires representative training instances, sample access to initial and terminal populations, and a meaningful cost functional. Once trained, the operator can support rapid replanning, scenario screening, and interactive simulation.

The framework assumes deterministic continuous dynamics and a variational formulation; it does not automatically solve stochastic diffusion games, non-variational equilibria, or multi-equilibrium selection. Attention can scale quadratically with sample count, and MMD depends on kernel and bandwidth choices. Unsupervised training removes labels but not optimization cost, while distribution shift may yield unsafe or physically implausible trajectories. Important next steps include stochastic extensions, sparse attention, finite-sample generalization bounds, safety constraints, and large-scale real-world validation.