Unsupervised solution operator learning for mean-field games

Abstract

Recent advances in deep learning has witnessed many innovative frameworks that solve high dimensional mean-field games (MFG) accurately and efficiently. These methods, however, are restricted to solving single-instance MFG and require extensive computational time per instance, limiting practicality. To overcome this, we develop a novel framework for learning the MFG solution operator. Our model takes MFG instances as input and outputs their solutions with one forward pass. To ensure that the proposed parametrization is well-suited for operator learning, we introduce and prove the notion of sampling consistency for our model, establishing its convergence to a continuous operator in the sampling limit. Our method has two key advantages. First, it is discretization-free, making it particularly suitable for learning operators of high-dimensional MFGs. Secondly, it can be trained without the need for access to superised labels, significantly reducing the overhead associated with creating training datasets in existing operator learning methods. We test our framework on synthetic and realistic datasets with varying complexity and dimensionality to substantiate its robustness. Compared to single-instance neural MFG solvers, our approach reduces the time to solve a MFG problem by more than five orders of magnitude without compromising the quality of computed solutions.