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 demands extensive computational time per instance, limiting practicality. To overcome this, we develop a novel framework to learn the MFG solution operator. Our model takes a MFG instances as input and output their solutions with one forward pass. To ensure the proposed parametrization is well-suited for operator learning, we introduce and prove the notion of sampling invariance for our model, establishing its convergence to a continuous operator in the sampling limit. Our method features 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 supervised labels, significantly reducing the computational 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.

翻译：近期深度学习进展催生了诸多创新框架，能够准确高效地求解高维平均场博弈问题。然而，这些方法局限于求解单实例平均场博弈，且每个实例需要大量计算时间，限制了其实际应用。为此，我们提出一种新型框架来学习平均场博弈解算算子。该模型将平均场博弈实例作为输入，通过一次前向传播即可输出其解。为确保所提出的参数化适用于算子学习，我们引入并证明了模型"采样不变性"概念，建立了其在采样极限下收敛为连续算子的理论。本方法具有两大关键优势：首先，它无需离散化，特别适用于学习高维平均场博弈的算子；其次，该模型可在无需监督标签的条件下训练，显著降低了现有算子学习方法中训练数据集生成的计算开销。我们在不同复杂度与维度的合成及真实数据集上验证了框架的鲁棒性。