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：深度学习的最新进展催生了许多能够精确高效求解高维平均场博弈的创新框架。然而，这些方法仅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，限制了其实用性。为克服这一局限，我们开发了一种学习平均场博弈解算子的新型框架。该模型以平均场博弈实例为输入，通过单次前向传播输出其解。为确保所提出的参数化方法适用于算子学习，我们引入并证明了模型采样不变性的概念，确立了其在采样极限下收敛到连续算子的性质。该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进行训练，显著降低了现有算子学习方法中构建训练数据集的计算开销。我们通过在具有不同复杂度和维度的合成数据集与真实数据集上的实验验证了该框架的鲁棒性。