We present a novel numerical method for solving McKean--Vlasov forward--backward stochastic differential equations (MV--FBSDEs) with common noise, combining Picard iterations, elicitability and deep learning. The key innovation involves elicitability to derive a pathwise loss function, enabling efficient training of neural networks to approximate both the backward process and the conditional expectations arising from common noise, without requiring computationally expensive nested Monte Carlo simulations. The mean-field interaction term is parameterized via a recurrent neural network trained to minimize an elicitable score, while the backward process is approximated through a hybrid feedforward and recurrent network representing the decoupling field. We validate the algorithm on a systemic-risk inter-bank borrowing and lending model, where analytical solutions exist, demonstrating accurate recovery of the true solution. We further extend the model to quantile-mediated interactions, showcasing the flexibility of the elicitability framework beyond conditional means or moments. Finally, we apply the method to a non-stationary Aiyagari--Bewley--Huggett economic growth model with endogenous interest rates, illustrating its applicability to complex mean-field games without closed-form solutions.

翻译：我们提出了一种求解含共同噪声的McKean-Vlasov正倒向随机微分方程（MV-FBSDEs）的新型数值方法，该方法融合了Picard迭代、可诱导性与深度学习。核心创新在于利用可诱导性导出路径损失函数，从而高效训练神经网络来逼近倒向过程及共同噪声引发的条件期望，避免了计算成本高昂的嵌套蒙特卡洛模拟。平均场相互作用项通过递归神经网络参数化，并以最小化可诱导得分为训练目标，而倒向过程则利用代表解耦场的混合前馈-递归网络进行近似。我们在存在解析解的系统性风险银行间借贷模型上验证了该算法，结果表明其能精确恢复真实解。进一步地，我们将模型扩展至分位数中介的相互作用，展示了可诱导性框架在条件均值或矩之外的灵活性。最后，我们将该方法应用于具有内生利率的非平稳Aiyagari-Bewley-Huggett经济增长模型，证明了其在无闭式解的复杂平均场博弈中的适用性。