Algorithm 1 GAN-based algorithm for complex-scenario MFGs

Require:$\sigma$ diffusion parameter, H Hamiltonian, g terminal cost, f interaction term. Require: Initialize neural networks $N_{\omega }$ and $N_{\theta }$, batch size B. Require: Set ${\varphi }_{\omega }$ and $G_{\theta }$ as in (29). while not converged do          train ${\varphi }_{\omega }$:          Sample batch ${\left\{\left({\mathbf{z}}_b,t_b\right)\right\}}_{b=1}^B$ where ${\mathbf{z}}_b\sim {\rho }_0$ and $t_b\sim \mathrm{Unif}(0,T)$          Obtaining generated data ${\left\{\left({\mathbf{x}}_b,t_b\right)\right\}}_{b=1}^B,{\mathbf{x}}_b\leftarrow G_{\theta }\left({\mathbf{z}}_b,t_b\right)$.          Update discriminator parameters $\omega$ to minimize ${\ell }_{\mathrm{total}}={\ell }_0+{\ell }_t+{\ell }_{\mathrm{HJB}}$                   $\omega \leftarrow \omega -{\eta }_1\nabla {\ell }_{\mathrm{total}}\left(\omega \right)$          train $G_{\theta }:$          Sample batch ${\left\{\left({\mathbf{z}}_b,t_b\right)\right\}}_{b=1}^B$ where ${\mathbf{z}}_b\sim {\rho }_0$ and $t_b\sim \mathrm{Unif}(0,T)$          Update generator parameters $\theta$ to minimize ${\zeta }_t$                   $\theta \leftarrow \theta -{\eta }_2\nabla {\zeta }_t\left(\theta \right)$ end while