Highlights: Deep Neural Networks for Mean-Field Stochastic Optimal Control with Applications to Mean Variance Optimization

 1. Three deep neural network-based algorithms are developed to solve a class of high-dimensional mean-field stochastic optimal control problems.

 2. The stochastic maximum principle is employed to reformulate the control problem as a variational optimization problem.

3. Deep neural network approaches are applied to solve the resulting variational problem efficiently.

4. The proposed algorithms are validated on the classical mean-variance portfolio optimization problem.

5. Numerical results demonstrate good accuracy and computational efficiency in high-dimensional settings, particularly when no analytical solution is available.