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Hamilton-Jacobi PDEs and optimal control problems are widely used in many practical problems in control engineering, physics, financial mathematics, and machine learning. For instance, controlling an autonomous system is important in everyday modern life, and it requires a scalable, robust, efficient, and data-driven algorithm for solving optimal control problems. Traditional grid-based numerical methods cannot solve these high-dimensional problems, because they are not scalable and may suffer from the curse of dimensionality. To overcome the curse of dimensionality, we developed several neural network methods for solving high-dimensional Hamilton-Jacobi PDEs and optimal control problems. This talk will contain two parts. In the first part, I will talk about SympOCNet method for solving multi-agent path planning problems, which solves a 512-dimensional path planning problem with training time of less than 1.5 hours. In the second part, I will show several neural network architectures with solid theoretical guarantees for solving certain classes of high-dimensional Hamilton-Jacobi PDEs. By leveraging dedicated efficient hardware designed for neural networks, these methods have the potential for real-time applications in the future. These are joint works with Jerome Darbon, George Em Karniadakis, Peter M. Dower, Gabriel P. Langlois, and Zhen Zhang.