Explore a comprehensive lecture on solving high-dimensional Hamilton-Jacobi equations using generalized Hopf-Lax formulas and machine learning techniques. Delve into the Lagrangian formulation, microscopic models, and Hamiltonian equations presented by Stanley Osher and Samy Wu Fung from UCLA. Examine existing work, including Monte Carlo methods and machine learning frameworks, while learning about record-based architecture, gradient trace cost, and improved reconstructions. Discover how to enforce physics in solutions and apply gradient penalization. Analyze numerical results and gain valuable insights into this complex topic in applied mathematics and control theory.