Revista de Matemática Aplicada e Computacional

Deep Learning for Solving High Dimensional Hamilton Jacobi Bellman Equations

Abstract

Reuther Benjamin*

Traditional methods for solving HJB equations face challenges, especially when dealing with high-dimensional spaces. However, deep learning offers a promising approach to overcome these limitations. The HJB equation, named after William Rowan Hamilton, Carl Gustav Jacob Jacobi, and Richard Bellman, provides a necessary condition for optimality. It is a partial differential equation (PDE) that characterizes the value function of the control problem, essentially describing the evolution of the optimal cost as a function of time and state. Solving the HJB equation is crucial for determining the optimal policy or strategy in various applications. However, as the dimensionality of the problem increases, traditional numerical methods like finite difference methods or finite element methods become computationally infeasible due to the curse of dimensionality. This is where deep learning techniques, particularly neural networks, come into play.