Deep neural network approximation for high-dimensional parabolic Hamilton-Jacobi-Bellman equations

Deep neural networks (DNNs) and optimization techniques are increasingly used in the search for approximation methods for high-dimensional problems in scientific computing. In the case of a partial differential equation (PDE), the approximation of the solution is challenging, since grid based approaches are burdened by the so called curse of dimension [2]. By this we mean that in order to achieve an accuracy ε 0, the computational cost has asymptotically an exponential dependence with respect to the dimension of the spatial domain of the PDE. As a result practical computations on dimensions larger than 4 or 6 become already very computationally intensive. A particularly relevant example of high dimensional PDEs are Hamilton-Jacobi-Bellman (HJB) equations associated with stochastic optimal control problems which are relevant for example in engineering and financial modelling.