Project Details
Projekt
Deep neural networks overcome the curse of dimensionality in the numerical approximation of stochastic control problems and of semilinear Poisson equations
Subject Area
Mathematics
Term
since 2021
Project identifier
Deutsche Forschungsgemeinschaft (DFG) - Project number 464101154
Partial differential equations (PDEs) are a key tool in the modeling of many real world phenomena. Several PDEs that arise in financial engineering, economics, quantum mechanics or statistical physics are nonlinear, high-dimensional, and cannot be solved explicitly. It is a highly challenging task to provably solve such high-dimensional nonlinear PDEs approximately without suffering from the so-called curse of dimensionality. Deep neural networks (DNNs) and other deep learning-based methods have recently been applied very successfully to a number of computational problems. In particular, simulations indicate that algorithms based on DNNs overcome the curse of dimensionality in the numerical approximation of solutions of certain nonlinear PDEs. For certain linear and nonlinear PDEs this has also been proven mathematically. The key goal of this project is to rigorously prove for the first time that DNNs overcome the curse of dimensionality for a class of nonlinear PDEs arising from stochastic control problems and for a class of semilinear Poisson equations with Dirichlet boundary conditions.
DFG Programme
Priority Programmes
Subproject of
SPP 2298:
Theoretical Foundations of Deep Learning
