Neural networks-based backward scheme for fully nonlinear PDEs
This work addresses computational challenges in high-dimensional PDEs for applications like control and optimization, but it is incremental as it extends an existing method to fully nonlinear cases.
The paper tackles solving high-dimensional fully nonlinear PDEs by proposing a neural network-based backward scheme that simultaneously estimates the solution and its gradient, extending a previous method for semi-linear PDEs, with numerical tests demonstrating performance on examples like Monge-Ampère and Hamilton-Jacobi-Bellman equations.
We propose a numerical method for solving high dimensional fully nonlinear partial differential equations (PDEs). Our algorithm estimates simultaneously by backward time induction the solution and its gradient by multi-layer neural networks, while the Hessian is approximated by automatic differentiation of the gradient at previous step. This methodology extends to the fully nonlinear case the approach recently proposed in \cite{HPW19} for semi-linear PDEs. Numerical tests illustrate the performance and accuracy of our method on several examples in high dimension with nonlinearity on the Hessian term including a linear quadratic control problem with control on the diffusion coefficient, Monge-Amp{è}re equation and Hamilton-Jacobi-Bellman equation in portfolio optimization.