# Neural networks-based backward scheme for fully nonlinear PDEs

arXiv: https://arxiv.org/abs/1908.00412 · HTML: https://mlsanity.com/paper/1908.00412/

## Key facts

- Cited by 23 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 31 Jul 2019.

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## Abstract (arXiv)

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 citeHPW19 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`ere equation and Hamilton-Jacobi-Bellman equation in portfolio optimization.