Some machine learning schemes for high-dimensional nonlinear PDEs

02/05/2019
by   Côme Huré, et al.
0

We propose new machine learning schemes for solving high dimensional nonlinear partial differential equations (PDEs). Relying on the classical backward stochastic differential equation (BSDE) representation of PDEs, our algorithms estimate simultaneously the solution and its gradient by deep neural networks. These approximations are performed at each time step from the minimization of loss functions defined recursively by backward induction. The methodology is extended to variational inequalities arising in optimal stopping problems. We analyze the convergence of the deep learning schemes and provide error estimates in terms of the universal approximation of neural networks. Numerical results show that our algorithms give very good results till dimension 50 (and certainly above), for both PDEs and variational inequalities problems. For the PDEs resolution, our results are very similar to those obtained by the recent method in weinan2017deep when the latter converges to the right solution or does not diverge. Numerical tests indicate that the proposed methods are not stuck in poor local minimaas it can be the case with the algorithm designed in weinan2017deep, and no divergence is experienced. The only limitation seems to be due to the inability of the considered deep neural networks to represent a solution with a too complex structure in high dimension.

READ FULL TEXT
research
07/31/2019

Neural networks-based backward scheme for fully nonlinear PDEs

We propose a numerical method for solving high dimensional fully nonline...
research
08/28/2023

Deep multi-step mixed algorithm for high dimensional non-linear PDEs and associated BSDEs

We propose a new multistep deep learning-based algorithm for the resolut...
research
02/24/2021

Learning optimal multigrid smoothers via neural networks

Multigrid methods are one of the most efficient techniques for solving l...
research
12/24/2022

Deep Quadratic Hedging

We present a novel computational approach for quadratic hedging in a hig...
research
12/13/2018

Deep neural networks algorithms for stochastic control problems on finite horizon, Part 2: numerical applications

This paper presents several numerical applications of deep learning-base...
research
02/24/2021

A learning scheme by sparse grids and Picard approximations for semilinear parabolic PDEs

Relying on the classical connection between Backward Stochastic Differen...

Please sign up or login with your details

Forgot password? Click here to reset