Deep backward schemes for high-dimensional nonlinear PDEs

Abstract

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 \cite{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 \cite{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.

arXiv comment34 pages

References in corpus (5)

• DGM: A deep learning algorithm for solving partial differential equations
• Solving high-dimensional partial differential equations using deep learning
• A regression-based Monte Carlo method to solve backward stochastic differential equations
• Convergence of the Deep BSDE Method for Coupled FBSDEs
• Monte Carlo for high-dimensional degenerated Semi Linear and Full Non Linear PDEs

Cited by in corpus (5)

• Rectified deep neural networks overcome the curse of dimensionality for nonsmooth value functions in zero-sum games of nonlinear stiff systems
• Overcoming the curse of dimensionality in the numerical approximation of high-dimensional semilinear elliptic partial differential equations
• Multilevel Picard approximations for high-dimensional semilinear second-order PDEs with Lipschitz nonlinearities
• Lax-Oleinik-type formulas and efficient algorithms for certain high-dimensional optimal control problems
• Hopf-type representation formulas and efficient algorithms for certain high-dimensional optimal control problems