Neural networks-based backward scheme for fully nonlinear PDEs

Huyên Pham (), Xavier Warin and Maximilien Germain
Additional contact information
Huyên Pham: LPSM, Université de Paris, CREST-ENSAE & FiME
Xavier Warin: EDF R&D & FiME
Maximilien Germain: EDF R&D, LPSM, Université de Paris

Partial Differential Equations and Applications, 2021, vol. 2, issue 1, 1-24

Abstract: Abstract 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 Huré et al. (Math Comput 89(324):1547–1579, 2020) for semi-linear PDEs. Numerical tests illustrate the performance and accuracy of our method on several examples in high dimension with non-linearity on the Hessian term including a linear quadratic control problem with control on the diffusion coefficient, Monge-Ampère equation and Hamilton–Jacobi–Bellman equation in portfolio optimization.

Keywords: Neural networks; Fully nonlinear PDEs in high dimension; Backward scheme; 60H35; 65C20; 65M12 (search for similar items in EconPapers)
Date: 2021
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (9)

Downloads: (external link)
http://link.springer.com/10.1007/s42985-020-00062-8 Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:pardea:v:2:y:2021:i:1:d:10.1007_s42985-020-00062-8

Ordering information: This journal article can be ordered from
https://www.springer.com/journal/42985/

DOI: 10.1007/s42985-020-00062-8

Access Statistics for this article

Partial Differential Equations and Applications is currently edited by Zhitao Zhang

More articles in Partial Differential Equations and Applications from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().