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Deep backward schemes for high-dimensional nonlinear PDEs


Authors: Côme Huré, Huyên Pham and Xavier Warin
Journal: Math. Comp. 89 (2020), 1547-1579
MSC (2010): Primary 60H35, 65C20, 65M12
DOI: https://doi.org/10.1090/mcom/3514
Published electronically: January 31, 2020
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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 [Commun. Math. Stat. 5 (2017), pp. 349-380] 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 minima as it can be the case with the algorithm designed in the work previously mentioned, 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.


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Additional Information

Côme Huré
Affiliation: Laboratoire de Probabilites Statistique et Modelisation, Paris-Diderot University, 75013 Paris, France
Email: hure@lpsm.paris

Huyên Pham
Affiliation: Laboratoire de Probabilites Statistique et Modelisation, Paris-Diderot University, 75013 Paris, France; and Center for Research in Economics and Statistics–ENSAE, Paris, France; and Finance et Marché de l’Energie, Paris, France
Email: pham@lpsm.paris

Xavier Warin
Affiliation: EDF R&D, site de Clamart, 1 avenue du General De Gaulle, 92141 Clamart, France; and Finance et Marché de l’Energie, Paris, France
Email: xavier.warin@edf.fr

DOI: https://doi.org/10.1090/mcom/3514
Keywords: Deep neural networks, nonlinear PDEs in high dimension, optimal stopping problem, backward stochastic differential equations.
Received by editor(s): February 6, 2019
Received by editor(s) in revised form: August 29, 2019, and November 8, 2019
Published electronically: January 31, 2020
Additional Notes: This work was supported by FiME, Laboratoire de Finance des Marchés de l’Energie, and the “Finance and Sustainable Development” EDF - CACIB Chair.
Article copyright: © Copyright 2020 American Mathematical Society