Deep backward schemes for high-dimensional nonlinear PDEs
| Autor: | Côme Huré, Huyên Pham, Xavier Warin |
|---|---|
| Přispěvatelé: | Laboratoire de Probabilités, Statistiques et Modélisations (LPSM (UMR_8001)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Université Paris Diderot - Paris 7 (UPD7), Laboratoire de Finance des Marchés d'Energie (FiME Lab), EDF R&D (EDF R&D), EDF (EDF)-EDF (EDF)-CREST-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), EDF (EDF), FiME, Laboratoire de Finance des Marchés de l'Energie, and the 'Finance and Sustainable Development' EDF - CACIB Chair. |
| Rok vydání: | 2019 |
| Předmět: |
FOS: Computer and information sciences
Machine Learning (stat.ML) 010103 numerical & computational mathematics [INFO.INFO-NE]Computer Science [cs]/Neural and Evolutionary Computing [cs.NE] 01 natural sciences Stochastic differential equation [STAT.ML]Statistics [stat]/Machine Learning [stat.ML] Statistics - Machine Learning Deep neural networks FOS: Mathematics Applied mathematics Optimal stopping Mathematics - Numerical Analysis Neural and Evolutionary Computing (cs.NE) 0101 mathematics nonlinear PDEs in high dimension Mathematics - Optimization and Control Mathematics Algebra and Number Theory Partial differential equation Artificial neural network business.industry Applied Mathematics Deep learning Probability (math.PR) Computer Science - Neural and Evolutionary Computing Numerical Analysis (math.NA) optimal stopping problem 010101 applied mathematics Maxima and minima [MATH.MATH-PR]Mathematics [math]/Probability [math.PR] Computational Mathematics Nonlinear system Optimization and Control (math.OC) Variational inequality backward stochastic differential equations Artificial intelligence [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC] business Mathematics - Probability [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA] |
| DOI: | 10.48550/arxiv.1902.01599 |
| Popis: | 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. Comment: 34 pages |
| Databáze: | OpenAIRE |
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