Solving high-dimensional partial differential equations using deep learning

Han, Jiequn; Jentzen, Arnulf; E, Weinan

doi:10.1073/pnas.1718942115

Title: Solving high-dimensional partial differential equations using deep learning

Journal Article · Mon Aug 06 00:00:00 EDT 2018 · Proceedings of the National Academy of Sciences of the United States of America

DOI:https://doi.org/10.1073/pnas.1718942115· OSTI ID:1540276

^[1]; Jentzen, Arnulf ^[2]; E, Weinan ^[3]

Princeton Univ., Princeton, NJ (United States)
ETH Zurich, Zurich (Switzerland)
Princeton Univ., Princeton, NJ (United States); Beijing Inst. of Big Data Research, Beijing (China)

Developing algorithms for solving high-dimensional partial differential equations (PDEs) has been an exceedingly difficult task for a long time, due to the notoriously difficult problem known as the “curse of dimensionality.” This paper introduces a deep learning-based approach that can handle general high-dimensional parabolic PDEs. To this end, the PDEs are reformulated using backward stochastic differential equations and the gradient of the unknown solution is approximated by neural networks, very much in the spirit of deep reinforcement learning with the gradient acting as the policy function. Numerical results on examples including the nonlinear Black–Scholes equation, the Hamilton–Jacobi–Bellman equation, and the Allen–Cahn equation suggest that the proposed algorithm is quite effective in high dimensions, in terms of both accuracy and cost. Furthermore, this opens up possibilities in economics, finance, operational research, and physics, by considering all participating agents, assets, resources, or particles together at the same time, instead of making ad hoc assumptions on their interrelationships.

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Research Organization:: Princeton Univ., NJ (United States)

Sponsoring Organization:: USDOE Office of Science (SC)

Grant/Contract Number:: SC0009248

OSTI ID:: 1540276

Journal Information:: Proceedings of the National Academy of Sciences of the United States of America, Vol. 115, Issue 34; ISSN 0027-8424

Publisher:: National Academy of SciencesCopyright Statement

Country of Publication:: United States

Language:: English

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Cited by: 638 works

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