Solving high-dimensional partial differential equations using deep learning
Abstract
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.
- Authors:
-
[1];
- Princeton Univ., Princeton, NJ (United States)
- ETH Zurich, Zurich (Switzerland)
- Princeton Univ., Princeton, NJ (United States); Beijing Inst. of Big Data Research, Beijing (China)
- Publication Date:
- Research Org.:
- Princeton Univ., NJ (United States)
- Sponsoring Org.:
- USDOE Office of Science (SC)
- OSTI Identifier:
- 1540276
- Grant/Contract Number:
- SC0009248
- Resource Type:
- Accepted Manuscript
- Journal Name:
- Proceedings of the National Academy of Sciences of the United States of America
- Additional Journal Information:
- Journal Volume: 115; Journal Issue: 34; Journal ID: ISSN 0027-8424
- Publisher:
- National Academy of Sciences
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 97 MATHEMATICS AND COMPUTING; Science & Technology; Other Topics; partial differential equations; backward stochastic differential equations; high dimension; deep learning; Feynman–Kac
Citation Formats
Han, Jiequn, Jentzen, Arnulf, and E, Weinan. Solving high-dimensional partial differential equations using deep learning. United States: N. p., 2018.
Web. doi:10.1073/pnas.1718942115.
Han, Jiequn, Jentzen, Arnulf, & E, Weinan. Solving high-dimensional partial differential equations using deep learning. United States. https://doi.org/10.1073/pnas.1718942115
Han, Jiequn, Jentzen, Arnulf, and E, Weinan. Mon .
"Solving high-dimensional partial differential equations using deep learning". United States. https://doi.org/10.1073/pnas.1718942115. https://www.osti.gov/servlets/purl/1540276.
@article{osti_1540276,
title = {Solving high-dimensional partial differential equations using deep learning},
author = {Han, Jiequn and Jentzen, Arnulf and E, Weinan},
abstractNote = {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.},
doi = {10.1073/pnas.1718942115},
journal = {Proceedings of the National Academy of Sciences of the United States of America},
number = 34,
volume = 115,
place = {United States},
year = {2018},
month = {8}
}
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