Solving high-dimensional partial differential equations using deep learning

Jiequn Han, Arnulf Jentzen, E. Weinan

Research output: Contribution to journalArticlepeer-review

855 Scopus citations


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. 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.

Original languageEnglish (US)
Pages (from-to)8505-8510
Number of pages6
JournalProceedings of the National Academy of Sciences of the United States of America
Issue number34
StatePublished - Aug 21 2018

All Science Journal Classification (ASJC) codes

  • General


  • Backward stochastic differential equations
  • Deep learning
  • Feynman–Kac
  • High dimension
  • Partial differential equations


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