Communications in Mathematics and Statistics

, Volume 5, Issue 4, pp 349–380 | Cite as

Deep Learning-Based Numerical Methods for High-Dimensional Parabolic Partial Differential Equations and Backward Stochastic Differential Equations

  • Weinan E
  • Jiequn Han
  • Arnulf Jentzen


We study a new algorithm for solving parabolic partial differential equations (PDEs) and backward stochastic differential equations (BSDEs) in high dimension, which is based on an analogy between the BSDE and reinforcement learning with the gradient of the solution playing the role of the policy function, and the loss function given by the error between the prescribed terminal condition and the solution of the BSDE. The policy function is then approximated by a neural network, as is done in deep reinforcement learning. Numerical results using TensorFlow illustrate the efficiency and accuracy of the studied algorithm for several 100-dimensional nonlinear PDEs from physics and finance such as the Allen–Cahn equation, the Hamilton–Jacobi–Bellman equation, and a nonlinear pricing model for financial derivatives.


PDEs High dimension Backward stochastic differential equations Deep learning Control Feynman-Kac 

Mathematics Subject Classification

65M75 60H35 65C30 



Christian Beck and Sebastian Becker are gratefully acknowledged for useful suggestions regarding the implementation of the deep BSDE method. This project has been partially supported through the Major Program of NNSFC under grant 91130005, the research grant ONR N00014-13-1-0338, and the research grant DOE DE-SC0009248.


  1. 1.
    Bellman, R.: Dynamic programming. Princeton Landmarks in Mathematics. Princeton University Press, Princeton, NJ. Reprint of the 1957 edition, with a new introduction by Stuart Dreyfus (2010)Google Scholar
  2. 2.
    Bender, C., Denk, R.: A forward scheme for backward SDEs. Stoch. Process. Appl. 117(12), 1793–1812 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bender, C., Schweizer, N., Zhuo, J.: A primal-dual algorithm for BSDEs. arXiv:1310.3694 (2014)
  4. 4.
    Bergman, Y.Z.: Option pricing with differential interest rates. Rev. Financ. Stud. 8(2), 475–500 (1995)CrossRefGoogle Scholar
  5. 5.
    Briand, P., Labart, C.: Simulation of BSDEs by Wiener chaos expansion. Ann. Appl. Probab. 24(3), 1129–1171 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chassagneux, J.-F.: Linear multistep schemes for BSDEs. SIAM J. Numer. Anal. 52(6), 2815–2836 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chassagneux, J.-F., Richou, A.: Numerical simulation of quadratic BSDEs. Ann. Appl. Probab. 26(1), 262–304 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Crisan, D., Manolarakis, K.: Solving backward stochastic differential equations using the cubature method: application to nonlinear pricing. SIAM J. Financ. Math. 3(1), 534–571 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Darbon, J., Osher, S.: Algorithms for overcoming the curse of dimensionality for certain Hamilton–Jacobi equations arising in control theory and elsewhere. Res. Math. Sci. 3(19), 26 (2016)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Debnath, L.: Nonlinear Partial Differential Equations for Scientists and Engineers, 3rd edn. Birkhäuser/Springer, New York (2012)CrossRefzbMATHGoogle Scholar
  11. 11.
    E, W., Han, J., Jentzen, A.: Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations. arXiv:1706.04702 (2017)
  12. 12.
    E, W., Hutzenthaler, M., Jentzen, A., Kruse, T.: Linear scaling algorithms for solving high-dimensional nonlinear parabolic differential equations. arXiv:1607.03295 (2017)
  13. 13.
    E, W., Hutzenthaler, M., Jentzen, A., Kruse, T.: On multilevel Picard numerical approximations for high-dimensional nonlinear parabolic partial differential equations and high-dimensional nonlinear backward stochastic differential equations. arXiv:1708.03223 (2017)
  14. 14.
    Gobet, E., Lemor, J.-P., Warin, X.: A regression-based Monte Carlo method to solve backward stochastic differential equations. Ann. Appl. Probab. 15(3), 2172–2202 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Gobet, E., Turkedjiev, P.: Linear regression MDP scheme for discrete backward stochastic differential equations under general conditions. Math. Comput. 85(299), 1359–1391 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Gobet, E., Turkedjiev, P.: Adaptive importance sampling in least-squares Monte Carlo algorithms for backward stochastic differential equations. Stoch. Process. Appl. 127(4), 1171–1203 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Goodfellow, I., Bengio, Y., Courville, A.: Deep Learning. MIT Press 2016.
  18. 18.
    Han, J., E, W.: Deep learning approximation for stochastic control problems. arXiv:1611.07422 (2016)
  19. 19.
    Han, J., Jentzen, A., E, W.: Overcoming the curse of dimensionality: solving high-dimensional partial differential equations using deep learning. arXiv:1707.02568 (2017)
  20. 20.
    Henry-Labordère, P.: Counterparty risk valuation: a marked branching diffusion approach. arXiv:1203.2369 (2012)
  21. 21.
    Henry-Labordère, P., Oudjane, N., Tan, X., Touzi, N., Warin, X.: Branching diffusion representation of semilinear PDEs and Monte Carlo approximation. arXiv:1603.01727 (2016)
  22. 22.
    Henry-Labordère, P., Tan, X., Touzi, N.: A numerical algorithm for a class of BSDEs via the branching process. Stoch. Process. Appl. 124(2), 1112–1140 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Hinton, G.E., Deng, L., Yu, D., Dahl, G., Mohamed, A., Jaitly, N., Senior, A., Vanhoucke, V., Nguyen, P., Sainath, T., Kingsbury, B.: Deep neural networks for acoustic modeling in speech recognition. Sig. Process. Mag. 29, 82–97 (2012)CrossRefGoogle Scholar
  24. 24.
    Ioffe, S., Szegedy, C.: Batch normalization: accelerating deep network training by reducing internal covariate shift. In: Proceedings of the International Conference on Machine Learning (ICML) (2015)Google Scholar
  25. 25.
    Kingma, D., Ba, J.: Adam: a method for stochastic optimization. In: Proceedings of the International Conference on Learning Representations (ICLR) (2015)Google Scholar
  26. 26.
    Krizhevsky, A., Sutskever, I., Hinton, G.E.: Imagenet classification with deep convolutional neural networks. Adv. Neural Inf. Process. Syst. 25, 1097–1105 (2012)Google Scholar
  27. 27.
    LeCun, Y., Bengio, Y., Hinton, G.E.: Deep learning. Nature 521, 436–444 (2015)CrossRefGoogle Scholar
  28. 28.
    Pardoux, É., Peng, S.: Adapted solution of a backward stochastic differential equation. Syst. Control Lett. 14(1), 55–61 (1990)Google Scholar
  29. 29.
    Pardoux, É., Peng, S.: Backward stochastic differential equations and quasilinear parabolic partial differential equations. In: Stochastic Partial Differential Equations and Their Applications (Charlotte, NC, 1991), vol. 176 of Lecture Notes in Control and Inform. Sci. Springer, Berlin, pp. 200–217 (1992)Google Scholar
  30. 30.
    Pardoux, É., Tang, S.: Forward-backward stochastic differential equations and quasilinear parabolic PDEs. Probab. Theory Relat. Fields 114(2), 123–150 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Peng, S.: Probabilistic interpretation for systems of quasilinear parabolic partial differential equations. Stoch. Stoch. Rep. 37(1–2), 61–74 (1991)MathSciNetzbMATHGoogle Scholar

Copyright information

© School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Beijing Institute of Big Data ResearchBeijingChina
  2. 2.Princeton UniversityPrincetonUSA
  3. 3.Peking UniversityBeijingChina
  4. 4.ETH ZurichZurichSwitzerland

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