Accessible Requires Authentication Published by De Gruyter December 11, 2020

Overcoming the curse of dimensionality in the numerical approximation of Allen–Cahn partial differential equations via truncated full-history recursive multilevel Picard approximations

Christian Beck, Fabian Hornung, Martin Hutzenthaler, Arnulf Jentzen and Thomas Kruse

Abstract

One of the most challenging problems in applied mathematics is the approximate solution of nonlinear partial differential equations (PDEs) in high dimensions. Standard deterministic approximation methods like finite differences or finite elements suffer from the curse of dimensionality in the sense that the computational effort grows exponentially in the dimension. In this work we overcome this difficulty in the case of reaction–diffusion type PDEs with a locally Lipschitz continuous coervice nonlinearity (such as Allen–Cahn PDEs) by introducing and analyzing truncated variants of the recently introduced full-history recursive multilevel Picard approximation schemes.

JEL Classification: 60H30; 65C05; 65M75

Acknowledgment

The second author gratefully acknowledges a Research Travel Grant by the Karlsruhe House of Young Scientists (KHYS) supporting his stay at ETH Zurich. The third author gratefully acknowledges funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) via research grant HU 1889/6-1. The fourth author gratefully acknowledges funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy EXC 2044-390685587, Mathematics Muenster: Dynamics–Geometry–Structure.

References

[1] C. Beck, S. Becker, P. Cheridito, A. Jentzen, and A. Neufeld, Deep splitting method for parabolic PDEs, arXiv:1907.03452 (2019), 40 p. Search in Google Scholar

[2] C. Beck, S. Becker, P. Grohs, N. Jaafari, and A. Jentzen, Solving stochastic differential equations and Kolmogorov equations by means of deep learning, arXiv:1806.00421 (2018), 56 p. Search in Google Scholar

[3] C. Beck, W. E, and A. Jentzen, Machine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order backward stochastic differential equations, J. Nonlinear Sci. 29 (2019), No. 4, 1563–1619. Search in Google Scholar

[4] C. Beck, L. Gonon, and A. Jentzen, Overcoming the curse of dimensionality in the numerical approximation of high-dimensional semilinear elliptic partial differential equations, arXiv:2003.00596 (2020), 50 p. Search in Google Scholar

[5] S. Becker, R. Braunwarth, M. Hutzenthaler, A. Jentzen, and P. von Wurstemberger, Numerical simulations for full history recursive multilevel Picard approximations for systems of high-dimensional partial differential equations, arXiv:2005.10206 (2020), 21 p. (to appear in Commun. Comput. Physics Search in Google Scholar

[6] S. Becker, P. Cheridito, and A. Jentzen, Deep optimal stopping, J. Mach. Learn. Res. 20 (2019), No. 74, 1–25. Search in Google Scholar

[7] J. Berg and K. Nyström, A unified deep artificial neural network approach to partial differential equations in complex geometries, Neurocomputing 317 (2018), 28–41. Search in Google Scholar

[8] B. Bouchard, X. Tan, X. Warin, and Y. Zou, Numerical approximation of BSDEs using local polynomial drivers and branching processes, Monte Carlo Methods Appl. 23 (2017), No. 4, 241–263. Search in Google Scholar

[9] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations Springer Science & Business Media, New York, 2010. Search in Google Scholar

[10] Q. Chan-Wai-Nam, J. Mikael, and X. Warin, Machine learning for semi linear PDEs, J. Sci. Comput. 79 (2019), No. 3, 1667–1712. Search in Google Scholar

[11] W. E, J. Han, and A. Jentzen, Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations, Commun. Math. Stat. 5 (2017), No. 4, 349–380. Search in Google Scholar

[12] W. E, M. Hutzenthaler, A. Jentzen, and T. Kruse, Multilevel Picard iterations for solving smooth semilinear parabolic heat equations, arXiv:1607.03295 (2016), 19 p. (revision requested from SN Part. Diff. Equ. Appl. Search in Google Scholar

[13] W. E, M. Hutzenthaler, A. Jentzen, and T. Kruse, On multilevel Picard numerical approximations for high-dimensional nonlinear parabolic partial differential equations and high-dimensional nonlinear backward stochastic differential equations, J. Sci. Comput. 79 (2019), No. 3, 1534–1571. Search in Google Scholar

[14] W. E and B. Yu, The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems, Commun. Math. Stat. 6 (2018), No. 1, 1–12. Search in Google Scholar

[15] A.-M. Farahmand, S. Nabi, and D. N. Nikovski, Deep reinforcement learning for partial differential equation control, 2017 American Control Conference (ACC) (2017), 3120–3127. Search in Google Scholar

[16] M. Fujii, A. Takahashi, and M. Takahashi, Asymptotic expansion as prior knowledge in deep learning method for high dimensional BSDEs, Asia-Pacific Financial Markets 26 (2019), No. 3, 391–408. Search in Google Scholar

[17] M. B. Giles, A. Jentzen, and T. Welti, Generalised multilevel Picard approximations, arXiv:1911.03188 (2019), 61 p. (revision requested from IMA J. Numer. Anal. Search in Google Scholar

[18] L. Goudenege, A.Molent, and A. Zanette, Machine Learning for Pricing American Options in High Dimension, arXiv:1903.11275v1 (2019), 11 p. Search in Google Scholar

[19] J. Han, A. Jentzen, and W. E, Solving high-dimensional partial differential equations using deep learning, Proc. of the National Academy of Sciences 115 (2018), No. 34, 8505–8510. Search in Google Scholar

[20] J. Han and J. Long, Convergence of the deep BSDE method for coupled FBSDEs, Probab. Uncertain. Quant. Risk 5 (2020), Paper No. 5, 33. Search in Google Scholar

[21] P. Henry-Labordère, Counterparty risk valuation: A marked branching diffusion approach, SSRN:1995503 (2012), 17 p. Search in Google Scholar

[22] P. Henry-Labordère, Deep Primal-Dual Algorithm for BSDEs: Applications of Machine Learning to CVA and IM, Preprint, SSRN–id3071506 (2017), 16 p. Search in Google Scholar

[23] P. Henry-Labordère, N. Oudjane, X. Tan, N. Touzi, and X.Warin, Branching diffusion representation of semilinear PDEs and Monte Carlo approximation, Ann. Inst. Henri Poincaré Probab. Stat. 55 (2019), No. 1, 184–210. Search in Google Scholar

[24] P. Henry-Labordère, X. Tan, and N. Touzi, A numerical algorithm for a class of BSDEs via the branching process, Stochastic Process. Appl. 124 (2014), No. 2, 1112–1140. Search in Google Scholar

[25] C. Huré, H. Pham, and X. Warin, Deep backward schemes for high-dimensional nonlinear PDEs, Math. Comp. 89 (2020), No. 324, 1547–1579. Search in Google Scholar

[26] M. Hutzenthaler, A. Jentzen, and T. Kruse, Overcoming the curse of dimensionality in the numerical approximation of parabolic partial differential equations with gradient-dependent nonlinearities, arXiv:1912.02571 (2019), 33 p. (revision requested from Found. Comput. Math. Search in Google Scholar

[27] M. Hutzenthaler, A. Jentzen, T. Kruse, and T. A. Nguyen, A proof that rectified deep neural networks overcome the curse of dimensionality in the numerical approximation of semilinear heat equations, SN Partial Differential Equations and Applications 1 (2020), 1–34. Search in Google Scholar

[28] M. Hutzenthaler, A. Jentzen, T. Kruse, T. A. Nguyen, and P. von Wurstemberger, Overcoming the curse of dimensionality in the numerical approximation of semilinear parabolic partial differential equations, arXiv:1807.01212v2 (2019), 30 p. (to appear in Proc. Royal Soc. A Search in Google Scholar

[29] M. Hutzenthaler, A. Jentzen, and P. von Wurstemberger, Overcoming the curse of dimensionality in the approximative pricing of financial derivatives with default risks, Electron. J. Probab. 25 (2020), Paper No. 101, 73. Search in Google Scholar

[30] M. Hutzenthaler and T. Kruse, Multilevel Picard approximations of high-dimensional semilinear parabolic differential equations with gradient-dependent nonlinearities, SIAM J. Numer. Anal. 58 (2020), No. 2, 929–961. Search in Google Scholar

[31] A. Jacquier and M. Oumgari, Deep PPDEs for rough local stochastic volatility, arXiv:1906.02551 (2019), 21 p. Search in Google Scholar

[32] F. John, Partial Differential Equations Applied mathematical sciences, Springer, 1982. Search in Google Scholar

[33] Z. Long, Y. Lu, X. Ma, and B. Dong, PDE-Net: Learning PDEs from Data, Proceedings of Machine Learning Research, Vol. 80, pp. 3208–3216, PMLR, Stockholmsmässan, Stockholm Sweden, July 10–15, 2018. Search in Google Scholar

[34] K. O. Lye, S. Mishra, and D. Ray, Deep learning observables in computational fluid dynamics, J. Comput. Phys. 410 (2020), 109339, 26. Search in Google Scholar

[35] M. Magill, F. Qureshi, and H. de Haan, Neural networks trained to solve differential equations learn general representations, Advances in Neural Information Processing Systems (2018), 4075–4085. Search in Google Scholar

[36] H. P. McKean, Application of Brownian motion to the equation of Kolmogorov–Petrovskii–Piskunov, Comm. Pure Appl. Math. 28 (1975), No. 3, 323–331. Search in Google Scholar

[37] M. Raissi, Deep hidden physics models: deep learning of nonlinear partial differential equations, J. Mach. Learn. Res. 19 (2018), Paper No. 25, 1–24. Search in Google Scholar

[38] J. Sirignano and K. Spiliopoulos, DGM: A deep learning algorithm for solving partial differential equations, J. Comp. Phys. 375 (2018), 1339–1364. Search in Google Scholar

[39] A. V. Skorohod, Branching diffusion processes, Teor. Verojatnost. i Primenen. 9 (1964), 492–497. Search in Google Scholar

[40] X. Warin, Nesting Monte Carlo for high-dimensional non-linear PDEs, Monte Carlo Methods and Applications 24 (2018), No. 4, 225–247. Search in Google Scholar

[41] S. Watanabe, On the branching process for Brownian particles with an absorbing boundary, J. Math. Kyoto Univ. 4 (1965), 385–398. Search in Google Scholar

Received: 2019-08-03
Revised: 2020-06-08
Accepted: 2020-06-18
Published Online: 2020-12-11
Published in Print: 2020-12-16

© 2020 Walter de Gruyter GmbH, Berlin/Boston