Numerical solution of the incompressible Navier-Stokes equation by a deep branching Algorithm. (English) Zbl 07738853
Summary: We present an algorithm for the numerical solution of systems of fully nonlinear PDEs using stochastic coded branching trees. This approach covers functional nonlinearities involving gradient terms of arbitrary orders, and it requires only a boundary condition over space at a given terminal time \(T\) instead of Dirichlet or Neumann boundary conditions at all times as in standard solvers. Its implementation relies on Monte Carlo estimation, and uses neural networks that perform a meshfree functional estimation on a space-time domain. The algorithm is applied to the numerical solution of the Navier-Stokes equation and is benchmarked to other implementations in the cases of the Taylor-Green vortex and Arnold-Beltrami-Childress flow.
MSC:
| 65-XX | Numerical analysis |
| 35Q30 | Navier-Stokes equations |
| 76M35 | Stochastic analysis applied to problems in fluid mechanics |
| 76D05 | Navier-Stokes equations for incompressible viscous fluids |
| 60H30 | Applications of stochastic analysis (to PDEs, etc.) |
| 60J85 | Applications of branching processes |
| 65C05 | Monte Carlo methods |
Keywords:
fully nonlinear PDEs; systems of PDEs; Navier-Stokes equations; Monte Carlo method; deep neural network; branching process; random treeReferences:
| [1] | S. Albeverio and Ya. Belopolskaya. Generalized solutions of the Cauchy problem for the Navier-Stokes system and diffusion processes. Cubo, 12(2):77-96, 2010. · Zbl 1221.60076 |
| [2] | P.-E. Angeli, M.-A. Puscas, G. Fauchet, and A. Cartalade. FVCA8 Benchmark for the Stokes and Navier-Stokes equations with the TrioCFD code-benchmark session. In FVCA 2017: Finite Volumes for Complex Applications VIII -Methods and Theoretical Aspects, volume 199 of Springer Proceedings in Mathematics & Statistics, pages 181-202. Springer Verlag, 2017. · Zbl 1391.76381 |
| [3] | V. Arnol’d. Sur la topologie desécoulements stationnaires des fluides parfaits. C. R. Acad. Sci. Paris, 261:17-20, 1965. · Zbl 0145.22203 |
| [4] | A.N. Borodin. Stochastic processes. Probability and its Applications. Birkhäuser/Springer, Cham, 2017. Original Russian edition published by LAN Publishing, St. Petersburg, 2013. |
| [5] | F. Cipriano and A.B. Cruzeiro. Navier-Stokes equation and diffusions on the group of home-omorphisms of the torus. Comm. Math. Phys., 275:255-269, 2007. · Zbl 1120.76013 |
| [6] | S. Childress. New solutions of the kinematic dynamo problem. J. Math. Phys., 11(10):3063-3076, 1970. |
| [7] | G.M. Constantine and T.H. Savits. A multivariate Faa di Bruno formula with applications. Trans. Amer. Math. Soc., 348(2):503-520, 1996. · Zbl 0846.05003 |
| [8] | A.B. Cruzeiro and E. Shamarova. Navier-Stokes equations and forward-backward SDEs on the group of diffeomorphisms of a torus. Stochastic Process. Appl., 119(12):4034-4060, 2009. · Zbl 1188.60036 |
| [9] | P. Cheridito, H.M. Soner, N. Touzi, and N. Victoir. Second-order backward stochastic dif-ferential equations and fully nonlinear parabolic PDEs. Comm. Pure Appl. Math., 60(7):1081-1110, 2007. · Zbl 1121.60062 |
| [10] | F. Delbaen, J. Qiu, and S. Tang. Forward-backward stochastic differential systems associated to Navier-Stokes equations in the whole space. Stochastic Process. Appl., 125(7):2516-2561, 2015. · Zbl 1323.35010 |
| [11] | A. Fahim, N. Touzi, and X. Warin. A probabilistic numerical method for fully nonlinear parabolic PDEs. Ann. Appl. Probab., 21(4):1322-1364, 2011. · Zbl 1230.65009 |
| [12] | W. Guo, J. Zhang, and J. Zhuo. A monotone scheme for high-dimensional fully nonlinear PDEs. Ann. Appl. Probab., 25(3):1540-1580, 2015. · Zbl 1321.65158 |
| [13] | J. Han, A. Jentzen, and W. E. Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences, 115(34):8505-8510, 2018. · Zbl 1416.35137 |
| [14] | P. Henry-Labordère. Counterparty risk valuation: A marked branching diffusion approach. Preprint arXiv:1203.2369, 2012. |
| [15] | P. Henry-Labordère, N. Oudjane, X. Tan, N. Touzi, and X. Warin. Branching diffusion repre-sentation of semilinear PDEs and Monte Carlo approximation. Ann. Inst. H. Poincaré Probab. Statist., 55(1):184-210, 2019. · Zbl 1467.60067 |
| [16] | S. Huang, G. Liang, and T. Zariphopoulou. An approximation scheme for semilinear parabolic PDEs with convex and coercive Hamiltonians. SIAM J. Control Optim., 58(1):165-191, 2020. · Zbl 1429.65217 |
| [17] | K. Hornik. Approximation capabilities of multilayer feedforward networks. Neural Net-works, 4(2):251-257, 1991. |
| [18] | K. He, X. Zhang, S. Ren, and J. Sun. Deep residual learning for image recognition. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 770-778, 2016. |
| [19] | N. Ikeda, M. Nagasawa, and S. Watanabe. Branching Markov processes I, II, III. J. Math. Kyoto Univ., 8-9:233-278, 365-410, 95-160, 1968-1969. · Zbl 0233.60068 |
| [20] | S. Ioffe and C. Szegedy. Batch normalization: Accelerating deep network training by re-ducing internal covariate shift. In Proceedings of the 32nd International Conference on Machine Learning, pages 448-456, 2015. |
| [21] | D.P. Kingma and J. Ba. Adam: A method for stochastic optimization. Preprint arXiv:1412.6980, 2014. |
| [22] | A. Lejay and H.M. González. A forward-backward probabilistic algorithm for the incom-pressible Navier-Stokes equations. Journal of Computational Physics, 420:109689, 19, 2020. · Zbl 07506618 |
| [23] | Y. Le Jan and A. S. Sznitman. Stochastic cascades and 3-dimensional Navier-Stokes equa-tions. Probab. Theory Related Fields, 109(3):343-366, 1997. · Zbl 0888.60072 |
| [24] | J. Li, J. Yue, W. Zhang, and W. Duan. The deep learning Galerkin method for the general Stokes equations. J. Sci. Comput., 93(1):Paper No. 5, 20, 2022. · Zbl 07590392 |
| [25] | M. Matsumoto. Application of Deep Galerkin Method to solve compressible Navier-Stokes equations. Trans. Japan Soc. Aero. Space Sci., 64(6):348-357, 2021. |
| [26] | A.J. Majda and A.L. Bertozzi. Vorticity and Incompressible Flow, volume 27 of Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 2002. · Zbl 0983.76001 |
| [27] | H.P. McKean. Application of Brownian motion to the equation of Kolmogorov-Petrovskii-Piskunov. Comm. Pure Appl. Math., 28(3):323-331, 1975. · Zbl 0316.35053 |
| [28] | J.Y. Nguwi, G. Penent, and N. Privault. A deep branching solver for fully nonlinear partial differential equations. Preprint arXiv:2203.03234, 17 pages, 2022. |
| [29] | J.Y. Nguwi, G. Penent, and N. Privault. A fully nonlinear Feynman-Kac formula with deriva-tives of arbitrary orders. Journal of Evolution Equations, 23(22), 2023. https://doi.org/10. 1007/s00028-023-00873-3. · Zbl 1510.35020 · doi:10.1007/s00028-023-00873-3 |
| [30] | É. Pardoux and S. Peng. Backward stochastic differential equations and quasilinear parabolic partial differential equations. In Stochastic Partial Differential Equations and Their Applications (Charlotte, NC, 1991), volume 176 of Lecture Notes in Control and Inform. Sci., pages 200-217. Springer, Berlin, 1992. · Zbl 0766.60079 |
| [31] | G. Penent and N. Privault. Numerical evaluation of ODE solutions by Monte Carlo enumer-ation of Butcher series. BIT Numerical Mathematics, 62:1921-1944, 2022. · Zbl 1503.65149 |
| [32] | M. Raissi, P. Perdikaris, and G. E. Karniadakis. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics, 378:686-707, 2019. · Zbl 1415.68175 |
| [33] | A.V. Skorokhod. Branching diffusion processes. Teor. Verojatnost. I. Primenen., 9:492-497, 1964. · Zbl 0264.60058 |
| [34] | J. Sirignano and K. Spiliopoulos. DGM: A deep learning algorithm for solving partial differ-ential equations. Journal of Computational Physics, 375:1339-1364, 2018. · Zbl 1416.65394 |
| [35] | H.M. Soner, N. Touzi, and J. Zhang. Wellposedness of second order backward SDEs. Probab. Theory Related Fields, 153(1-2):149-190, 2012. · Zbl 1252.60056 |
| [36] | X. Tan. A splitting method for fully nonlinear degenerate parabolic PDEs. Electron. J. Probab., 18:no. 15, 24, 2013. · Zbl 1282.65103 |
| [37] | G.I. Taylor and A.E. Green. Mechanism of the production of small eddies from large ones. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 158(895):499-521, 1937. · JFM 63.1358.03 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
