×
Xu, Xingjian; Chen, Minghua

Discovery of subdiffusion problem with noisy data via deep learning. (English) Zbl 07549611

J. Sci. Comput. 92, No. 1, Paper No. 23, 24 p. (2022).
Summary: Data-driven discovery of partial differential equations (PDEs) from observed data in machine learning has been developed by embedding the discovery problem. Recently, the discovery of traditional ODEs dynamics using linear multistep methods in deep learning have been discussed in [Racheal and Du, SIAM J. Numer. Anal. 59 (2021) 429–455; Du et al. arXiv:2103.1148]. We extend this framework to the data-driven discovery of the time-fractional PDEs, which can effectively characterize the ubiquitous power-law phenomena. In this paper, identifying source function of subdiffusion with noisy data using \(L_1\) approximation in deep neural network is presented. In particular, two types of networks for improving the generalization of the subdiffusion problem are designed with noisy data. The numerical experiments are given to illustrate the availability with high noise levels using deep learning. To the best of our knowledge, this is the first topic on the discovery of subdiffusion in deep learning with noisy data.

Cited in 3 Documents

MSC:

65Mxx Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
68Txx Artificial intelligence
35Rxx Miscellaneous topics in partial differential equations

Keywords:

deep learning; discovery of subdiffusion; noisy data

Software:

DGM
Cite Review PDF
Full Text: DOI arXiv

References:

[1] Cao, RJ; Chen, MH; Ng, MK; Wu, YJ, Fast and high-order accuracy numerical methods for time-dependent nonlocal problems in \({\mathbb{R}}^2\), J. Sci. Comput., 84, 8 (2020) · Zbl 1445.74054 · doi:10.1007/s10915-020-01260-7
[2] Chen, MH; Ekström, SE; Serra-Capizzano, S., A Multigrid method for nonlocal problems: non-diagonally dominant or Toeplitz-plus-tridiagonal systems, SIAM J. Matrix Anal. Appl., 41, 1546-1570 (2020) · Zbl 1461.65236 · doi:10.1137/18M1210460
[3] Chen, MH; Jiang, SZ; Bu, WP, Two \(L1\) schemes on graded meshes for fractional Feynman-Kac equation, J. Sci. Comput., 88, 58 (2021) · Zbl 1500.65039 · doi:10.1007/s10915-021-01581-1
[4] Chen, J.R., Jin, S., Lyu, L.Y.: A deep learning based discotinuous Galerkin method for hyperbolic equations with discontinuous solutions and random uncertainties. arXiv:2107.01127
[5] Chen, WQ; Wang, Q.; Hesthaven, JS; Zhang, CH, Physics-informed machine learning for reduced-order modeling of nonlinear problems, J. Comput. Phys., 446, 110666 (2021) · Zbl 07516475 · doi:10.1016/j.jcp.2021.110666
[6] Dattorro. J.: Conves Optimization & Euclidean Distance Geometry. Meboo Publishing USA, California (2015)
[7] Du. Q., Gu, Y.Q., Yang, H.Z., Zhou, C.: The discovery of dynamics via linear multistep methods and deep learning:error estimation. arXiv:2103.11488
[8] Duan, C.G., Jiao, Y.L, Lai, Y.M., Lu, X.L.,Yang, Z.J.: Convergence rate analysis for deep Ritz method. arXiv:2103.13330 · Zbl 1491.65117
[9] E, W.N., Yu, B.: The Deep Ritz method:A deep learning-based numerical algorithm for solving variational problems. Commun. Math. Stat. 6, 1-12 (2018) · Zbl 1392.35306
[10] Eliazar, I.; Klafter, J., Anomalous is ubiquitous, Ann. Physies, 326, 2517-2531 (2011) · Zbl 1238.60086 · doi:10.1016/j.aop.2011.07.006
[11] Goodfellow, I.; Bengio, Y.; Courville, A., Deep Learning: Adaptive Computation and Machine Learning (2016), Cambridge: MIT Press, Cambridge · Zbl 1373.68009
[12] Gu, Y.Q., Ng, M.K.: Deep Ritz method for the spectral fractional Laplacian equation using the Caffarelli-Silvestre extension. arXiv:2108.11592
[13] Gulian, MK; Raissi, M.; Perdikaris, P.; Karniadakis, G., Machine learning of space-fractional differential equations, SIAM J. Sci. Comput., 41, A2485-A2509 (2019) · Zbl 1419.35209 · doi:10.1137/18M1204991
[14] Jiao, Y.L., Lai, Y.M., Lo,Y.S., Wang.Y., Yang.Y.F.: Error analysis of Deep Ritz Methods for elliptic equations. arXiv:2107.14478
[15] Keller, R.; Du, Q., Discovery of dynamics using linear multistep methods, SIAM J. Numer. Anal., 59, 429-455 (2021) · Zbl 1466.65050 · doi:10.1137/19M130981X
[16] Li, YJ; Wang, YJ; Deng, WH, Galerkin finite element approximations for stochastic space-time fractional wave equations, SIAM J. Numer. Anal., 55, 3173-3202 (2017) · Zbl 1380.65017 · doi:10.1137/16M1096451
[17] Li, Z.Y., Kovachki, N., Azizzadenesheli., K., Liu, B. ,Bhattacharya, K., Stuart, A., Anandkumar, A.: Markov neural operators for learning chaotic systems. arXiv:2106.06898
[18] Lin, YM; Xu, CJ, Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys., 225, 1533-1552 (2007) · Zbl 1126.65121 · doi:10.1016/j.jcp.2007.02.001
[19] Meerschaert, M.M., Sikorskii, A.: Stochastic Models for Fractional Calculus. de Gruyter GmbH Berlin (2012) · Zbl 1247.60003
[20] Metzler, R.; Klafter, J., The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep., 339, 1-77 (2000) · Zbl 0984.82032 · doi:10.1016/S0370-1573(00)00070-3
[21] Podlubny, I., Fractional Differential Equations (1999), New York: Academic Press, New York · Zbl 0924.34008
[22] Qin, T.; Wu, K.; Xiu, D., Data driven governing equations approximation using deep neural network, J. Comput. Phys., 395, 620-635 (2019) · Zbl 1455.65125 · doi:10.1016/j.jcp.2019.06.042
[23] Raissi, M., Deep hidden physics models deep learning of nonlinear partial differential equations, The Journal of Machine Learning Research., 19, 1-24 (2018) · Zbl 1439.68021
[24] Rudy, SH; Kutz, JN; Brunton, SL, Deep learning of dynamics and signal-noise decompoisition with time-stepping constraints, J. Comput. Phys., 396, 483-506 (2019) · Zbl 1452.68170 · doi:10.1016/j.jcp.2019.06.056
[25] Shen, ZW; Yang, HZ; Zhang, SJ, Deep network approximation characterized by number of neurons Commun, Comput. Phys., 28, 1768-1811 (2019) · Zbl 1507.68276 · doi:10.4208/cicp.OA-2020-0149
[26] Shen, X., Cheng, X.L., Liang, .K.W.: Deep Euler Method: Solving ODEs by approximating the local truncation error of the euler method. arXiv:2003.09573
[27] Sirignano, J.; Spiliopoulos, K., DGM: A deep learning algorithm for solving partial differental equations, J. Comput. Phys., 375, 1339-1364 (2018) · Zbl 1416.65394 · doi:10.1016/j.jcp.2018.08.029
[28] Shi, JK; Chen, MH, Correction of high-order BDF convolution quadrature for fractional Feynman-Kac equation with Lévy flight, J. Sci. Comput., 85, 28 (2020) · Zbl 1480.76088 · doi:10.1007/s10915-020-01331-9
[29] Stynes, M.; O’riordan, E.; Gracia, JL, Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation, SIAM J. Numer. Anal., 55, 1057-1079 (2017) · Zbl 1362.65089 · doi:10.1137/16M1082329
[30] Tipireddy, R., Perdikaris, P., Stinis , P., Tartakovsky, A.: A comparative study of physics-informed neural network models for learning unknown dynamics and constitutive relations. arXiv:1904.04058
[31] Wang, C., Chen, M.H., Deng, W.H., Bu, W.P., Dai, X.J.: A sharp error estimate of Euler-Maruyama method for stochastic Volterra integral equations. Math. Meth. Appl. Sci. 1-25 (2022)
[32] Xu, Y.; Zhang, H.; Li, YG; Zhou, K.; Liu, Q.; Jurgen, K., Solving Fokker-Planck equation using deep learning, Chaos., 30, 013133 (2020) · Zbl 1431.35210 · doi:10.1063/1.5132840
[33] Zhang, ZH; Karniadakis, GE, Numerical Methods for Stochastic Partial Differential Equations with White Noise (2017), New York: Springer, New York · Zbl 1380.65021 · doi:10.1007/978-3-319-57511-7
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.