Peridynamics enabled learning partial differential equations. (English) Zbl 07508517
Summary: This study presents an approach to discover the significant terms in partial differential equations (PDEs) that describe particular phenomena based on the measured data. The relationship between the known field data and its continuous representation of PDEs is achieved through a linear regression model. It specifically employs the peridynamic differential operator (PDDO) and sparse linear regression learning algorithm. The PDEs are approximated by constructing a feature matrix, velocity vector and unknown coefficient vector. Each candidate term (derivatives) appearing in the feature matrix is evaluated numerically by using the PDDO. The solution to the regression model with regularization is achieved through Douglas-Rachford (D-R) algorithm which is based on proximal operators. This coupling performs well due to their robustness to noisy data and the calculation of accurate derivatives. Its effectiveness is demonstrated by considering several fabricated data associated with challenging nonlinear PDEs such as Burgers, Swift-Hohenberg (S-H), Korteweg-de Vries (KdV), Kuramoto-Sivashinsky (K-S), nonlinear Schrödinger (NLS) and Cahn-Hilliard (C-H) equations.
MSC:
| 35Qxx | Partial differential equations of mathematical physics and other areas of application |
| 65Fxx | Numerical linear algebra |
| 74Axx | Generalities, axiomatics, foundations of continuum mechanics of solids |
References:
| [1] | Crutchfield, J. P.; McNamara, B., Equations of motion from a data series, Complex Syst., 1, 417-452 (1987) · Zbl 0675.58026 |
| [2] | Bongard, J.; Lipson, H., Automated reverse engineering of nonlinear dynamical systems, Proc. Natl. Acad. Sci., 104, 9943-9948 (2007) · Zbl 1155.37044 |
| [3] | Schmidt, M.; Lipson, H., Distilling free-form natural laws from experimental data, Science, 324, 81-85 (2009) |
| [4] | Schaeffer, H.; Caflisch, R.; Hauck, C. D.; Osher, S., Sparse dynamics for partial differential equations, Proc. Natl. Acad. Sci., 110, 6634-6639 (2013) · Zbl 1292.35012 |
| [5] | Brunton, S. L.; Proctor, J. L.; Kutz, J. N., Discovering governing equations from data by sparse identification of nonlinear dynamical systems, Proc. Natl. Acad. Sci., 113, 3932-3937 (2016) · Zbl 1355.94013 |
| [6] | Rudy, S. H.; Brunton, S. L.; Proctor, J. L.; Kutz, J. N., Data-driven discovery of partial differential equations, Sci. Adv., 3, Article e1602614 pp. (2017) |
| [7] | Schaeffer, H., Learning partial differential equations via data discovery and sparse optimization, Proc. R. Soc. A, 473, Article 20160446 pp. (2017) · Zbl 1404.35397 |
| [8] | Tibshirani, R., Regression shrinkage and selection via the Lasso, J. R. Stat. Soc., Ser. B, Methodol., 58, 267-288 (1996) · Zbl 0850.62538 |
| [9] | Parikh, N.; Boyd, S., Proximal algorithms, Found. Trends Optim., 1, 123-231 (2014) |
| [10] | Combettes, P. L.; Pesquet, J.-C., Proximal splitting methods in signal processing, (Bauschke, H. H.; Burachik, R. S.; Combettes, P. L., Fixed-Point Algorithms for Inverse Problems in Science and Engineering (2011), Springer: Springer New York, NY), 185-212 · Zbl 1242.90160 |
| [11] | Madenci, E.; Barut, A.; Futch, M., Peridynamic differential operator and its applications, Comput. Methods Appl. Mech. Eng., 304, 408-451 (2016) · Zbl 1425.74043 |
| [12] | Madenci, E.; Barut, A.; Dorduncu, M.; Futch, M., Numerical solution of linear and nonlinear partial differential equations by using the peridynamic differential operator, Numer. Methods Partial Differ. Equ., 33, 1726-1753 (2017) · Zbl 1375.65124 |
| [13] | Madenci, E.; Barut, A.; Dorduncu, M., Peridynamic Differential Operators for Numerical Analysis (2019), Springer: Springer Boston, MA · Zbl 07658003 |
| [14] | He, B.; Yuan, X., On the O(1/n) convergence rate of the Douglas-Rachford alternating direction method, SIAM J. Numer. Anal., 50, 700-709 (2012) · Zbl 1245.90084 |
| [15] | Silling, S. A., Reformulation of elasticity theory for discontinuities and long-range forces, J. Mech. Phys. Solids, 48, 175-209 (2000) · Zbl 0970.74030 |
| [16] | Silling, S. A.; Epton, M.; Weckner, O.; Xu, J.; Askari, E., Peridynamic states and constitutive modeling, J. Elast., 88, 151-184 (2007) · Zbl 1120.74003 |
| [17] | Silling, S. A.; Lehoucq, R. B., Convergence of peridynamics to classical elasticity theory, J. Elast., 93, 13-37 (2008) · Zbl 1159.74316 |
| [18] | Madenci, E.; Barut, A.; Phan, N., Peridynamic unit cell homogenization for thermoelastic properties of heterogenous microstructures with defects, Compos. Struct., 188, 104-115 (2017) |
| [19] | Bentley, J. L., Multidimensional binary search trees used for associative searching, Commun. ACM, 18, 509-517 (1975) · Zbl 0306.68061 |
| [20] | Burgers, J. M., A mathematical model illustrating the theory of turbulence, Adv. Appl. Mech., 1, 171-199 (1948) |
| [21] | Swift, J.; Hohenberg, P. C., Hydrodynamic fluctuations at the convective instability, Phys. Rev. A, 15, 319-328 (1977) |
| [22] | Korteweg, D. J.; de Vries, G., On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Philos. Mag., 39, 422-443 (1895) · JFM 26.0881.02 |
| [23] | Sivashinsky, G. I., Nonlinear analysis of hydrodynamic instability in laminar flames - I. Derivation of basic equations, Acta Astronaut., 4, 1177-1206 (1977) · Zbl 0427.76047 |
| [24] | Zakharov, V. E.; Manakov, S. V., On the complete integrability of a nonlinear Schrödinger equation, J. Theor. Math. Phys., 19, 551-559 (1974) · Zbl 0298.35016 |
| [25] | Kim, J., Basic principles and practical applications of the Cahn-Hilliard equation, Math. Probl. Eng., Article 9532608 pp. (2016) · Zbl 1400.35157 |
| [26] | Landajuela, M., Burgers equation (2011), Basque Center for Applied Mathematics, BCAM Internship report |
| [27] | Pérez-Moreno, S. S.; Chavarría, S. R.; Chavarría, G. R., Numerical solution of the Swift-Hohenberg equation, (Klapp, J.; Medina, A., Experimental and Computational Fluid Mechanics (2013), Springer), 409-416 |
| [28] | Goda, K., On stability of some finite difference schemes for the Korteweg-de Vries equation, J. Phys. Soc. Jpn., 39, 229-236 (1975) · Zbl 1337.65136 |
| [29] | Raissi, M.; Karniadakis, G. E., Hidden physics models: machine learning of nonlinear partial differential equations, J. Comput. Phys., 357, 125-141 (2018) · Zbl 1381.68248 |
| [30] | Raissi, M.; Perdikaris, P.; Karniadakis, G. E., Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, J. Comput. Phys., 378, 686-707 (2019) · Zbl 1415.68175 |
| [31] | Broyden, C. G., A class of methods for solving nonlinear simultaneous equations, Math. Comput., 19, 577-593 (1965) · Zbl 0131.13905 |
| [32] | Marwil, E., Convergence results for Schubert’s method for solving sparse nonlinear equations, SIAM J. Numer. Anal., 16, 588-604 (1979) · Zbl 0453.65033 |
| [33] | Baydin, A. G.; Pearlmutter, B. A.; Radul, A. A.; Siskind, J. M., Automatic differentiation in machine learning: a survey, J. Mach. Learn. Res., 18, 1-43 (2018) · Zbl 06982909 |
| [34] | Saad, Y.; Schultz, M. H., GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comput., 7, 856-869 (1986) · Zbl 0599.65018 |
| [35] | Chan, T. F.; der Vorst, H. A.V., Approximate and incomplete factorizations, (Keyes, D. E.; Sameh, A.; Venkatakrishnan, V., Parallel Numerical Algorithms. Parallel Numerical Algorithms, ICASE/LaRC Interdisciplinary Series in Science and Engineering, vol. 4 (1997)), 167-202 · Zbl 0865.65015 |
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.
