Abstract
Many branches of the modern computational science and engineering are based on numerical simulations, for which we must prepare appropriate equations that well reflect the behavior of real-world phenomena and numerically solve them. For these purposes, we may utilize the data-driven identification and simulation technique of nonlinear partial differential equations (NPDEs) using deep neural networks (DNNs). A potential issue of the DNN-based identification and simulation in practice is the high variance due to the complexity of DNNs. To alleviate it, we propose a simple yet efficient way to incorporate prior knowledge of phenomena. Specifically, we can often anticipate what kinds of terms are present in a part of an appropriate NPDE, which should be utilized as prior knowledge for identifying the remaining part of the NPDE. To this end, we design DNN’s inputs and the loss function for identification according to the prior knowledge. We present the results of the experiments conducted using three different types of NPDEs: the Korteweg–de Vries equation, the Navier–Stokes equation, and the Kuramoto–Sivashinsky equation. The experimental results show the effectiveness of the proposed method, i.e., utilizing known terms of an NPDE.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
We only consider the derivatives with regard to x for ease of discussion in this paper, albeit it is straightforward to add other derivatives such as \(u_{tt}\) and \(u_{xt}\) to T.
References
Anderson, J.S., Kevrekidisi, I.G., Rico-Martínez, R.: A comparison of recurrent training algorithms for time series analysis and system identification. Comput. Chem. Eng. 20, S751–S756 (1996)
Bongard, J., Lipson, H.: Automated reverse engineering of nonlinear dynamical systems. Proc. Nat. Acad. Sci. U.S.A. 104(24), 9943–9948 (2007)
Brunton, S.L., Proctor, J.L., Kutz, J.N.: Discovering governing equations from data: sparse identification of nonlinear dynamical systems. Proc. Nat. Acad. Sci. U.S.A. 113(15), 3932–3937 (2015)
Crutchfield, J.P., McNamara, B.: Equations of motion from a data series. Complex Syst. 1, 417–452 (1987)
Daniels, B.C., Nemenman, I.: Automated adaptive inference of phenomenological dynamical models. Nat. Commun. 6, 8133 (2015a)
Daniels, B.C., Nemenman, I.: Efficient inference of parsimonious phenomenological models of cellular dynamics using S-systems and alternating regression. PLoS ONE 10(3), e0119821 (2015b)
Driscoll, T.A., Hale, N., Trefethen, L.N. (eds.): Chebfun Guide. Pafnuty Publications, Oxford (2014)
González-García, R., Rico-Martínez, R., Kevrekidis, I.G.: Identification of distributed parameter systems: a neural net based approach. Comput. Chem. Eng. 22, S965–S968 (1998)
Hastie, T., Tibshirani, R., Wainwright, M.: Statistical Learning with Sparsity: The Lasso and Generalizations. CRC Press, Boca Raton (2018)
Kevrekidis, I.G., Gear, C.W., Hyman, J.M., Kevrekidis, P.G., Runborg, O., Theodoropoulos, C.: Equation-free multiscale computation: enabling microscopic simulators to perform system-level tasks. Commun. Math. Sci. 1(4), 715–762 (2003)
Mangan, N.M., Brunton, S.L., Proctor, J.L., Kutz, J.N.: Inferring biological networks by sparse identification of nonlinear dynamics. IEEE Trans. Mol. Biol. Multi-Scale Commun. 2(1), 52–63 (2016)
Raissi, M., Perdikaris, P., Karniadakis, G.: 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)
Raissi, M.: Deep hidden physics models: deep learning of nonlinear partial differential equations. J. Mach. Learn. Res. 19(25), 1–24 (2018)
Rico-Martínes, R., Kevrekidis, I.G., Kube, M.C., Hudson, J.L.: Discrete- vs. continuous-time nonlinear signal processing: attractors, transitions and parallel implementation issues. Chem. Eng. Commun. 118(1), 25–48 (1992)
Roberts, A.J.: Model Emergent Dynamics in Complex Systems. SIAM (2015)
Rudy, S.H., Brunton, S.L., Proctor, J.L., Kutz, J.N.: Data-driven discovery of partial differential equations. Sci. Adv. 3(4), e1602614 (2017)
Schaeffer, H., Osher, S., Caflisch, R., Hauck, C.: Sparse dynamics for partial differential equations. Proc. Nat. Acad. Sci. U.S.A. 110(17), 6634–6639 (2013)
Schmidt, M., Lipson, H.: Distilling free-form natural laws from experimental data. Science 324(5923), 81–85 (2012)
Schmidt, M.D., et al.: Automated refinement and inference of analytical models for metabolic networks. Phys. Biol. 8(5), 055011 (2011)
Strang, G.: Computational Science and Engineering. Wellesley-Cambridge Press, Wellesley (2007)
Sugihara, G., et al.: Detecting causality in complex ecosystems. Science 338(6106), 496–500 (2012)
Tran, G., Ward, R.: Exact recovery of chaotic systems from highly corrupted data. Multiscale Model. Simul. 15(3), 1108–1129 (2017)
Ye, H., et al.: Equation-free mechanistic ecosystem forecasting using empirical dynamic modeling. Proc. Nat. Acad. Sci. U.S.A. 112(13), E1569–E1576 (2015)
Acknowledgements
This work was supported by JSPS KAKENHI Grant Numbers JP18H06487, JP19K21550 and JP19K12094.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Sasaki, R., Takeishi, N., Yairi, T., Hori, K. (2019). Neural Gray-Box Identification of Nonlinear Partial Differential Equations. In: Nayak, A., Sharma, A. (eds) PRICAI 2019: Trends in Artificial Intelligence. PRICAI 2019. Lecture Notes in Computer Science(), vol 11671. Springer, Cham. https://doi.org/10.1007/978-3-030-29911-8_24
Download citation
DOI: https://doi.org/10.1007/978-3-030-29911-8_24
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-29910-1
Online ISBN: 978-3-030-29911-8
eBook Packages: Computer ScienceComputer Science (R0)