Journal of Korea Water Resources Association (한국수자원학회논문집)

Korea Water Resources Association (한국수자원학회)
권호 표지
DOI QR코드

DOI QR Code

Exploring the power of physics-informed neural networks for accurate and efficient solutions to 1D shallow water equations

물리 정보 신경망을 이용한 1차원 천수방정식의 해석

  • Nguyen, Van Giang (Department of Advanced Science and Technology Convergence, Kyungpook National University) ;
  • Nguyen, Van Linh (Department of Advanced Science and Technology Convergence, Kyungpook National University) ;
  • Jung, Sungho (Disaster Prevention and Emergency Management Institute, Kyungpook National University) ;
  • An, Hyunuk (Department of Agricultural. and Rural Engineering, Chungnam National University) ;
  • Lee, Giha (Department of Advanced Science and Technology Convergence, Kyungpook National University)
  • 응웬반지앙 (경북대학교 미래과학기술융합학과) ;
  • 응웬반링 (경북대학교 미래과학기술융합학과) ;
  • 정성호 (경북대학교 재난대응전략연구소) ;
  • 안현욱 (충남대학교 지역환경토목학과) ;
  • 이기하 (경북대학교 미래과학기술융합학과)
  • Received : 2023.07.24
  • Accepted : 2023.11.30
  • Published : 2023.12.31
Download PDF
⟨ Previous Next ⟩

Abstract

Shallow water equations (SWE) serve as fundamental equations governing the movement of the water. Traditional numerical approaches for solving these equations generally face various challenges, such as sensitivity to mesh generation, and numerical oscillation, or become more computationally unstable around shock and discontinuities regions. In this study, we present a novel approach that leverages the power of physics-informed neural networks (PINNs) to approximate the solution of the SWE. PINNs integrate physical law directly into the neural network architecture, enabling the accurate approximation of solutions to the SWE. We provide a comprehensive methodology for formulating the SWE within the PINNs framework, encompassing network architecture, training strategy, and data generation techniques. Through the results obtained from experiments, we found that PINNs could be an accurate output solution of SWE when its results were compared with the analytical method. In addition, PINNs also present better performance over the Artificial Neural Network. This study highlights the transformative potential of PINNs in revolutionizing water resources research, offering a new paradigm for accurate and efficient solutions to the SVE.

천수방정식(shallow water equations, SWE)은 물의 거동을 수치적으로 해석하기 위한 지배방정식으로 수리수문 분야에 널리 활용되고 있으며, 비선형 연립방정식으로 일반적으로 수치적으로 해석할 수 있다. 하지만 기존의 여러 수치 해석법은 격자망 생성에 민감하며 복잡한 지형에서의 해석에 한계가 발생할 수 있다. 이러한 한계점을 극복하기 위하여 본 연구에서는 물리 정보 신경망(Physics-Informed Neural Networks, PINNs)을 사용하고자 하였다. PINNs은 물리 법칙을 신경망에 직접적으로 도입하여 지배방정식을 해석하고자 하는 기법이며 지배 방정식에 대한 물리적, 수학적 정보를 손실함수로 변환하여 최적화하고 해를 산정할 수 있다. 본 연구에서는 지배방정식을 PINNs 구조 내에서 사용할 수 있도록 신경망 구조, 학습 전략, 데이터 생성 기술과 같은 포괄적인 방법론을 제시하고 결과를 ANN 기법과 비교하였다. 물리적 사전지식이 반영되지 않은 ANN과 달리 PINNs은 천수방정식에 대하여 매우 정확한 수치적 솔루션을 효과적으로 제공하는 것으로 나타났다. 따라서 PINNs은 지배방정식의 수치해석적 연구에 많은 잠재력이 있는 것으로 판단되며, 정확하고 효율적인 천수방정식의 솔루션을 위한 기법으로 활용될 수 있을 것으로 기대된다.

Keywords

Acknowledgement

This work was supported by Korea Environmental Industry & Technology Institute (KEITI) through R&D Program for Innovative Flood Protection Technologies against Climate Crisis Project, funded by Korea Ministry of Environment (MOE)(2022003460002)

References

  1. An, H., and Yu, S. (2014). "Finite volume integrated surface-subsurface flow modeling on nonorthogonal grids." Water Resources Research, Vol. 50, No. 3, pp. 2312-2328. https://doi.org/10.1002/2013WR013828
  2. Ayog, J.L., Kesserwani, G., Shaw, J., Sharifian, M.K., and Bau, D. (2021). "Second-order discontinuous Galerkin flood model: Comparison with industry-standard finite volume models." Journal of Hydrology, Vol. 594, 125924.
  3. Bandai, T., and Ghezzehei, T.A. (2022). "Forward and inverse modeling of water flow in unsaturated soils with discontinuous hydraulic conductivities using physics-informed neural networks with domain decomposition." Hydrology and Earth System Sciences, Vol. 26, No. 16, pp. 4469-4495. https://doi.org/10.5194/hess-26-4469-2022
  4. Bates, P.D., Dawson, R.J., Hall, J.W., Horritt, M.S., Nicholls, R.J., Wicks, J., and Hassan, M.A.A.M. (2005). "Simplified two-dimensional numerical modelling of coastal flooding and example applications." Coastal Engineering, Vol. 52, No. 9, pp. 793-810. https://doi.org/10.1016/j.coastaleng.2005.06.001
  5. Bates, P.D., Horritt, M.S., and Fewtrell, T.J. (2010). "A simple inertial formulation of the shallow water equations for efficient two-dimensional flood inundation modelling." Journal of Hydrology, Vol. 387, No. 1-2, pp. 33-45. https://doi.org/10.1016/j.jhydrol.2010.03.027
  6. Bec k, C., E., W., and Jentzen, A. (2019). "Mac hine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order backward stochastic differential equations." Journal of Nonlinear Science, Vol. 29, pp. 1563-1619.
  7. Bermudez, A., Dervieux, A., Desideri, J.-A., and Vazquez, M.E. (1998). "Upwind schemes for the two-dimensional shallow water equations with variable depth using unstructured meshes." Computer Methods in Applied Mechanics and Engineering, Vol. 155, No. 1-2, pp. 49-72. https://doi.org/10.1016/S0045-7825(97)85625-3
  8. Cai, S., Mao, Z., Wang, Z., Yin, M., and Karniadakis, G.E. (2021). "Physics-informed neural networks (PINNs) for fluid mechanics: A review." Acta Mechanica Sinica, Vol. 37 No. 12, pp. 1727-1738. https://doi.org/10.1007/s10409-021-01148-1
  9. Cea, L., and Blade, E. (2015). "A simple and efficient unstructured finite volume scheme for solving the shallow water equations in overland flow applications." Water Resources Research, Vol. 51 No. 7, pp. 5464-5486. https://doi.org/10.1002/2014WR016547
  10. Chen, Y., Xu, Y., Wang, L., and Li, T. (2023). "Modeling water flow in unsaturated soils through physics-informed neural network with principled loss function." Computers and Geotechnics, Vol. 161, 105546.
  11. de Saint-Venant, A.J.-C. (1871). "Theorie du mouvement nonpermanent des eaux, avec application aux crues des rivieres et a l'introduction des marees dans leur lit." Academic de Sci. Comptes Redus, Vol. 73 No. 147-154, pp. 237-240.
  12. Feng, D., Tan, Z., and He, Q. (2023). "Physics-informed neural networks of the Saint-Venant equations for downscaling a large-scale river model." Water Resources Research, Vol. 59 No. 2, e2022WR033168.
  13. Ferrari, A., Viero, D.P., Vacondio, R., Defina, A., and Mignosa, P. (2019). "Flood inundation modeling in urbanized areas: A mesh-independent porosity approach with anisotropic friction." Advances in Water Resources, Vol. 125, pp. 98-113. https://doi.org/10.1016/j.advwatres.2019.01.010
  14. Gao, H., Sun, L., and Wang, J.-X. (2021). "PhyGeoNet: Physics-informed geometry-adaptive convolutional neural networks for solving parameterized steady-state PDEs on irregular domain." Journal of Computational physics, Vol. 428, 110079. https://doi.org/10.1016/j.jcp.2020.110079
  15. Guo, Z., Leitao, J.P., Simoes, N.E., and Moosavi, V. (2021). "Data-driven flood emulation: Speeding up urban flood predictions by deep convolutional neural networks." Journal of Flood Risk Management, Vol. 14, No. 1, e12684. https://doi.org/10.1111/jfr3.12684
  16. Hai, P.T., Masumoto, T., and Shimizu, K. (2008). "Development of a two-dimensional finite element model for inundation processes in the Tonle Sap and its environs." Hydrological Processes: An International Journal, Vol. 22, No. 9, pp. 1329-1336. https://doi.org/10.1002/hyp.6942
  17. Han, J., Jentzen, A., and E.W. (2018). "Solving high-dimensional partial differential equations using deep learning." Proceedings of the National Academy of Sciences, Vol. 115, No. 34, pp. 8505-8510. https://doi.org/10.1073/pnas.1718942115
  18. Hodges, B.R. (2019). "Conservative finite-volume forms of the Saint-Venant equations for hydrology and urban drainage." Hydrology and Earth System Sciences, Vol. 23, No. 3, pp. 1281-1304. https://doi.org/10.5194/hess-23-1281-2019
  19. Hunter, N.M., Horritt, M.S., Bates, P.D., Wilson, M.D., and Werner, M.G. (2005). "An adaptive time step solution for raster-based storage cell modelling of floodplain inundation." Advances in Water Resources, Vol. 28, No. 9, pp. 975-991. https://doi.org/10.1016/j.advwatres.2005.03.007
  20. Jagtap, A.D., Mao, Z., Adams, N., and Karniadakis, G.E. (2022). "Physics-informed neural networks for inverse problems in supersonic flows." Journal of Computational physics, Vol. 466, 111402.
  21. Jin, X., Cai, S., Li, H., and Karniadakis, G.E. (2021). "NSFnets (Navier-Stokes flow nets): Physics-informed neural networks for the incompressible Navier-Stokes equations." Journal of Computational physics, Vol. 426, 109951.
  22. Kabir, S., Patidar, S., Xia, X., Liang, Q., Neal, J., and Pender, G. (2020). "A deep convolutional neural network model for rapid prediction of fluvial flood inundation." Journal of Hydrology, Vol. 590, 125481.
  23. Karniadakis, G.E., Kevrekidis, I.G., Lu, L., Perdikaris, P., Wang, S., and Yang, L. (2021). "Physics-informed machine learning." Nature Reviews Physics, Vol. 3, No. 6, pp. 422-440. https://doi.org/10.1038/s42254-021-00314-5
  24. Kuffour, B.N., Engdahl, N.B., Woodward, C.S., Condon, L.E., Kollet, S., and Maxwell, R.M. (2020). "Simulating coupled surface-subsurface flows with ParFlow v3. 5.0: capabilities, applications, and ongoing development of an open-source, massively parallel, integrated hydrologic model." Geoscientific Model Development, Vol. 13, No. 3, pp. 1373-1397. https://doi.org/10.5194/gmd-13-1373-2020
  25. Lee, S.Y., Park, C.-S., Park, K., Lee, H.J., and Lee, S. (2022). "A Physics-informed and data-driven deep learning approach for wave propagation and its scattering characteristics." Engineering with Computers, Vol. 39, pp. 1-17. https://doi.org/10.1108/EC-02-2022-759
  26. Liu, Y., and Pender, G. (2015). "A flood inundation modelling using v-support vector machine regression model." Engineering Applications of Artificial Intelligence, Vol. 46, pp. 223-231. https://doi.org/10.1016/j.engappai.2015.09.014
  27. Lu, L., Meng, X., Mao, Z., and Karniadakis, G.E. (2021). "DeepXDE: A deep learning library for solving differential equations." SIAM Review, Vol. 63, No. 1, pp. 208-228. https://doi.org/10.1137/19M1274067
  28. Lu, X., Mao, B., Zhang, X., and Ren, S. (2020). "Well-balanced and shock-capturing solving of 3D shallow-water equations involving rapid wetting and drying with a local 2D transition approach." Computer Methods in Applied Mechanics and Engineering, Vol. 364, 112897.
  29. Lundgren, L., and Mattsson, K. (2020). "An efficient finite difference method for the shallow water equations." Journal of Computational physics, Vol. 422, 109784.
  30. Mao, Z., Jagtap, A. D., and Karniadakis, G.E. (2020). "Physicsinformed neural networks for high-speed flows." Computer Methods in Applied Mechanics and Engineering, Vol. 360, 112789.
  31. Marras, S., Kopera, M.A., Constantinescu, E.M., Suckale, J., and Giraldo, F.X. (2018). "A residual-based shock capturing scheme for the continuous/discontinuous spectral element solution of the 2D shallow water equations." Advances in Water Resources, Vol. 114, pp. 45-63. https://doi.org/10.1016/j.advwatres.2018.02.003
  32. McClenny, L., and Braga-Neto, U. (2020). "Self-adaptive physics-informed neural networks using a soft attention mechanism." arXiv Preprint, arXiv:2009.04544.
  33. Ni, Y., Cao, Z., Liu, Q., and Liu, Q. (2020). "A 2D hydrodynamic model for shallow water flows with significant infiltration losses." Hydrological Processes, Vol. 34, No. 10, pp. 2263-2280. https://doi.org/10.1002/hyp.13722
  34. Peng, G.C., Alber, M., Buganza Tepole, A., Cannon, W.R., De, S., Dura-Bernal, S., Garikipati, K., Karniadakis, G., Lytton, W.W., and Perdikaris, P. (2021). "Multiscale modeling meets machine learning: What can we learn?." Archives of Computational Methods in Engineering, Vol. 28, pp. 1017-1037. https://doi.org/10.1007/s11831-020-09405-5
  35. Raissi, M., Perdikaris, P., and Karniadakis, G.E. (2019). "Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations." Journal of Computational Physics, Vol. 378, pp. 686-707. https://doi.org/10.1016/j.jcp.2018.10.045
  36. Ren, S., Wu, S., and Weng, Q. (2023). "Physics-informed machine learning methods for biomass gasification modeling by considering monotonic relationships." Bioresource Technology, Vol. 369, 128472.
  37. Riley, P. (2019). "Three pitfalls to avoid in machine learning." Nature, Vol. 572, No. 7767, pp. 27-29. https://doi.org/10.1038/d41586-019-02307-y
  38. Sirignano, J., and Spiliopoulos, K. (2018). "DGM: A deep learning algorithm for solving partial differential equations." Journal of Computational physics, Vol. 375, pp. 1339-1364. https://doi.org/10.1016/j.jcp.2018.08.029
  39. Soares-Frazao, S., Canelas, R., Cao, Z., Cea, L., Chaudhry, H.M., Die Moran, A., El Kadi, K., Ferreira, R., Cadorniga, I.F., and Gonzalez-Ramirez, N. (2012). "Dam-break flows over mobile beds: experiments and benchmark tests for numerical models." Journal of Hydraulic Research, Vol. 50, No. 4, pp. 364-375. https://doi.org/10.1080/00221686.2012.689682
  40. Sun, L., and Wang, J.-X. (2020). "Physics-constrained bayesian neural network for fluid flow reconstruction with sparse and noisy data." Theoretical and Applied Mechanics Letters, Vol. 10, No. 3, pp. 161-169. https://doi.org/10.1016/j.taml.2020.01.031
  41. Tartakovsky, A.M., Marrero, C.O., Perdikaris, P., Tartakovsky, G.D., and Barajas-Solano, D. (2020). "Physics-informed deep neural networks for learning parameters and constitutive relationships in subsurface flow problems." Water Resources Research, Vol. 56, No. 5, e2019WR026731.
  42. Vacondio, R., Dal Palu, A., and Mignosa, P. (2014). "GPU-enhanced finite volume shallow water solver for fast flood simulations." Environmental Modelling and Software, Vol. 57, pp. 60-75. https://doi.org/10.1016/j.envsoft.2014.02.003
  43. Valiani, A., and Caleffi, V. (2019). "Dam break in rectangular channels with different upstream-downstream widths." Advances in Water Resources, Vol. 132, 103389.
  44. Van, C.P., Gourgue, O., Sassi, M., Hoitink, A., Deleersnijder, E., and Soares-Frazao, S. (2016). "Modelling fine-grained sediment transport in the Mahakam land-sea continuum, Indonesia." Journal of Hydro-environment Research, Vol. 13, pp. 103-120. https://doi.org/10.1016/j.jher.2015.04.005
  45. Viero, D.P., and Valipour, M. (2017). "Modeling anisotropy in freesurface overland and shallow inundation flows." Advances in Water Resources, Vol. 104, pp. 1-14. https://doi.org/10.1016/j.advwatres.2017.03.007
  46. Vijayaraghavan, S., Wu, L., Noels, L., Bordas, S., Natarajan, S., and Beex, L.A. (2023). "A data-driven reduced-order surrogate model for entire elastoplastic simulations applied to representative volume elements." Scientific Reports, Vol. 13, No. 1, 12781.
  47. Wang, N., Zhang, D., Chang, H., and Li, H. (2020). "Deep learning of subsurface flow via theory-guided neural network." Journal of Hydrology, Vol. 584, 124700.
  48. Wang, S., Teng, Y., and Perdikaris, P. (2021). "Understanding and mitigating gradient flow pathologies in physics-informed neural networks." SIAM Journal on Scientific Computing, Vol. 43, No. 5, pp. A3055-A3081. https://doi.org/10.1137/20M1318043
  49. West, D.W., Kubatko, E.J., Conroy, C.J., Yaufman, M., and Wood, D. (2017). "A multidimensional discontinuous Galerkin modeling framework for overland flow and channel routing." Advances in Water Resources, Vol. 102, pp. 142-160. https://doi.org/10.1016/j.advwatres.2017.02.008
  50. Wu, C., Zhu, M., Tan, Q., Kartha, Y., and Lu, L. (2023). "A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks." Computer Methods in Applied Mechanics and Engineering, Vol. 403, 115671.
  51. Xu, Y., Kohtz, S., Boakye, J., Gardoni, P., and Wang, P. (2022). "Physics-informed machine learning for reliability and systems safety applications: State of the art and challenges." Reliability Engineering and System Safety, Vol. 230, 108900.
  52. Zhang, Z. (2022). "A physics-informed deep convolutional neural network for simulating and predicting transient Darcy flows in heterogeneous reservoirs without labeled data." Journal of Petroleum Science and Engineering, Vol. 211, 110179.
  53. Zobeiry, N., and Humfeld, K.D. (2021). "A physics-informed machine learning approach for solving heat transfer equation in advanced manufacturing and engineering applications." Engineering Applications of Artificial Intelligence, Vol. 101, 104232.