# A Hybrid Theory-Driven and Data-Driven Modeling Method for Solving the Shallow Water Equations

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{7}

^{8}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Governing Equations

## 3. Convection Flux Term Calculation Based on the Data-Driven Method

#### 3.1. Generation of the Training Dataset

#### 3.2. Design of the Data-Driven Model

#### 3.3. Model Training

^{−8}. The batch size we used was 128 times the resampling factor.

#### 3.4. The Trained Model

## 4. Construction of the Hybrid Solver

## 5. Model Validation

#### 5.1. Comparison with the Analytical Solution Results

^{−6}, and the MAE of the simulation results of our proposed solver was about 0.004. From the above figure, we can see that the solver we propose can guarantee the stability well in the shock wave environment, and the format constructed in this paper can guarantee the high precision of the solution.

#### 5.2. Comparison of Solver Simulation Results with Real Landslide Cases

^{6}m

^{3}. This catastrophic event led to the destruction of 22 houses, claiming the lives of 42 people, with nine individuals reported as missing. In this study, we utilized simulation data obtained from Guo [34] for our analysis. This case serves as a case study to assess both the stability and generalization capabilities of the proposed hybrid SWE solver (Figure 10).

#### 5.3. Model Application in Woda Town Landslide

^{7}m

^{3}. We employed the Coulomb friction model as the stress constitutive model, with a friction coefficient of 0.3 as suggested by Liu and He [63].

## 6. Discussion and Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Liu, D.; Cui, Y.; Wang, H.; Jin, W.; Wu, C.; Bazai, N.A.; Zhang, G.; Carling, P.A.; Chen, H. Assessment of local outburst flood risk from successive landslides: Case study of Baige landslide-dammed lake, upper Jinsha river, eastern Tibet. J. Hydrol.
**2021**, 599, 126294. [Google Scholar] [CrossRef] - Yan, Y.; Cui, Y.; Liu, D.; Tang, H.; Li, Y.; Tian, X.; Zhang, L.; Hu, S. Seismic signal characteristics and interpretation of the 2020 “6.17” Danba landslide dam failure hazard chain process. Landslides
**2021**, 18, 2175–2192. [Google Scholar] [CrossRef] - Kan, G.; Yao, C.; Li, Q.; Li, Z.; Yu, Z.; Liu, Z.; Ding, L.; He, X.; Liang, K. Improving event-based rainfall-runoff simulation using an ensemble artificial neural network based hybrid data-driven model. Stoch. Environ. Res. Risk Assess.
**2015**, 29, 1345–1370. [Google Scholar] [CrossRef] - Kan, G.; He, X.; Ding, L.; Li, J.; Hong, Y.; Zuo, D.; Ren, M.; Lei, T.; Liang, K. Fast hydrological model calibration based on the heterogeneous parallel computing accelerated shuffled complex evolution method. Eng. Optim.
**2017**, 50, 106–119. [Google Scholar] [CrossRef] - Kan, G.; Li, J.; Zhang, X.; Ding, L.; He, X.; Liang, K.; Jiang, X.; Ren, M.; Li, H.; Wang, F. A new hybrid data-driven model for event-based rainfall–runoff simulation. Neural Comput. Appl.
**2017**, 28, 2519–2534. [Google Scholar] [CrossRef] - Shunyu, Y.; Bazai, N.A.; Jinbo, T.; Hu, J.; Shujian, Y.; Qiang, Z.; Ahmed, T.; Jian, G. Dynamic process of a typical slope debris flow: A case study of the wujia gully, Zengda, Sichuan Province, China. Nat. Hazards
**2022**, 112, 565–586. [Google Scholar] [CrossRef] - Yan, Y.; Tang, H.; Hu, K.; Turowski, J.M.; Wei, F. Deriving Debris-Flow Dynamics From Real-Time Impact-Force Measurements. J. Geophys. Res. Earth Surf.
**2023**, 128, e2022JF006715. [Google Scholar] [CrossRef] - Yan, Y.; Cui, Y.; Huang, X.; Zhang, W.; Yin, S.; Zhou, J.; Hu, S. Combining seismic signal dynamic inversion and numerical modeling improves landslide process reconstruction. EGUsphere
**2022**, 10, 1233–1252. [Google Scholar] [CrossRef] - Cao, Z.; Pender, G.; Wallis, S.; Carling, P. Computational Dam-Break Hydraulics over Erodible Sediment Bed. J. Hydraul. Eng.
**2004**, 130, 689–703. [Google Scholar] [CrossRef] - Ouyang, C.; He, S.; Xu, Q. MacCormack-TVD Finite Difference Solution for Dam Break Hydraulics over Erodible Sediment Beds. J. Hydraul. Eng.
**2015**, 141, 06014026. [Google Scholar] [CrossRef] - Bazai, N.A.; Cui, P.; Carling, P.A.; Wang, H.; Hassan, J.; Liu, D.; Zhang, G.; Wen, J. Increasing glacial lake outburst flood hazard in response to surge glaciers in the Karakoram. Earth-Sci. Rev.
**2020**, 212, 103432. [Google Scholar] [CrossRef] - Touma, R. Central unstaggered finite volume schemes for hyperbolic systems: Applications to unsteady shallow water equations. Appl. Math. Comput.
**2009**, 213, 47–59. [Google Scholar] [CrossRef] - Murillo, J.; Navas-Montilla, A. A comprehensive explanation and exercise of the source terms in hyperbolic systems using Roe type solutions. Application to the 1D-2D shallow water equations. Adv. Water Resour.
**2016**, 98, 70–96. [Google Scholar] [CrossRef] - Ricchiuto, M. Contributions to the Development of Residual Discretizations for Hyperbolic Conservation Laws with Application to Shallow Water Flows; Université Sciences et Technologies-Bordeaux I: Gradignan, France, 2011. [Google Scholar]
- Ouyang, C.; He, S.; Xu, Q.; Luo, Y.; Zhang, W. A MacCormack-TVD finite difference method to simulate the mass flow in mountainous terrain with variable computational domain. Comput. Geosci.
**2013**, 52, 1–10. [Google Scholar] [CrossRef] - Ouyang, C.; He, S.; Tang, C. Numerical analysis of dynamics of debris flow over erodible beds in Wenchuan earthquake-induced area. Eng. Geol.
**2015**, 194, 62–72. [Google Scholar] [CrossRef] - Ouyang, C.; Wang, Z.; An, H.; Liu, X.; Wang, D. An example of a hazard and risk assessment for debris flows—A case study of Niwan Gully, Wudu, China. Eng. Geol.
**2019**, 263, 105351. [Google Scholar] [CrossRef] - Ouyang, C.; An, H.; Zhou, S.; Wang, Z.; Su, P.; Wang, D.; Cheng, D.; She, J. Insights from the failure and dynamic characteristics of two sequential landslides at Baige village along the Jinsha River, China. Landslides
**2019**, 16, 1397–1414. [Google Scholar] [CrossRef] - Ouyang, C.; Zhou, K.; Xu, Q.; Yin, J.; Peng, D.; Wang, D.; Li, W. Dynamic analysis and numerical modeling of the 2015 catastrophic landslide of the construction waste landfill at Guangming, Shenzhen, China. Landslides
**2016**, 14, 705–718. [Google Scholar] [CrossRef] - Iverson, R.M.; Ouyang, C. Entrainment of bed material by Earth-surface mass flows: Review and reformulation of depth-integrated theory. Rev. Geophys.
**2015**, 53, 27–58. [Google Scholar] [CrossRef] - Rickenmann, D.; Laigle, D.; McArdell, B.W.; Hübl, J. Comparison of 2D debris-flow simulation models with field events. Comput. Geosci.
**2006**, 10, 241–264. [Google Scholar] [CrossRef] - Peng, S.-H.; Lu, S.-C. FLO-2D simulation of mudflow caused by large landslide due to extremely heavy rainfall in southeastern Taiwan during Typhoon Morakot. J. Mt. Sci.
**2013**, 10, 207–218. [Google Scholar] [CrossRef] - Neglia, F.; Sulpizio, R.; Dioguardi, F.; Capra, L.; Sarocchi, D. Shallow-water models for volcanic granular flows: A review of strengths and weaknesses of TITAN2D and FLO2D numerical codes. J. Volcanol. Geotherm. Res.
**2021**, 410, 107146. [Google Scholar] [CrossRef] - Nocentini, M.; Tofani, V.; Gigli, G.; Fidolini, F.; Casagli, N. Modeling debris flows in volcanic terrains for hazard mapping: The case study of Ischia Island (Italy). Landslides
**2014**, 12, 831–846. [Google Scholar] [CrossRef] - Luppichini, M.; Favalli, M.; Isola, I.; Nannipieri, L.; Giannecchini, R.; Bini, M. Influence of Topographic Resolution and Accuracy on Hydraulic Channel Flow Simulations: Case Study of the Versilia River (Italy). Remote Sens.
**2019**, 11, 1630. [Google Scholar] [CrossRef] - Pratomo, R.A. Sensitivity analysis of flash-flood modelling in Grenada, as a small island Caribbean states. AIP Conf. Proc.
**2016**, 1730, 070002. [Google Scholar] - Pérez-Molina, E.; Sliuzas, R.; Flacke, J.; Jetten, V. Developing a cellular automata model of urban growth to inform spatial policy for flood mitigation: A case study in Kampala, Uganda. Comput. Environ. Urban Syst.
**2017**, 65, 53–65. [Google Scholar] [CrossRef] - Van den Bout, B.; Lombardo, L.; Chiyang, M.; van Westen, C.; Jetten, V. Physically-based catchment-scale prediction of slope failure volume and geometry. Eng. Geol.
**2021**, 284, 105942. [Google Scholar] [CrossRef] - Bout, B.; Lombardo, L.; van Westen, C.J.; Jetten, V.G. Integration of two-phase solid fluid equations in a catchment model for flashfloods, debris flows and shallow slope failures. Environ. Model. Softw.
**2018**, 105, 1–16. [Google Scholar] [CrossRef] - van den Bout, B.; van Asch, T.; Hu, W.; Tang, C.X.; Mavrouli, O.; Jetten, V.G.; van Westen, C.J. Towards a model for structured mass movements: The OpenLISEM hazard model 2.0a. Geosci. Model Dev.
**2021**, 14, 1841–1864. [Google Scholar] [CrossRef] - Pratomo, R.A.; Jetten, V.; Alkema, D. A comparison of flash flood response at two different watersheds in Grenada, Caribbean Islands. IOP Conf. Ser. Earth Environ. Sci.
**2016**, 29, 012004. [Google Scholar] [CrossRef] - Bout, B.; Jetten, V.G. The validity of flow approximations when simulating catchment-integrated flash floods. J. Hydrol.
**2018**, 556, 674–688. [Google Scholar] [CrossRef] - Umer, Y.M.; Jetten, V.G.; Ettema, J. Sensitivity of flood dynamics to different soil information sources in urbanized areas. J. Hydrol.
**2019**, 577, 123945. [Google Scholar] [CrossRef] - Guo, J.; Yi, S.; Yin, Y.; Cui, Y.; Qin, M.; Li, T.; Wang, C. The effect of topography on landslide kinematics: A case study of the Jichang town landslide in Guizhou, China. Landslides
**2020**, 17, 959–973. [Google Scholar] [CrossRef] - Shen, W.; Li, T.; Li, P.; Guo, J. A modified finite difference model for the modeling of flowslides. Landslides
**2018**, 15, 1577–1593. [Google Scholar] [CrossRef] - Bai, X.; He, S. Dynamic process of the massive Aru glacier collapse in Tibet. Landslides
**2020**, 17, 1353–1361. [Google Scholar] [CrossRef] - Denlinger, R.P.; Iverson, R.M. Granular avalanches across irregular three-dimensional terrain: 1. Theory and computation. J. Geophys. Res. Earth Surf.
**2004**, 109. [Google Scholar] [CrossRef] - LeCun, Y.; Bengio, Y.; Hinton, G. Deep learning. Nature
**2015**, 521, 436–444. [Google Scholar] [CrossRef] - Marx, V. The big challenges of big data. Nature
**2013**, 498, 255–260. [Google Scholar] [CrossRef] - Reichstein, M.; Camps-Valls, G.; Stevens, B.; Jung, M.; Denzler, J.; Carvalhais, N.; Prabhat. Deep learning and process understanding for data-driven Earth system science. Nature
**2019**, 566, 195–204. [Google Scholar] [CrossRef] - Jordan, M.I.; Mitchell, T.M. Machine learning: Trends, perspectives, and prospects. Science
**2015**, 349, 255–260. [Google Scholar] [CrossRef] - Tompson, J.; Schlachter, K.; Sprechmann, P.; Perlin, K. Accelerating eulerian fluid simulation with convolutional networks. In Proceedings of the International Conference on Machine Learning, Sydney, NSW, Australia, 6–11 August 2017; pp. 3424–3433. [Google Scholar]
- Bar-Sinai, Y.; Hoyer, S.; Hickey, J.; Brenner, M.P. Learning data-driven discretizations for partial differential equations. Proc. Natl. Acad Sci. USA
**2019**, 116, 15344–15349. [Google Scholar] [CrossRef] [PubMed] - Kochkov, D.; Smith, J.A.; Alieva, A.; Wang, Q.; Brenner, M.P.; Hoyer, S. Machine learning-accelerated computational fluid dynamics. Proc. Natl. Acad Sci. USA
**2021**, 118, e2101784118. [Google Scholar] [CrossRef] - Khoo, Y.; Lu, J.; Ying, L. Solving parametric PDE problems with artificial neural networks. Eur. J. Appl. Math.
**2021**, 32, 421–435. [Google Scholar] [CrossRef] - Adler, J.; Öktem, O. Solving ill-posed inverse problems using iterative deep neural networks. Inverse Probl.
**2017**, 33, 124007. [Google Scholar] [CrossRef] - Bhatnagar, S.; Afshar, Y.; Pan, S.; Duraisamy, K.; Kaushik, S. Prediction of aerodynamic flow fields using convolutional neural networks. Comput. Mech.
**2019**, 64, 525–545. [Google Scholar] [CrossRef] - Guo, X.; Li, W.; Iorio, F. Convolutional neural networks for steady flow approximation. In Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, San Francisco, CA, USA, 13–17 August 2016; pp. 481–490. [Google Scholar]
- Zhu, Y.; Zabaras, N. Bayesian deep convolutional encoder–decoder networks for surrogate modeling and uncertainty quantification. J. Comput. Phys.
**2018**, 366, 415–447. [Google Scholar] [CrossRef] - Bar, L.; Sochen, N. Unsupervised deep learning algorithm for PDE-based forward and inverse problems. arXiv
**2019**, arXiv:1904.05417. [Google Scholar] - Pan, S.; Duraisamy, K. Physics-informed probabilistic learning of linear embeddings of nonlinear dynamics with guaranteed stability. SIAM J. Appl. Dyn. Syst.
**2020**, 19, 480–509. [Google Scholar] [CrossRef] - 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.
**2019**, 378, 686–707. [Google Scholar] [CrossRef] - Smith, J.D.; Azizzadenesheli, K.; Ross, Z.E. Eikonet: Solving the eikonal equation with deep neural networks. IEEE Trans. Geosci. Remote Sens.
**2020**, 59, 10685–10696. [Google Scholar] [CrossRef] - Yu, B. The deep Ritz method: A deep learning-based numerical algorithm for solving variational problems. Commun. Math. Stat.
**2018**, 6, 1–12. [Google Scholar] - Smith, J.D. Stability of a sand bed subjected to a shear flow of low Froude number. J. Geophys. Res.
**1970**, 75, 5928–5940. [Google Scholar] [CrossRef] - Abadi, M.; Barham, P.; Chen, J.; Chen, Z.; Davis, A.; Dean, J.; Devin, M.; Ghemawat, S.; Irving, G.; Isard, M. Tensorflow: A system for large-scale machine learning. In Proceedings of the 12th USENIX Symposium on Operating Systems Design and Implementation (OSDI 16), Savannah, GA, USA, 2–4 November 2016; pp. 265–283. [Google Scholar]
- Kingma, D.P.; Ba, J. Adam: A method for stochastic optimization. arXiv
**2014**, arXiv:1412.6980. [Google Scholar] - Kan, G.; He, X.; Li, J.; Ding, L.; Hong, Y.; Zhang, H.; Liang, K.; Zhang, M. Computer aided numerical methods for hydrological model calibration: An overview and recent development. Arch. Comput. Methods Eng.
**2019**, 26, 35–59. [Google Scholar] [CrossRef] - Kan, G.; Lei, T.; Liang, K.; Li, J.; Ding, L.; He, X.; Yu, H.; Zhang, D.; Zuo, D.; Bao, Z. A multi-core CPU and many-core GPU based fast parallel shuffled complex evolution global optimization approach. IEEE Trans. Parallel Distrib. Syst.
**2016**, 28, 332–344. [Google Scholar] [CrossRef] - Liu, D.; Tang, J.; Wang, H.; Cao, Y.; Bazai, N.A.; Chen, H.; Liu, D. A New Method for Wet-Dry Front Treatment in Outburst Flood Simulation. Water
**2021**, 13, 221. [Google Scholar] [CrossRef] - Li, B.; Jiang, W.; Li, Y.; Luo, Y.; Jiao, Q.; Wang, X.; Zhang, J. Comparison of different atmospheric phase screen correction models in ground-based radar interferometry for landslide and open-pit mine monitoring. Int. J. Remote Sens.
**2021**, 42, 5925–5942. [Google Scholar] [CrossRef] - Li, B.; Jiang, W.; Li, Y.; Luo, Y.; Qian, H.; Wang, Y.; Jiao, Q.; Zhang, Q.; Zhou, Z.; Zhang, J. Monitoring and analysis of Woda landslide stability (China) combined with InSAR, GNSS and meteorological data. Nat. Hazards Earth Syst. Sci. Discuss.
**2021**, 2021, 1–23. [Google Scholar] - Liu, W.; He, S. Numerical Simulation of the Evolution Process of Disaster Chain Induced by Potential Landslide in Woda of Jinsha River Basin. Adv. Eng. Sci.
**2020**, 52, 38–46. [Google Scholar] [CrossRef]

**Figure 2.**Figures showing comparison results of predicted ${\alpha}_{i+1/2}$ values generated by upwind scheme and CNN (The first set of data).

**Figure 3.**Figures showing comparison results of predicted ${\alpha}_{i+1/2}$ values generated by upwind scheme and CNN (The second set of data).

**Figure 7.**Under the condition of smooth bed at T = 10.00 s, demonstrating a solver to simulate water depth and analytical water depth.

**Figure 8.**Under the condition of smooth bed at T = 10.00 s, this paper proposes a solver to simulate water depth and analyze water depth.

**Figure 9.**Under the condition of smooth bed at T = 20.00 s, demonstrating a solver to simulate water depth and analytical water depth.

**Figure 11.**The flow depth of simulation results generated by the proposed solver ((

**a**) at t = 0 s, (

**b**) t = 20 s, (

**c**) t = 50 s) in Jichang town.

**Figure 12.**Figure demonstrating the simulation results generated by the Massflow ((

**b**) at t = 0 s, (

**d**) t = 20 s, (

**f**) t = 60 s) and the proposed solver ((

**a**) at t = 0 s, (

**c**) t = 20 s, (

**e**) t = 60 s) in Woda town.

**Figure 13.**The flow depth histogram of simulation results generated by the Massflow ((

**b**) at t = 0 s, (

**d**) t = 20 s, (

**f**) t = 60 s) and the proposed solver ((

**a**) at t = 0 s, (

**c**) t = 20 s, (

**e**) t = 60 s) in Woda town.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Yao, S.; Kan, G.; Liu, C.; Tang, J.; Cheng, D.; Guo, J.; Jiang, H.
A Hybrid Theory-Driven and Data-Driven Modeling Method for Solving the Shallow Water Equations. *Water* **2023**, *15*, 3140.
https://doi.org/10.3390/w15173140

**AMA Style**

Yao S, Kan G, Liu C, Tang J, Cheng D, Guo J, Jiang H.
A Hybrid Theory-Driven and Data-Driven Modeling Method for Solving the Shallow Water Equations. *Water*. 2023; 15(17):3140.
https://doi.org/10.3390/w15173140

**Chicago/Turabian Style**

Yao, Shunyu, Guangyuan Kan, Changjun Liu, Jinbo Tang, Deqiang Cheng, Jian Guo, and Hu Jiang.
2023. "A Hybrid Theory-Driven and Data-Driven Modeling Method for Solving the Shallow Water Equations" *Water* 15, no. 17: 3140.
https://doi.org/10.3390/w15173140