# Approximating the Solution of Surface Wave Propagation Using Deep Neural Networks

## Abstract

Partial differential equations formalise the understanding of the behaviour of the physical world that humans acquire through experience and observation. Through their numerical solution, such equations are used to model and predict the evolution of dynamical systems. However, such techniques require extensive computational resources and assume the physics are prescribed *a priori*. Here, we propose a neural network capable of predicting the evolution of a specific physical phenomenon: propagation of surface waves enclosed in a tank, which, mathematically, can be described by the Saint-Venant equations. The existence of reflections and interference makes this problem non-trivial. Forecasting of future states (i.e. spatial patterns of rendered wave amplitude) is achieved from a relatively small set of initial observations. Using a network to make approximate but rapid predictions would enable the active, real-time control of physical systems, often required for engineering design. We used a deep neural network comprising of three main blocks: an encoder, a propagator with three parallel Long Short-Term Memory layers, and a decoder. Results on a novel, custom dataset of simulated sequences produced by a numerical solver show reasonable predictions for as long as 80 time steps into the future on a hold-out dataset. Furthermore, we show that the network is capable of generalising to two other initial conditions that are qualitatively different from those seen at training time.

## Keywords

Deep learning for physical systems Recurrent neural networks Representation learning## References

- 1.Cellier, N.: Locie/triflow: Triflow v0.5.0., May 2018. https://doi.org/10.5281/zenodo.1239703
- 2.Ehrhardt, S., Monszpart, A., Mitra, N.J., Vedaldi, A.: Learning a physical long-term predictor. ArXiv e-prints arXiv:1703.00247, March 2017
- 3.Finn, C., Goodfellow, I., Levine, S.: Unsupervised learning for physical interaction through video prediction. In: NeurIPS, pp. 64–72 (2016)Google Scholar
- 4.Guevara, T.L., Gutmann, M.U., Taylor, N.K., Ramamoorthy, S., Subr, K.: Adaptable pouring: teaching robots not to spill using fast but approximate fluid simulation. ArXiv e-prints arXiv:1708.01465v2, August 2017
- 5.Kim, B., Azevedo, V.C., Thuerey, N., Kim, T., Gross, M., Solenthaler, B.: Deep fluids: a generative network for parameterized fluid simulations. ArXiv e-prints arXiv:1806.02071, June 2018
- 6.Long, Y., She, X., Mukhopadhyay, S.: HybridNet: integrating model-based and data-driven learning to predict evolution of dynamical systems. ArXiv e-prints arXiv:1806.07439, June 2018
- 7.Mangeney-Castelnau, A., Vilotte, J.P., Bristeau, M.O., Perthame, B., Bouchut, F., Simeoni, C., Yerneni, S.: Numerical modeling of avalanches based on Saint Venant equations using a kinetic scheme. J. Geophys. Res. Solid Earth
**108**(B11) (2003)Google Scholar - 8.Oezgen, I., Zhao, J., Liang, D., Hinkelmann, R.: Urban flood modeling using shallow water equations with depth-dependent anisotropic porosity. J. Hydrol.
**541**, 1165–1184 (2016)CrossRefGoogle Scholar - 9.Pathak, J., Lu, Z., Hunt, B.R., Girvan, M., Ott, E.: Using machine learning to replicate chaotic attractors and calculate lyapunov exponents from data. Chaos Interdisc. J. Nonlinear Sci.
**27**(12), 121102 (2017)MathSciNetCrossRefGoogle Scholar - 10.Sanchez-Gonzalez, A., Heess, N., Springenberg, J.T., Merel, J., Riedmiller, M., Hadsell, R., Battaglia, P.: Graph networks as learnable physics engines for inference and control. ArXiv e-prints arXiv:1806.01242, June 2018
- 11.Schenck, C., Fox, D.: SPNets: differentiable fluid dynamics for deep neural networks. In: Billard, A., Dragan, A., Peters, J., Morimoto, J. (eds.) Proceedings of The 2nd Conference on Robot Learning. Proceedings of Machine Learning Research, PMLR, 29–31 October 2018, vol. 87, pp. 317–335 (2018)Google Scholar
- 12.Tompson, J., Schlachter, K., Sprechmann, P., Perlin, K.: Accelerating Eulerian fluid simulation with convolutional networks. ArXiv e-prints arXiv:1607.03597v6, June 2017
- 13.Wang, Z., Bovik, A.C., Sheikh, H.R., Simoncelli, E.P.: Image quality assessment: from error visibility to structural similarity. IEEE Trans. Image Process.
**13**(4), 600–612 (2004)CrossRefGoogle Scholar - 14.Wiewel, S., Becher, M., Thuerey, N.: Latent-space physics: towards learning the temporal evolution of fluid flow. ArXiv e-prints arXiv:1802.10123v2, June 2018
- 15.Yang, C., Yang, X., Xiao, X.: Data-driven projection method in fluid simulation. Comput. Animation Virtual Worlds
**27**, 415–424 (2016)CrossRefGoogle Scholar