# Surrogate Model for Shallow Water Equations Solvers with Deep Learning

arXiv: https://arxiv.org/abs/2112.10889 · HTML: https://mlsanity.com/paper/2112.10889/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 46 references that are in the map.
- First posted on arXiv 20 Dec 2021.

## Builds on (most cited first, 15 of 15)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [TensorFlow: Large-Scale Machine Learning on Heterogeneous Distributed Systems](https://mlsanity.com/paper/1603.04467/) (2016)
- [Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1708.00588/) (2017)
- [Bayesian Deep Convolutional Encoder-Decoder Networks for Surrogate Modeling and Uncertainty Quantification](https://mlsanity.com/paper/1801.06879/) (2018)
- [Prediction of Aerodynamic Flow Fields Using Convolutional Neural Networks](https://mlsanity.com/paper/1905.13166/) (2019)
- [Accelerating Eulerian Fluid Simulation With Convolutional Networks](https://mlsanity.com/paper/1607.03597/) (2016)
- [Physics-informed deep learning for incompressible laminar flows](https://mlsanity.com/paper/2002.10558/) (2020)
- [A trans-disciplinary review of deep learning research for water resources scientists](https://mlsanity.com/paper/1712.02162/) (2017)
- [Physics-informed neural networks for the shallow-water equations on the sphere](https://mlsanity.com/paper/2104.00615/) (2021)
- [From calibration to parameter learning: Harnessing the scaling effects of big data in geoscientific modeling](https://mlsanity.com/paper/2007.15751/) (2020)
- [DeepCFD: Efficient Steady-State Laminar Flow Approximation with Deep Convolutional Neural Networks](https://mlsanity.com/paper/2004.08826/) (2020)
- [Physics Informed Data Driven model for Flood Prediction: Application of Deep Learning in prediction of urban flood development](https://mlsanity.com/paper/1908.10312/) (2019)
- [A machine learning framework for LES closure terms](https://mlsanity.com/paper/2010.03030/) (2020)
- [Application of deep learning to large scale riverine flow velocity estimation](https://mlsanity.com/paper/2012.02620/) (2020)
- [Reconstruction of Hydraulic Data by Machine Learning](https://mlsanity.com/paper/1903.01123/) (2019)

## Built on by (most cited first, 2 of 2)

- [Bathymetry Inversion using a Deep-Learning-Based Surrogate for Shallow Water Equations Solvers](https://mlsanity.com/paper/2203.02821/) (2022)
- [Scientific Machine Learning of Flow Resistance Using Universal Shallow Water Equations with Differentiable Programming](https://mlsanity.com/paper/2502.12396/) (2025)

## Abstract (arXiv)

Shallow water equations are the foundation of most models for flooding and river hydraulics analysis. These physics-based models are usually expensive and slow to run, thus not suitable for real-time prediction or parameter inversion. An attractive alternative is surrogate model. This work introduces an efficient, accurate, and flexible surrogate model, NN-p2p, based on deep learning and it can make point-to-point predictions on unstructured or irregular meshes. The new method was evaluated and compared against existing methods based on convolutional neural networks (CNNs), which can only make image-to-image predictions on structured or regular meshes. In NN-p2p, the input includes both spatial coordinates and boundary features that can describe the geometry of hydraulic structures, such as bridge piers. All surrogate models perform well in predicting flow around different types of piers in the training domain. However, only NN-p2p works well when spatial extrapolation is performed. The limitations of CNN-based methods are rooted in their raster-image nature which cannot capture boundary geometry and flow features exactly, which are of paramount importance to fluid dynamics. NN-p2p also has good performance in predicting flow around piers unseen by the neural network. The NN-p2p model also respects conservation laws more strictly. The application of the proposed surrogate model was demonstrated by calculating the drag coefficient CD for piers and a new linear relationship between CD and the logarithmic transformation of pier's length/width ratio was discovered.