Bathymetry Inversion using a Deep-Learning-Based Surrogate for Shallow Water Equations Solvers

River bathymetry is critical for many aspects of water resources management. We propose and demonstrate a bathymetry inversion method using a deep-learning-based surrogate for shallow water equations solvers. The surrogate uses the convolutional autoencoder with a shared-encoder, separate-decoder architecture. It encodes the input bathymetry and decodes to separate outputs for flow-field variables. A gradient-based optimizer is used to perform bathymetry inversion with the trained surrogate. Two physically-based constraints on both bed elevation value and slope have to be added as inversion loss regularizations to obtain usable inversion results. Using the "L-curve" criterion, a heuristic approach was proposed to determine the regularization parameters. Both the surrogate model and the inversion algorithm show good performance. We found the bathymetry inversion process has two distinctive stages, which resembles the sculptural process of initial broad-brush calving and final detailing. The inversion loss due to flow prediction error reaches its minimum in the first stage and remains almost constant afterward. The bed elevation value and slope regularizations play the dominant role in the second stage in selecting the most probable solution. We also found the surrogate architecture (whether with both velocity and water surface elevation or velocity only as outputs) does not show significant impact on inversion result.

Surrogate Model for Shallow Water Equations Solvers with Deep Learning

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 $C_D$ for piers and a new linear relationship between $C_D$ and the logarithmic transformation of pier's length/width ratio was discovered.