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. Utilizing the differentiability of the surrogate, a gradient-based optimizer is used to perform bathymetry inversion. Two physically based constraints on the ranges of both bed elevation and slope have to be added as inversion loss regularizations to contract the solution space. Using the “L-curve” criterion, a heuristic approach is proposed to determine the regularization parameters. Both the surrogate model and the inversion algorithm show good performance. The bathymetry inversion progresses in two distinctive stages, which resembles the sculptural process of initial broad-brush calving and final fine detailing. The inversion loss due to flow prediction error and the two regularizations play dominant roles in the initial and final stages, respectively. The surrogate architecture (whether with both velocity and water surface elevation or velocity only as outputs) does not have significant impact on inversion result. The methodology proposed in this work, an example of differentiable parameter learning, can be similarly used in the inversion of other important distributed parameters such as roughness coefficient.