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

arXiv: https://arxiv.org/abs/2203.02821 · HTML: https://mlsanity.com/paper/2203.02821/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 65 references that are in the map.
- First posted on arXiv 5 Mar 2022.

## Builds on (most cited first, 10 of 10)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [A trans-disciplinary review of deep learning research for water resources scientists](https://mlsanity.com/paper/1712.02162/) (2017)
- [Benchmarking deep inverse models over time, and the neural-adjoint method](https://mlsanity.com/paper/2009.12919/) (2020)
- [From calibration to parameter learning: Harnessing the scaling effects of big data in geoscientific modeling](https://mlsanity.com/paper/2007.15751/) (2020)
- [Inversion using a new low-dimensional representation of complex binary geological media based on a deep neural network](https://mlsanity.com/paper/1710.09196/) (2017)
- [Markov Chain Monte Carlo with Neural Network Surrogates: Application to Contaminant Source Identification](https://mlsanity.com/paper/2003.02322/) (2020)
- [Application of deep learning to large scale riverine flow velocity estimation](https://mlsanity.com/paper/2012.02620/) (2020)
- [Surrogate Model for Shallow Water Equations Solvers with Deep Learning](https://mlsanity.com/paper/2112.10889/) (2021)

## Built on by (most cited first, 6 of 6)

- [Differentiable modeling to unify machine learning and physical models and advance Geosciences](https://mlsanity.com/paper/2301.04027/) (2023)
- [Scientific Machine Learning of Flow Resistance Using Universal Shallow Water Equations with Differentiable Programming](https://mlsanity.com/paper/2502.12396/) (2025)
- [A Neural Operator Emulator for Coastal and Riverine Shallow Water Dynamics](https://mlsanity.com/paper/2502.14782/) (2025)
- [PETAL: Physics Emulation Through Averaged Linearizations for Solving Inverse Problems](https://mlsanity.com/paper/2305.11056/) (2023)
- [Harnessing AI for Inverse Partial Differential Equation Problems: Past, Present, and Prospects](https://mlsanity.com/paper/2605.16966/) (2026)
- [Finite Volume-Informed Neural Network Framework for 2D Shallow Water Equations: Rugged Loss Landscapes and the Importance of Data Guidance](https://mlsanity.com/paper/2605.11001/) (2026)

## Abstract (arXiv)

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.