# Physics-informed neural networks for the shallow-water equations on the sphere

arXiv: https://arxiv.org/abs/2104.00615 · HTML: https://mlsanity.com/paper/2104.00615/

## Key facts

- Cited by 32 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 58 references that are in the map.
- First posted on arXiv 1 Apr 2021.

## Builds on (most cited first, 20 of 20)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
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- [Playing Atari with Deep Reinforcement Learning](https://mlsanity.com/paper/1312.5602/) (2013)
- [Convolutional LSTM Network: A Machine Learning Approach for Precipitation Nowcasting](https://mlsanity.com/paper/1506.04214/) (2015)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [Gradient Surgery for Multi-Task Learning](https://mlsanity.com/paper/2001.06782/) (2020)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [Multi-Task Learning as Multi-Objective Optimization](https://mlsanity.com/paper/1810.04650/) (2018)
- [Large Batch Optimization for Deep Learning: Training BERT in 76 minutes](https://mlsanity.com/paper/1904.00962/) (2019)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [Lookahead Optimizer: k steps forward, 1 step back](https://mlsanity.com/paper/1907.08610/) (2019)
- [Understanding and mitigating gradient pathologies in physics-informed neural networks](https://mlsanity.com/paper/2001.04536/) (2020)
- [PPINN: Parareal Physics-Informed Neural Network for time-dependent PDEs](https://mlsanity.com/paper/1909.10145/) (2019)
- [Improving data-driven global weather prediction using deep convolutional neural networks on a cubed sphere](https://mlsanity.com/paper/2003.11927/) (2020)
- [Machine learning in cardiovascular flows modeling: Predicting arterial blood pressure from non-invasive 4D flow MRI data using physics-informed neural networks](https://mlsanity.com/paper/1905.04817/) (2019)
- [On the convergence of physics informed neural networks for linear second-order elliptic and parabolic type PDEs](https://mlsanity.com/paper/2004.01806/) (2020)
- [A generative adversarial network approach to (ensemble) weather prediction](https://mlsanity.com/paper/2006.07718/) (2020)
- [Precipitation nowcasting using a stochastic variational frame predictor with learned prior distribution](https://mlsanity.com/paper/1905.05037/) (2019)
- [Ensemble methods for neural network-based weather forecasts](https://mlsanity.com/paper/2002.05398/) (2020)

## Built on by (most cited first, 30 of 32)

- [Physics-Informed Machine Learning: A Survey on Problems, Methods and Applications](https://mlsanity.com/paper/2211.08064/) (2022)
- [A unified scalable framework for causal sweeping strategies for Physics-Informed Neural Networks (PINNs) and their temporal decompositions](https://mlsanity.com/paper/2302.14227/) (2023)
- [Lagrangian PINNs: A causality-conforming solution to failure modes of physics-informed neural networks](https://mlsanity.com/paper/2205.02902/) (2022)
- [Improving physics-informed neural networks with meta-learned optimization](https://mlsanity.com/paper/2303.07127/) (2023)
- [Exactly conservative physics-informed neural networks and deep operator networks for dynamical systems](https://mlsanity.com/paper/2311.14131/) (2023)
- [Physics-aware Machine Learning Revolutionizes Scientific Paradigm for Machine Learning and Process-based Hydrology](https://mlsanity.com/paper/2310.05227/) (2023)
- [Improving physics-informed DeepONets with hard constraints](https://mlsanity.com/paper/2309.07899/) (2023)
- [Exact Enforcement of Temporal Continuity in Sequential Physics-Informed Neural Networks](https://mlsanity.com/paper/2403.03223/) (2024)
- [Deep learning for improved global precipitation in numerical weather prediction systems](https://mlsanity.com/paper/2106.12045/) (2021)
- [Computing the ensemble spread from deterministic weather predictions using conditional generative adversarial networks](https://mlsanity.com/paper/2205.09182/) (2022)
- [Large-scale flood modeling and forecasting with FloodCast](https://mlsanity.com/paper/2403.12226/) (2024)
- [ControlSynth Neural ODEs: Modeling Dynamical Systems with Guaranteed Convergence](https://mlsanity.com/paper/2411.02292/) (2024)
- [M-ENIAC: A machine learning recreation of the first successful numerical weather forecasts](https://mlsanity.com/paper/2304.09070/) (2023)
- [Scientific Machine Learning of Flow Resistance Using Universal Shallow Water Equations with Differentiable Programming](https://mlsanity.com/paper/2502.12396/) (2025)
- [Surrogate Model for Shallow Water Equations Solvers with Deep Learning](https://mlsanity.com/paper/2112.10889/) (2021)
- [Short-range forecasts of global precipitation using deep learning-augmented numerical weather prediction](https://mlsanity.com/paper/2206.11669/) (2022)
- [A Survey of Physics-Informed AI for Complex Urban Systems](https://mlsanity.com/paper/2506.13777/) (2025)
- [Towards replacing precipitation ensemble predictions systems using machine learning](https://mlsanity.com/paper/2304.10251/) (2023)
- [PinnDE: Physics-Informed Neural Networks for Solving Differential Equations](https://mlsanity.com/paper/2408.10011/) (2024)
- [ICODE: Modeling Dynamical Systems with Extrinsic Input Information](https://mlsanity.com/paper/2411.13914/) (2024)
- [A comprehensive analysis of PINNs: Variants, Applications, and Challenges](https://mlsanity.com/paper/2505.22761/) (2025)
- [Toward AI-Driven Digital Twins for Metropolitan Floods: A Conditional Latent Dynamics Network Surrogate of the Shallow Water Equations](https://mlsanity.com/paper/2605.13761/) (2026)
- [Improving trajectory calculations using deep learning inspired single image superresolution](https://mlsanity.com/paper/2206.04015/) (2022)
- [Number Theoretic Accelerated Learning of Physics-Informed Neural Networks](https://mlsanity.com/paper/2307.13869/) (2023)
- [Low-rank adaptive physics-informed HyperDeepONets for solving differential equations](https://mlsanity.com/paper/2507.18346/) (2025)
- [Reduced Order Modeling for Tsunami Forecasting with Bayesian Hierarchical Pooling](https://mlsanity.com/paper/2512.19804/) (2025)
- [A Physics Informed Neural Network For Deriving MHD State Vectors From Global Active Regions Observations](https://mlsanity.com/paper/2512.20747/) (2025)
- [A solver-in-the-loop framework for end-to-end differentiable coastal hydrodynamics](https://mlsanity.com/paper/2604.07129/) (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)
- [Comparison of a Parametric Physics-Informed Neural Network and a Tensorial Reduced-Order Model for the Shallow-Water Dam-Break Problem](https://mlsanity.com/paper/2607.27433/) (2026)

## Abstract (arXiv)

We propose the use of physics-informed neural networks for solving the shallow-water equations on the sphere in the meteorological context. Physics-informed neural networks are trained to satisfy the differential equations along with the prescribed initial and boundary data, and thus can be seen as an alternative approach to solving differential equations compared to traditional numerical approaches such as finite difference, finite volume or spectral methods. We discuss the training difficulties of physics-informed neural networks for the shallow-water equations on the sphere and propose a simple multi-model approach to tackle test cases of comparatively long time intervals. Here we train a sequence of neural networks instead of a single neural network for the entire integration interval. We also avoid the use of a boundary value loss by encoding the boundary conditions in a custom neural network layer. We illustrate the abilities of the method by solving the most prominent test cases proposed by Williamson et al. [J. Comput. Phys. 102 (1992), 211-224].