mlsanityData to 7 Oct 2026

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

arXiv 2104.00615 ↗First posted 1 Apr 2021physics.comp-ph cs.AI cs.LGcited by 32

Not on the map. Why?

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].

Key facts. Cited by 32 papers in the map (data through 7 Oct 2026). Builds on 20 of its 58 references that are in the map. First posted on arXiv 1 Apr 2021.

Builds on

20 of 58 references.

References · 20

  1. 2014Adam: A Method for Stochastic Optimizationcited by 36,960
  2. 2020Language Models are Few-Shot Learnerscited by 32,199
  3. 2016Densely Connected Convolutional Networkscited by 6,691
  4. 2013Playing Atari with Deep Reinforcement Learningcited by 3,211
  5. 2015Convolutional LSTM Network: A Machine Learning Approach for Precipitation Nowcastingcited by 1,428
  6. 2018On the Spectral Bias of Neural Networkscited by 1,321
  7. 2020Gradient Surgery for Multi-Task Learningcited by 1,210
  8. 2015Automatic differentiation in machine learning: a surveycited by 846
  9. 2018Multi-Task Learning as Multi-Objective Optimizationcited by 809
  10. 2019Large Batch Optimization for Deep Learning: Training BERT in 76 minutescited by 671
  11. 2020When and why PINNs fail to train: A neural tangent kernel perspectivecited by 629
  12. 2019Lookahead Optimizer: k steps forward, 1 step backcited by 333
8 more
  1. 2020Understanding and mitigating gradient pathologies in physics-informed neural networkscited by 134
  2. 2019PPINN: Parareal Physics-Informed Neural Network for time-dependent PDEscited by 116
  3. 2020Improving data-driven global weather prediction using deep convolutional neural networks on a cubed spherecited by 95
  4. 2019Machine learning in cardiovascular flows modeling: Predicting arterial blood pressure from non-invasive 4D flow MRI data using physics-informed neural networkscited by 76
  5. 2020On the convergence of physics informed neural networks for linear second-order elliptic and parabolic type PDEscited by 31
  6. 2020A generative adversarial network approach to (ensemble) weather predictioncited by 18
  7. 2019Precipitation nowcasting using a stochastic variational frame predictor with learned prior distributioncited by 4
  8. 2020Ensemble methods for neural network-based weather forecastscited by 3

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 2; 2022: 5; 2023: 8; 2024: 5; 2025: 6; 2026: 6.

32 papers.

Sort:

  1. 2022Physics-Informed Machine Learning: A Survey on Problems, Methods and Applicationscited by 88
  2. 2023A unified scalable framework for causal sweeping strategies for Physics-Informed Neural Networks (PINNs) and their temporal decompositionscited by 48
  3. 2022Lagrangian PINNs: A causality-conforming solution to failure modes of physics-informed neural networkscited by 26
  4. 2023Improving physics-informed neural networks with meta-learned optimizationcited by 19
  5. 2023Exactly conservative physics-informed neural networks and deep operator networks for dynamical systemscited by 13
  6. 2023Physics-aware Machine Learning Revolutionizes Scientific Paradigm for Machine Learning and Process-based Hydrologycited by 11
  7. 2023Improving physics-informed DeepONets with hard constraintscited by 9
  8. 2024Exact Enforcement of Temporal Continuity in Sequential Physics-Informed Neural Networkscited by 9
  9. 2021Deep learning for improved global precipitation in numerical weather prediction systemscited by 5
  10. 2022Computing the ensemble spread from deterministic weather predictions using conditional generative adversarial networkscited by 5
  11. 2024Large-scale flood modeling and forecasting with FloodCastcited by 5
  12. 2024ControlSynth Neural ODEs: Modeling Dynamical Systems with Guaranteed Convergencecited by 5
20 more
  1. 2023M-ENIAC: A machine learning recreation of the first successful numerical weather forecastscited by 3
  2. 2025Scientific Machine Learning of Flow Resistance Using Universal Shallow Water Equations with Differentiable Programmingcited by 3
  3. 2021Surrogate Model for Shallow Water Equations Solvers with Deep Learningcited by 2
  4. 2022Short-range forecasts of global precipitation using deep learning-augmented numerical weather predictioncited by 2
  5. 2025A Survey of Physics-Informed AI for Complex Urban Systemscited by 2
  6. 2023Towards replacing precipitation ensemble predictions systems using machine learningcited by 1
  7. 2024PinnDE: Physics-Informed Neural Networks for Solving Differential Equationscited by 1
  8. 2024ICODE: Modeling Dynamical Systems with Extrinsic Input Informationcited by 1
  9. 2025A comprehensive analysis of PINNs: Variants, Applications, and Challengescited by 1
  10. 2026Toward AI-Driven Digital Twins for Metropolitan Floods: A Conditional Latent Dynamics Network Surrogate of the Shallow Water Equationscited by 1
  11. 2022Improving trajectory calculations using deep learning inspired single image superresolutioncited by 0
  12. 2023Number Theoretic Accelerated Learning of Physics-Informed Neural Networkscited by 0
  13. 2025Low-rank adaptive physics-informed HyperDeepONets for solving differential equationscited by 0
  14. 2025Reduced Order Modeling for Tsunami Forecasting with Bayesian Hierarchical Poolingcited by 0
  15. 2025A Physics Informed Neural Network For Deriving MHD State Vectors From Global Active Regions Observationscited by 0
  16. 2026A solver-in-the-loop framework for end-to-end differentiable coastal hydrodynamicscited by 0
  17. 2026Finite Volume-Informed Neural Network Framework for 2D Shallow Water Equations: Rugged Loss Landscapes and the Importance of Data Guidancecited by 0
  18. 2026Comparison of a Parametric Physics-Informed Neural Network and a Tensorial Reduced-Order Model for the Shallow-Water Dam-Break Problemcited by 0
  19. 2026Physics-Informed Neural Networks for Depth-Averaged Granular Avalanche Dynamics on Curved Topographycited by 0
  20. 2026Physics-Informed Neural Networks for Depth-Averaged Avalanche Dynamicscited by 0