Articles | Volume 16, issue 22
https://doi.org/10.5194/gmd-16-6671-2023
© Author(s) 2023. This work is distributed under
the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
https://doi.org/10.5194/gmd-16-6671-2023
© Author(s) 2023. This work is distributed under
the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Development and technical paper
| Highlight paper
| 
16 Nov 2023
Development and technical paper | Highlight paper |
| 16 Nov 2023
| 16 Nov 2023
Universal differential equations for glacier ice flow modelling
Institute for Marine and Atmospheric research Utrecht, Utrecht University, Utrecht, the Netherlands
Faculty of Civil Engineering and Geosciences, Technische Universiteit Delft, Delft, the Netherlands
Facundo Sapienza
CORRESPONDING AUTHOR
Department of Statistics, University of California, Berkeley, CA, USA
Fabien Maussion
Department of Atmospheric and Cryospheric Sciences, Universität Innsbruck, Innsbruck, Austria
Bristol Glaciology Centre, School of Geographical Sciences, University of Bristol, Bristol, UK
Redouane Lguensat
Institut Pierre-Simon Laplace, IRD, Sorbonne Université, Paris, France
Bert Wouters
Institute for Marine and Atmospheric research Utrecht, Utrecht University, Utrecht, the Netherlands
Faculty of Civil Engineering and Geosciences, Technische Universiteit Delft, Delft, the Netherlands
Fernando Pérez
Department of Statistics, University of California, Berkeley, CA, USA
Viewed
Total article views: 6,839 (including HTML, PDF, and XML)
Cumulative views and downloads
(calculated since 15 Jun 2023)
| HTML | XML | Total | BibTeX | EndNote | |
|---|---|---|---|---|---|
| 5,357 | 1,381 | 101 | 6,839 | 126 | 136 |
- HTML: 5,357
- PDF: 1,381
- XML: 101
- Total: 6,839
- BibTeX: 126
- EndNote: 136
Total article views: 4,854 (including HTML, PDF, and XML)
Cumulative views and downloads
(calculated since 16 Nov 2023)
| HTML | XML | Total | BibTeX | EndNote | |
|---|---|---|---|---|---|
| 4,142 | 635 | 77 | 4,854 | 111 | 121 |
- HTML: 4,142
- PDF: 635
- XML: 77
- Total: 4,854
- BibTeX: 111
- EndNote: 121
Total article views: 1,985 (including HTML, PDF, and XML)
Cumulative views and downloads
(calculated since 15 Jun 2023)
| HTML | XML | Total | BibTeX | EndNote | |
|---|---|---|---|---|---|
| 1,215 | 746 | 24 | 1,985 | 15 | 15 |
- HTML: 1,215
- PDF: 746
- XML: 24
- Total: 1,985
- BibTeX: 15
- EndNote: 15
Viewed (geographical distribution)
Total article views: 6,839 (including HTML, PDF, and XML)
Thereof 6,781 with geography defined
and 58 with unknown origin.
Total article views: 4,854 (including HTML, PDF, and XML)
Thereof 4,835 with geography defined
and 19 with unknown origin.
Total article views: 1,985 (including HTML, PDF, and XML)
Thereof 1,946 with geography defined
and 39 with unknown origin.
| Country | # | Views | % |
|---|
| Country | # | Views | % |
|---|
| Country | # | Views | % |
|---|
| Total: | 0 |
| HTML: | 0 |
| PDF: | 0 |
| XML: | 0 |
- 1
1
| Total: | 0 |
| HTML: | 0 |
| PDF: | 0 |
| XML: | 0 |
- 1
1
| Total: | 0 |
| HTML: | 0 |
| PDF: | 0 |
| XML: | 0 |
- 1
1
Cited
19 citations as recorded by crossref.
- Snapshot and time-dependent inversions of basal sliding using automatic generation of adjoint code on graphics processing units I. Utkin et al. 10.1017/jog.2025.40
- TICOI: an operational Python package to generate regular glacier velocity time series L. Charrier et al. 10.5194/tc-19-4555-2025
- Using neural ordinary differential equations to predict complex ecological dynamics from population density data J. Arroyo-Esquivel et al. 10.1098/rsif.2023.0604
- Deep dive into hydrologic simulations at global scale: harnessing the power of deep learning and physics-informed differentiable models (δHBV-globe1.0-hydroDL) D. Feng et al. 10.5194/gmd-17-7181-2024
- Physiology-informed regularisation enables training of universal differential equation systems for biological applications M. de Rooij et al. 10.1371/journal.pcbi.1012198
- A Python library for solving ice sheet modeling problems using physics-informed neural networks, PINNICLE v1.0 G. Cheng et al. 10.5194/gmd-18-5311-2025
- Evaluating the affecting factors of glacier mass balance in Tanggula Mountains using explainable machine learning and the open global glacier model Q. Xu et al. 10.1007/s11629-024-9047-4
- A global perspective on the development and application of glacio-hydrological model C. Yang et al. 10.1016/j.jhydrol.2025.132797
- A framework for three-dimensional dynamic modeling of mountain glaciers in the Community Ice Sheet Model (CISM v2.2) S. Minallah et al. 10.5194/gmd-18-5467-2025
- A model for ice sheets and glaciers in fractal dimensions R. El-Nabulsi 10.1016/j.polar.2025.101171
- Filling the GRACE/-FO gap of mass balance observation in central west Greenland by data-driven modelling A. Puggaard et al. 10.1017/aog.2025.10019
- Advancing cryospheric studies: a historical perspective on radio-echo soundgram analysis techniques A. Awati et al. 10.1007/s12145-025-01996-6
- Twenty-first century global glacier evolution under CMIP6 scenarios and the role of glacier-specific observations H. Zekollari et al. 10.5194/tc-18-5045-2024
- DIFFICE-jax: Differentiable neural-network solver for data assimilation of ice shelves in JAX Y. Wang & C. Lai 10.21105/joss.07254
- A minimal machine-learning glacier mass balance model M. van der Meer et al. 10.5194/tc-19-805-2025
- Computationally efficient subglacial drainage modelling using Gaussian process emulators: GlaDS-GP v1.0 T. Hill et al. 10.5194/gmd-18-4045-2025
- Numerical simulations of recent and future evolution of Monte Perdido glacier A. Mateos-García et al. 10.18172/cig.5816
- Finding the underlying viscoelastic constitutive equation via universal differential equations and differentiable physics E. Rodrigues et al. 10.1016/j.engappai.2025.111788
- Scientific machine learning in hydrology: a unified perspective A. Adombi 10.1007/s12145-025-02021-6
19 citations as recorded by crossref.
- Snapshot and time-dependent inversions of basal sliding using automatic generation of adjoint code on graphics processing units I. Utkin et al. 10.1017/jog.2025.40
- TICOI: an operational Python package to generate regular glacier velocity time series L. Charrier et al. 10.5194/tc-19-4555-2025
- Using neural ordinary differential equations to predict complex ecological dynamics from population density data J. Arroyo-Esquivel et al. 10.1098/rsif.2023.0604
- Deep dive into hydrologic simulations at global scale: harnessing the power of deep learning and physics-informed differentiable models (δHBV-globe1.0-hydroDL) D. Feng et al. 10.5194/gmd-17-7181-2024
- Physiology-informed regularisation enables training of universal differential equation systems for biological applications M. de Rooij et al. 10.1371/journal.pcbi.1012198
- A Python library for solving ice sheet modeling problems using physics-informed neural networks, PINNICLE v1.0 G. Cheng et al. 10.5194/gmd-18-5311-2025
- Evaluating the affecting factors of glacier mass balance in Tanggula Mountains using explainable machine learning and the open global glacier model Q. Xu et al. 10.1007/s11629-024-9047-4
- A global perspective on the development and application of glacio-hydrological model C. Yang et al. 10.1016/j.jhydrol.2025.132797
- A framework for three-dimensional dynamic modeling of mountain glaciers in the Community Ice Sheet Model (CISM v2.2) S. Minallah et al. 10.5194/gmd-18-5467-2025
- A model for ice sheets and glaciers in fractal dimensions R. El-Nabulsi 10.1016/j.polar.2025.101171
- Filling the GRACE/-FO gap of mass balance observation in central west Greenland by data-driven modelling A. Puggaard et al. 10.1017/aog.2025.10019
- Advancing cryospheric studies: a historical perspective on radio-echo soundgram analysis techniques A. Awati et al. 10.1007/s12145-025-01996-6
- Twenty-first century global glacier evolution under CMIP6 scenarios and the role of glacier-specific observations H. Zekollari et al. 10.5194/tc-18-5045-2024
- DIFFICE-jax: Differentiable neural-network solver for data assimilation of ice shelves in JAX Y. Wang & C. Lai 10.21105/joss.07254
- A minimal machine-learning glacier mass balance model M. van der Meer et al. 10.5194/tc-19-805-2025
- Computationally efficient subglacial drainage modelling using Gaussian process emulators: GlaDS-GP v1.0 T. Hill et al. 10.5194/gmd-18-4045-2025
- Numerical simulations of recent and future evolution of Monte Perdido glacier A. Mateos-García et al. 10.18172/cig.5816
- Finding the underlying viscoelastic constitutive equation via universal differential equations and differentiable physics E. Rodrigues et al. 10.1016/j.engappai.2025.111788
- Scientific machine learning in hydrology: a unified perspective A. Adombi 10.1007/s12145-025-02021-6
Latest update: 21 Oct 2025
Executive editor
The integration of neural networks into PDE solvers to simulate systems for which the PDE models are incomplete is a key advance at the cutting edge of geoscientific modelling. The approach presented here is applicable far beyond the realm of ice modelling, and will be of interest to model developers and users across geoscience and beyond.
The integration of neural networks into PDE solvers to simulate systems for which the PDE models...
Short summary
We developed a new modelling framework combining numerical methods with machine learning. Using this approach, we focused on understanding how ice moves within glaciers, and we successfully learnt a prescribed law describing ice movement for 17 glaciers worldwide as a proof of concept. Our framework has the potential to discover important laws governing glacier processes, aiding our understanding of glacier physics and their contribution to water resources and sea-level rise.
We developed a new modelling framework combining numerical methods with machine learning. Using...
