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  • Perspective
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Differentiable modelling to unify machine learning and physical models for geosciences

Nature Reviews Earth & Environment volume 4, pages 552–567 (2023)Cite this article

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Abstract

Process-based modelling offers interpretability and physical consistency in many domains of geosciences but struggles to leverage large datasets efficiently. Machine-learning methods, especially deep networks, have strong predictive skills yet are unable to answer specific scientific questions. In this Perspective, we explore differentiable modelling as a pathway to dissolve the perceived barrier between process-based modelling and machine learning in the geosciences and demonstrate its potential with examples from hydrological modelling. ‘Differentiable’ refers to accurately and efficiently calculating gradients with respect to model variables or parameters, enabling the discovery of high-dimensional unknown relationships. Differentiable modelling involves connecting (flexible amounts of) prior physical knowledge to neural networks, pushing the boundary of physics-informed machine learning. It offers better interpretability, generalizability, and extrapolation capabilities than purely data-driven machine learning, achieving a similar level of accuracy while requiring less training data. Additionally, the performance and efficiency of differentiable models scale well with increasing data volumes. Under data-scarce scenarios, differentiable models have outperformed machine-learning models in producing short-term dynamics and decadal-scale trends owing to the imposed physical constraints. Differentiable modelling approaches are primed to enable geoscientists to ask questions, test hypotheses, and discover unrecognized physical relationships. Future work should address computational challenges, reduce uncertainty, and verify the physical significance of outputs.

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Fig. 1: Synergies from combining machine learning and process-based modelling.
Fig. 2: Why differentiable models are interpretable and performant.
Fig. 3: An example differentiable hydrological model.
Fig. 4: Differentiable parameter learning.
Fig. 5: Physics-informed neural networks for learning constitutive relationships in unsaturated flow models.

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Acknowledgements

The authors are grateful for the discussion at the HydroML symposium, University Park, PA, May 2022, https://bit.ly/3g3DQNX, sponsored by National Science Foundation EAR #2015680 and Penn State Institute for Computational and Data Sciences. C.S., Y.S. and T.B. were supported by National Science Foundation Award EAR-2221880, Office of Science, US Department of Energy under award DE-SC0016605, and Cooperative Institute for Research to Operations in Hydrology (CIROH), award number A22-0307-S003, respectively. P.G. acknowledges funding from the National Science Foundational Science and Technology Center, Learning the Earth with Artificial intelligence and Physics (LEAP), award #2019625, and USMILE European Research Council grant. M. Wernimont at the US Geological Survey (USGS) greatly improved the presentation of Figs. 1 and 2. A.P.A. was supported by the USGS Water Mission Area, Water Availability and Use Science Program. Any use of trade, firm or product names is for descriptive purposes only and does not imply endorsement by the US Government.

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Authors and Affiliations

  1. Civil and Environmental Engineering, The Pennsylvania State University, University Park, PA, USA

    Chaopeng Shen, Li Li, Xiaofeng Liu, Dapeng Feng, Doaa Aboelyazeed, Farshid Rahmani, Yalan Song, Tadd Bindas & Kathryn Lawson

  2. US Geological Survey, Reston, VA, USA

    Alison P. Appling

  3. National Science Foundation Science and Technology Center for Learning the Earth with Artificial Intelligence and Physics (LEAP), Columbia University, New York, NY, USA

    Pierre Gentine

  4. Energy Geoscience Divisions, Earth and Environmental Sciences Area, Lawrence Berkeley National Laboratory, Berkeley, CA, USA

    Toshiyuki Bandai

  5. Hydrology and Atmospheric Sciences, The University of Arizona, Tucson, AZ, USA

    Hoshin Gupta

  6. Civil and Environmental Engineering, University of Illinois, Urbana Champaign, IL, USA

    Alexandre Tartakovsky & Praveen Kumar

  7. Eawag: Swiss Federal Institute of Aquatic Science and Technology, Dübendorf, Switzerland

    Marco Baity-Jesi, Fabrizio Fenicia & Marvin Höge

  8. Computer Science and Engineering, The Pennsylvania State University, University Park, PA, USA

    Daniel Kifer

  9. Department of Natural Resources and the Environment, University of Connecticut, Storrs, CT, USA

    Wei Ren

  10. Southern University of Science and Technology, Shenzhen, Guangdong Province, China

    Yi Zheng

  11. Department of Environmental Health and Engineering, Johns Hopkins University, Baltimore, MD, USA

    Ciaran J. Harman

  12. Global Institute for Water Security, University of Saskatchewan, Canmore, Alberta, Canada

    Martyn Clark

  13. US Army Engineer Research and Development Center, Vicksburg, MS, USA

    Matthew Farthing

  14. Prairie Research Institute, University of Illinois, Urbana Champaign, IL, USA

    Praveen Kumar

  15. Physical Science and Engineering Division, King Abdullah University of Science and Technology, Thuwal, Saudi Arabia

    Hylke E. Beck

  16. Climate and Ecosystem Sciences Divisions, Earth and Environmental Sciences Area, Lawrence Berkeley National Laboratory, Berkeley, CA, USA

    Dipankar Dwivedi

  17. Department of Earth System Science, Stanford University, Stanford, CA, USA

    Kuai Fang

  18. Computer Science and Artificial Intelligence Laboratory (CSAIL), Massachusetts Institute of Technology, Cambridge, MA, USA

    Chris Rackauckas

  19. Department of Biological and Agricultural Engineering, Texas A&M University, College Station, TX, USA

    Binayak Mohanty

  20. Civil and Environmental Engineering, University of Nebraska-Lincoln, Lincoln, NE, USA

    Tirthankar Roy

  21. Earth and Environmental Sciences Division, Los Alamos National Laboratory, New Mexico, NM, USA

    Chonggang Xu

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Contributions

C.S. researched data for the article. C.S., A.P.A., P.G., T.B., H.G., Y.Z., A.T., M.B.-J., F.F., D.K., L.L., X.L. and W.R. contributed substantially to discussion of the content. C.S. wrote the initial article. All authors reviewed and/or edited the manuscript before submission.

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Correspondence to Chaopeng Shen.

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K.L. and C.S. have financial interests in HydroSapient, Inc.

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Shen, C., Appling, A.P., Gentine, P. et al. Differentiable modelling to unify machine learning and physical models for geosciences. Nat Rev Earth Environ 4, 552–567 (2023). https://doi.org/10.1038/s43017-023-00450-9

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