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(DLR)"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Sci Rep"],"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>Cloud-related parameterizations remain a leading source of uncertainty in climate projections. Although machine learning holds promise for Earth system models (ESMs), many data-driven parameterizations lack interpretability, physical consistency, and smooth integration into ESMs. Here, a two-step method is presented to improve a climate model with data-driven parameterizations. First, we incorporate a physically consistent cloud cover parameterization\u2014derived from storm-resolving simulations via symbolic regression, preserving interpretability while enhancing accuracy\u2014into the ICON global atmospheric model. Second, we apply the gradient-free Nelder\u2013Mead optimizer to automatically recalibrate the hybrid model against Earth observations, tuning in nested stages (2-, 7-, 30- and 365-day runs) to ensure stability and tractability. The tuned hybrid model substantially reduces long-standing biases in cloud cover\u2014particularly over the Southern Ocean (by 75%) and subtropical stratocumulus regions (by 44%)\u2014and remains robust under +4K surface warming. These results demonstrate that interpretable machine-learned parameterizations, paired with practical tuning, can efficiently and transparently strengthen ESM fidelity.<\/jats:p>","DOI":"10.1038\/s41598-025-29155-3","type":"journal-article","created":{"date-parts":[[2025,12,13]],"date-time":"2025-12-13T11:34:38Z","timestamp":1765625678000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Reduced cloud cover errors in a hybrid AI-climate model through equation discovery and automatic tuning"],"prefix":"10.1038","volume":"15","author":[{"given":"Arthur","family":"Grundner","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Tom","family":"Beucler","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Julien","family":"Savre","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Axel","family":"Lauer","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Manuel","family":"Schlund","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Veronika","family":"Eyring","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2025,12,13]]},"reference":[{"key":"29155_CR1","unstructured":"Stensrud, D.\u00a0J. Parameterization Schemes: Keys to Understanding Numerical Weather Prediction Models (Cambridge University Press, 2009)."},{"key":"29155_CR2","doi-asserted-by":"crossref","unstructured":"Eyring, V. et al. Pushing the frontiers in climate modelling and analysis with machine learning. Nat. Clim. Change 14(9), 916\u2013928 (2024).","DOI":"10.1038\/s41558-024-02095-y"},{"key":"29155_CR3","doi-asserted-by":"publisher","first-page":"963","DOI":"10.1038\/s41561-024-01527-w","volume":"17","author":"V Eyring","year":"2024","unstructured":"Eyring, V., Gentine, P., Camps-Valls, G., Lawrence, D. M. & Reichstein, M. AI-empowered next-generation multiscale climate modelling for mitigation and adaptation. Nat. Geosci. 17, 963\u2013971. https:\/\/doi.org\/10.1038\/s41561-024-01527-w (2024).","journal-title":"Nat. Geosci."},{"key":"29155_CR4","doi-asserted-by":"crossref","unstructured":"Gentine, P., Pritchard, M., Rasp, S., Reinaudi, G., & Yacalis, G. Could machine learning break the convection parameterization deadlock? Geophys. Res. Lett. 45(1), 5742\u20135751 (2018).","DOI":"10.1029\/2018GL078202"},{"key":"29155_CR5","doi-asserted-by":"crossref","unstructured":"Grundner, A., Beucler, T., Gentine, P., Iglesias-Suarez, F., Giorgetta, M.\u00a0A., & Eyring, V. Deep learning based cloud cover parameterization for ICON. J. Adv. Model. Earth Syst. 14(12), e2021MS002959 (2022).","DOI":"10.1029\/2021MS002959"},{"key":"29155_CR6","doi-asserted-by":"crossref","unstructured":"Heuer, H., Schwabe, M., Gentine, P., Giorgetta, M.\u00a0A., & Eyring, V. Interpretable multiscale machine learning-based parameterizations of convection for ICON. J. Adv. Model. Earth Syst. 16(8), e2024MS004398 (2024).","DOI":"10.1029\/2024MS004398"},{"key":"29155_CR7","doi-asserted-by":"crossref","unstructured":"O\u2019Gorman, P.\u00a0A. & Dwyer, J.\u00a0G. Using machine learning to parameterize moist convection: Potential for modeling of climate, climate change, and extreme events. J. Adv. Model. Earth Syst. 10(10), 2548\u20132563 (2018).","DOI":"10.1029\/2018MS001351"},{"key":"29155_CR8","doi-asserted-by":"publisher","unstructured":"Rasp, S., Pritchard, M.\u00a0S., & Gentine, P. Deep learning to represent subgrid processes in climate models. Proc. Natl. Acad. Sci. 115(39), 9684\u20139689. https:\/\/doi.org\/10.1073\/pnas.1810286115 (2018).","DOI":"10.1073\/pnas.1810286115"},{"key":"29155_CR9","doi-asserted-by":"crossref","unstructured":"Reichstein, M. et al. Deep learning and process understanding for data-driven Earth system science. Nature. 566(7743), 195\u2013204 (2019).","DOI":"10.1038\/s41586-019-0912-1"},{"key":"29155_CR10","doi-asserted-by":"crossref","unstructured":"Grundner, A. Data-driven cloud cover parameterizations for the ICON earth system model using deep learning and symbolic regression. Accessed: 2024-10-16. PhD thesis. Universit\u00e4t Bremen, Jan. 2024. 10.26092\/elib\/2821. https:\/\/media.suub.uni-bremen.de\/handle\/elib\/7739.","DOI":"10.1029\/2023MS003763"},{"key":"29155_CR11","doi-asserted-by":"crossref","unstructured":"Bracco, A., Brajard, J., Dijkstra, H.\u00a0A., Hassanzadeh, P., Lessig, C., & Monteleoni, C. Machine learning for the physics of climate. Nat. Rev. Phys. 7(1), 6\u201320 (2025).","DOI":"10.1038\/s42254-024-00776-3"},{"key":"29155_CR12","doi-asserted-by":"crossref","unstructured":"Frezat, H., Le Sommer, J., Fablet, R., Balarac, G., & Lguensat, R. A posteriori learning for quasi-geostrophic turbulence parametrization. J. Adv. Model. Earth Syst. 14(11), e2022MS003124 (2022).","DOI":"10.1029\/2022MS003124"},{"key":"29155_CR13","doi-asserted-by":"publisher","unstructured":"Christopoulos, C. et al. Online learning of entrainment closures in a hybrid machine learning parameterization. J. Adv. Model. 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