- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Probing forced responses and causality in data-driven climate emulators: Conceptual limitations and the role of reduced-order models

Fabrizio Falasca*

  • *Contact author: fabri.falasca@nyu.edu
PDF

Phys. Rev. Research 7, 043314 – Published 19 December, 2025

DOI: https://doi.org/10.1103/2f4r-k8lr

Abstract

This article is part of a Physical Review Collection on the Physics of Changing Climate.

A central challenge in climate science and applied mathematics is developing data-driven models of multiscale systems that capture both stationary statistics and responses to external perturbations. Current neural climate emulators aim to resolve the atmosphere-ocean system in all its complexity but often struggle to reproduce forced responses, limiting their use in causal studies such as Green's function experiments. To explore the origin of these limitations, we first examine a simplified dynamical system that retains key features of climate variability. We interpret the results through linear response theory, providing a rigorous framework to evaluate neural models beyond stationary statistics and probe causal mechanisms. We argue that the ability of emulators of multiscale systems to reproduce perturbed statistics depends critically on (1) the choice of an appropriate coarse-grained representation and (2) careful parameterizations of unresolved processes. These insights highlight reduced-order models, tailored to specific goals, processes, and scales, as valuable alternatives to general-purpose emulators. We next consider a real-world application, developing a neural model to investigate the joint variability of the surface temperature field and radiative fluxes. The model infers a multiplicative noise process directly from data, largely reproduces the system's probability distribution, and enables causal studies through forced responses. We discuss its limitations and outline directions for future work. These results expose key challenges in data-driven modeling of multiscale physical systems and underscore the value of coarse-grained, stochastic approaches, with response theory as a principled framework to guide model design and enhance causal understanding.

View figure in article

Physics Subject Headings (PhySH)

Corrections

20 January, 2026

Correction: The first sentence of the second paragraph of Sec. II B entitled “Impulse response operator” contained an error in wording and has been fixed.

5 March, 2026

Second Correction: The first sentence of the second paragraph of Sec. II B has been further modified. The first sentence of the paragraph numbered (2) in Appendix A 1 has been modified.

Collections

This article appears in the following collection:

Physics in a Changing Climate

Climate Science is a growing field that is attracting colleagues from many disciplines. This Collection, a joint effort by Physical Review E, Physical Review Research, and PRX Energy curated by Prof. Justin Burton from Emory University, aims at providing a survey of the diverse range of topics that scientists with a connection to Physics are exploring in this exciting area of research. This includes climate and related geosciences, sustainability, energy, and material science, and may include fundamental as well as applied research.

Every article in this Collection was subjected to a rigorous peer review process, upholding the high standards applied to all papers. The Physical Review E, Physical Review Research, and PRX Energy editorial teams managed the peer review and made all editorial decisions.

Article Text

References (153)

  1. M. Ghil and V. Lucarini, The physics of climate variability and climate change, Rev. Mod. Phys. 92, 035002 (2020).
  2. V. Lucarini and M. Chekroun, Theoretical tools for understanding the climate crisis from Hasselmann's programme and beyond, Nat. Rev. Phys. 5, 744 (2023).
  3. C. Penland, Random forcing and forecasting using principal oscillation pattern analysis, Mon. Weather Rev. 117, 2165 (1989).
  4. C. Penland and P. Sardeshmukh, The optimal growth of tropical sea surface temperature anomalies, J. Clim. 8, 1999 (1995).
  5. C. Penland, A stochastic model of IndoPacific sea surface temperature anomalies, Phys. D 98, 534 (1996).
  6. S. Kravtsov, D. Kondrashov, and M. Ghil, Multilevel regression modeling of nonlinear processes: Derivation and applications to climatic variability, J. Clim. 18, 4404 (2005).
  7. D. Kondrashov, M. Chekroun, and M. Ghil, Data-driven non-Markovian closure models, Phys. D 297, 33 (2015).
  8. K. Strounine, S. Kravtsov, D. Kondrashov, and M. Ghil, Reduced models of atmospheric low-frequency variability: Parameter estimation and comparative performance, Phys. D 239, 145 (2010).
  9. N. Agarwal, D. Kondrashov, P. Dueben, E. Ryzhov, and P. Berloff, A comparison of data-driven approaches to build low-dimensional ocean models, J. Adv. Model. Earth Syst. 13, e2021MS002537 (2005).
  10. A. J. Majda and Y. Yuan, Fundamental limitations of ad hoc linear and quadratic multi-level regression models for physical systems, Discrete Contin. Dyn. Syst.–Ser. B 17, 1333 (2012).
  11. A. Majda and J. Harlim, Physics constrained nonlinear regression models for time series, Nonlinearity 26, 201 (2013).
  12. A. Souza, Representing turbulent statistics with partitions of state space. Part 1. Theory and methodology, J. Fluid Mech. 997, A1 (2024).
  13. A. Souza, Representing turbulent statistics with partitions of state space. Part 2. The compressible Euler equations, J. Fluid Mech. 997, A2 (2024).
  14. L. T. Giorgini, A. N. Souza, and P. J. Schmid, Reduced Markovian models of dynamical systems, Phys. D 470, 134393 (2024).
  15. L. Giorgini, Data-driven decomposition of conservative and non-conservative dynamics in multiscale systems, arXiv:2505.01895.
  16. L. Giorgini, T. Bischoff, and A. Souza, Reduced-order modeling of cyclo-stationary time series using score-based generative methods, arXiv:2508.19448.
  17. H. Fan, Y. Xiao, Y. Qu, F. Ling, B. Fei, L. Bai, and P. Gentine, Incorporating multivariate consistency in ML-based weather forecasting with latent-space constraints, arXiv:2510.04006.
  18. A. Majda and D. Qi, Strategies for reduced-order models for predicting the statistical responses and uncertainty quantification in complex turbulent dynamical systems, SIAM Rev. 60, 491 (2018).
  19. D. Qi and A. Majda, Linear and nonlinear statistical response theories with prototype applications to sensitivity analysis and statistical control of complex turbulent dynamical systems, Chaos 29, 103131 (2019).
  20. M. Baldovin, F. Cecconi, and A. Vulpiani, Understanding causation via correlations and linear response theory, Phys. Rev. Res. 2, 043436 (2020).
  21. F. Falasca, P. Perezhogin, and L. Zanna, Data-driven dimensionality reduction and causal inference for spatiotemporal climate fields, Phys. Rev. E 109, 044202 (2024).
  22. J. Pathak, S. Subramanian, P. Harrington, S. Raja, A. Chattopadhyay, M. Mardani, T. Kurth, D. Hall, Z. Li, K. Azizzadenesheli, P. Hassanzadeh, K. Kashinath, and A. Anandkumar, FourCastNet: A global data-driven high-resolution weather model using adaptive Fourier neural operators, arXiv:2202.11214.
  23. I. Price, A. Sanchez-Gonzalez, F. Alet, T. R. Andersson, A. El-Kadi, D. Masters, T. Ewalds, J. Stott, S. Mohamed, P. Battaglia, R. Lam, and M. Willson, Probabilistic weather forecasting with machine learning, Nature (London) 637, 84 (2025).
  24. R. Lam, A. Sanchez-Gonzalez, M. Willson, P. Wirnsberger, M. Fortunato, F. Alet, S. Ravuri, T. Ewalds, Z. Eaton-Rosen, W. Hu, A. Merose, S. Hoyer, G. Holland, O. Vinyals, J. Stott, A. Pritzel, S. Mohamed, and P. Battaglia, Learning skillful medium-range global weather forecasting, Science 382, 1416 (2023).
  25. K. Bi, L. Xie, H. Zhang, X. Chen, X. Gu, and Q. Tian, Accurate medium-range global weather forecasting with 3D neural networks, Nature (London) 619, 533 (2023).
  26. B. Bonev et al., FourCastNet 3: A geometric approach to probabilistic machine-learning weather forecasting at scale, arXiv:2507.12144.
  27. D. Kochkov, J. Yuval, I. Langmore, P. Norgaard, J. Smith, G. Mooers, M. Klöwer, J. Lottes, S. Rasp, P. Düben, S. Hatfield, P. Battaglia, A. Sanchez-Gonzalez, M. Willson, M. P. Brenner, and S. Hoyer, Neural general circulation models for weather and climate, Nature (London) 632, 1060 (2024).
  28. O. Watt-Meyer, G. Dresdner, J. McGibbon, S. K. Clark, B. Henn, J. Duncan, N. D. Brenowitz, K. Kashinath, M. S. Pritchard, B. Bonev, M. E. Peters, and C. S. Bretherton, ACE: A fast, skillful learned global atmospheric model for climate prediction, arXiv:2310.02074.
  29. O. Watt-Meyer, B. Henn, J. McGibbon, S. K. Clark, A. Kwa, W. A. Perkins, and C. S. Bretherton, ACE2: Accurately learning subseasonal to decadal atmospheric variability and forced responses, arXiv:2411.11268.
  30. H. Guan, T. Arcomano, A. Chattopaday, and R. Maulik, LUCIE: A lightweight uncoupled climate emulator with long-term stability and physical consistency for O(1000)-member ensembles, arXiv:2405.16297.
  31. A. Chattopadhyay, M. Gray, T. Wu, A. B. Lowe, and R. He, OceanNet: A principled neural operator-based digital twin for regional oceans, Sci. Rep. 14, 21181 (2024).
  32. S. Bire, B. Lütjens, K. Azizzadenesheli, A. Anandkumar, and C. Hill, Ocean emulation with Fourier neural operators: Double gyre, J. Adv. Model. Earth Syst. 17, e2023MS004137 (2025).
  33. S. Dheeshjith, A. Subel, A. Adcroft, J. Busecke, C. Fernandez‐Granda, S. Gupta, and L. Zanna, Samudra: An AI global ocean emulator for climate, Geophys. Res. Lett. 52, e2024GL114318 (2025).
  34. W. Chapman, J. Schreck, Y. Sha, D. Gagne II, D. Kimpara, L. Zanna, K. Mayer, and J. Berner, CAMulator: Fast emulation of the community atmosphere model, arXiv:2504.06007.
  35. H. Guan, T. Arcomano, A. Chattopadhyay, and R. Maulik, LUCIE-3D: A three-dimensional climate emulator for forced responses, arXiv:2509.02061.
  36. J. Duncan et al., SamudrACE: Fast and accurate coupled climate modeling with 3D ocean and atmosphere emulators, arXiv:2509.12490.
  37. L. Zanna et al., A framework for hybrid physics-AI coupled ocean models, arXiv:2510.22676.
  38. A. Lancelin et al., AI-boosted rare event sampling to characterize extreme weather, arXiv:2510.27066.
  39. C. Pedersen, L. Zanna, and J. Bruna, Thermalizer: Stable autoregressive neural emulation of spatiotemporal chaos, arXiv:2503.18731.
  40. R. Jiang, X. Zhang, K. Jakhar, P. Lu, P. Hassanzadeh, M. Maire, and R. Willett, Hierarchical implicit neural emulators, arXiv:2506.04528.
  41. Y. Sun, P. Hassanzadeh, M. Zand, and D. Abbot, Can AI weather models predict out-of-distribution gray swan tropical cyclones? Proc. Natl. Acad. Sci. USA 122, e2420914122 (2025).
  42. I. Baxter, H. Pahlavan, P. Hassanzadeh, K. Rucker, and T. Shaw, Benchmarking atmospheric circulation variability in an AI emulator, ACE2, and a hybrid model, neuralgcm, arXiv:2510.04466.
  43. S. Van Loon, M. Rugenstein, and E. A. Barnes, Reanalysis-based global radiative response to sea surface temperature patterns: Evaluating the ai2 Climate emulator, Geophys. Res. Lett. 52, e2025GL115432 (2025).
  44. J. Bloch-Johnson, M. A. A. Rugenstein, M. J. Alessi, C. Proistosescu, M. Zhao, B. Zhang, A. I. L. Williams, J. M. Gregory, J. Cole, Y. Dong, M. L. Duffy, S. M. Kang, and C. Zhou, The Green's function model intercomparison project (gfmip) protocol, J. Adv. Model. Earth Syst. 16, e2023MS003700 (2024).
  45. Y. Dong, C. Proistosescu, K. C. Armour, and D. S. Battisti, Attributing historical and future evolution of radiative feedbacks to regional warming patterns using a Green's function approach: The preeminence of the western Pacific, J. Clim. 32, 5471 (2019).
  46. H. Hersbach, B. Bell, P. Berrisford, S. Hirahara, A. Horányi, J. Muñoz‐Sabater, J. Nicolas, C. Peubey, R. Radu, D. Schepers, A. Simmons, C. Soci, S. Abdalla, X. Abellan, G. Balsamo, P. Bechtold, G. Biavati, J. Bidlot, M. Bonavita, G. De Chiara et al., The ERA5 global reanalysis, Q. J. R. Meteorol. Soc. 146, 1999 (2020).
  47. B. Zhang and T. Merlis, The equilibrium response of atmospheric machine-learning models to uniform sea surface temperature warming, arXiv:2510.02415.
  48. E. Wu, F. Rebassoo, P. Paul, C. Proistosescu, J. Nugent, D. McCoy et al., Applying the ACE2 emulator to SST Green's functions for the E3SMv3 global atmosphere model, J. Geophys. Res.: Mach. Learn. Comput. 2, e2025JH000774 (2025).
  49. F. Cecconi, M. Cencini, M. Falcioni, and A. Vulpiani, Predicting the future from the past: An old problem from a modern perspective, Am. J. Phys. 80, 1001 (2012).
  50. H. Hosni and A. Vulpiani, Forecasting in light of big data, Philos. Technol. 31, 557 (2018).
  51. M. Baldovin, F. Cecconi, M. Cencini, A. Puglisi, and A. Vulpiani, The role of data in model building and prediction: A survey through examples, Phys. Rev. Res. 20, 807 (2018).
  52. N. Chen and A. J. Majda, Conditional Gaussian systems for multiscale nonlinear stochastic systems: Prediction, state estimation and uncertainty quantification, Entropy 20, 509 (2018).
  53. A. Majda, C. Franzke, and D. Crommelin, Normal forms for reduced stochastic climate models, Proc. Natl. Acad. Sci. USA 106, 3649 (2010).
  54. A. Majda, B. Gershgorin, and Y. Yuan, Low-frequency climate response and fluctuation–dissipation theorems: Theory and practice, J. Atmos. Sci. 67, 1186 (2010).
  55. A. J. Majda, R. V. Abramov, and M. J. Grote, Information Theory and Stochastics for Multiscale Nonlinear Systems, CRM Monograph Series (American Mathematical Society, Providence, RI, 2005).
  56. U. Marconi, A. Puglisi, L. Rondoni, and A. Vulpiani, Fluctuation-dissipation: Response theory in statistical physics, Phys. Rep. 461, 111 (2008).
  57. G. L. and A. Vulpiani, Prediction and inference: From models and data to artificial intelligence, Found. Phys. 54, 67 (2024).
  58. A. Dalmedico, History and epistemology of models: Meteorology (1946–1963) as a case study, Arch. Hist. Exact Sci. 55, 395 (2001).
  59. L. A. Smith, What might we learn from climate forecasts? Proc. Natl. Acad. Sci. USA 99, 2487 (2002).
  60. J. G. Charney, On a physical basis for numerical prediction of large-scale motions in the atmosphere, J. Meteorol. 6, 372 (1949).
  61. J. G. Charney, R. Fjörtoft, and J. Von Neumann, Numerical integration of the barotropic vorticity equation, Tellus 2, 237 (1950).
  62. A. Timmermann et al., El Niño-Southern Oscillation complexity, Nature (London) 559, 535 (2018).
  63. B. Dubrulle, F. Daviaud, D. Faranda, L. Marié, and B. Saint-Michel, How many modes are needed to predict climate bifurcations? Lessons from an experiment, Nonlin. Processes Geophys. 29, 17 (2022).
  64. J. Wouters and V. Lucarini, Cluster-based reduced-order modelling of a mixing layer, J. Stat. Mech. (2012) P03003.
  65. G. Vissio and V. Lucarini, A proof of concept for scale-adaptive parametrizations: The case of the Lorenz ’96 model, Q. J. R. Meteorol. Soc. 144, 63 (2018).
  66. I. Held, The gap between simulation and understanding in climate modeling, Bull. Am. Meteorol. Soc. 86, 1609 (2005).
  67. K. Hasselmann, Stochastic climate models part I. Theory, Tellus 28, 473 (1976).
  68. N. Chen and H. Liu, Minimum reduced-order models via causal inference, Nonlinear Dyn. 113, 11327 (2025).
  69. A. J. Majda, I. Timofeyev, and E. Vanden-Eijnden, Models for stochastic climate prediction, Proc. Natl. Acad. Sci. USA 96, 14687 (1999).
  70. A. J. Majda, I. Timofeyev, and Vanden-Eijnden, A mathematical framework for stochastic climate models, Communications on Pure and Applied Mathematics 54, 891 (2001).
  71. A. J. Majda, I. Timofeyev, and Vanden-Eijnden, A priori test of a stochastic mode reduction strategy, Phys. D 170, 206 (2002).
  72. A. J. Majda, I. Timofeyev, and Vanden-Eijnden, Systematic strategies for stochastic mode reduction in climate, J. Atmos. Sci. 60, 1705 (2003).
  73. T. J. O’Kane, D. Harries, and M. A. Collier, Bayesian structure learning for climate model evaluation, J. Adv. Model. Earth Syst. 16, e2023MS004034 (2024).
  74. J. Donges, Y. Zou, N. Marwan, and J. Kurths, Complex networks in climate dynamics, Eur. Phys. J. Spec. Top. 174, 157 (2009).
  75. I. Fountalis, C. Dovrolis, A. Bracco, B. Dilkina, and S. Keilholz, δ-MAPS from spatio-temporal data to a weighted and lagged network between functional domain, Appl. Netw. Sci. 3, 21 (2018).
  76. A. Majda, C. Franzke, and K. Boualem, An applied mathematics perspective on stochastic modelling for climate, Phil. Trans. R. Soc. A. 366, 2427 (2008).
  77. D. Crommelin and A. Majda, Strategies for model reduction: Comparing different optimal bases, J. Atmos. Sci. 61, 2206 (2004).
  78. H. Fan, B. Fei, P. Gentine, Y. Xiao, K. Chen, Y. Liu, Y. Qu, F. Ling, and L. Bai, Physically consistent global atmospheric data assimilation with machine learning in a latent space, arXiv:2502.02884.
  79. N. Lutsko, I. Held, and P. Zurita-Gotor, Applying the fluctuation–dissipation theorem to a two-layer model of quasigeostrophic turbulence, J. Atmos. Sci. 72, 3161 (2015).
  80. P. Hassanzadeh and Z. Kuang, The linear response function of an idealized atmosphere. Part II: Implications for the practical use of the fluctuation–dissipation theorem and the role of operator's nonnormality, J. Atmos. Sci. 73, 3441 (2016).
  81. C. Penland, M. D. Fowler, D. L. Jackson, and R. Cifelli, Forecasts of opportunity for northern California soil moisture, Land 10, 713 (2021).
  82. Y. Zhao and A. Capotondi, The role of the tropical atlantic in tropical Pacific climate variability, npj Clim. Atmos. Sci. 7, 140 (2024).
  83. J. Lien, Y.-N. Kuo, H. Ando, and S. Kido, Colored linear inverse model: A data-driven method for studying dynamical systems with temporally correlated stochasticity, Phys. Rev. Res. 7, 023042 (2025).
  84. J. Lien and H. Ando, Nonlinear extensions of linear inverse models under memoryless or persistent random forcing, Phys. Rev. Res. 7, L032038 (2025).
  85. D. Kondrashov, S. Kratsov, A. Robertson, and M. Ghil, A hierarchy of data-based enso models, J. Clim. 18, 4425 (2005).
  86. H. Risken, The Fokker-Planck Equation–Methods of Solution and Applications (Springer, New York, 1996).
  87. J. Ismael, Reflections on the asymmetry of causation, Interface Focus 13, 20220081 (2023).
  88. E. Aurell and G. Del Ferraro, Causal analysis, correlation-response, and dynamic cavity, J. Phys.: Conf. Ser. 699, 012002 (2016).
  89. V. Lucarini, Revising and extending the linear response theory for statistical mechanical systems: Evaluating observables as predictors and predictands, J. Stat. Phys. 173, 1698 (2018).
  90. V. Lucarini and M.D. Chekroun, Detecting and attributing change in climate and complex systems: Foundations, Green's functions, and nonlinear fingerprints, Phys. Rev. Lett. 133, 244201 (2024).
  91. A. Majda, R. Abramov, and B. Gershgorin, High skill in low-frequency climate response through fluctuation dissipation theorems despite structural instability, Proc. Natl. Acad. Sci. USA 107, 581 (2010).
  92. C. Franzke and A. Majda, Low-order stochastic mode reduction for a prototype atmospheric GCM, J. Atmos. Sci. 63, 457 (2006).
  93. J. Berner and G. Branstator, Linear and nonlinear signatures in the planetary wave dynamics of an agcm: Probability density functions, J. Atmos. Sci. 64, 117 (2007).
  94. P. Sura, M. Newman, C. Penland, and P. Sardeshmukh, Multiplicative noise and non-Gaussianity: A paradigm for atmospheric regimes? J. Clim. 62, 1391 (2005).
  95. G. Lacorata and A. Vulpiani, Fluctuation-response relation and modeling in systems with fast and slow dynamics, Nonlin. Processes Geophys. 14, 681 (2007).
  96. M. Baldovin, A. Puglisi, and A. Vulpiani, Langevin equations from experimental data: The case of rotational diffusion in granular media, PLoS ONE 14, e0212135 (2019).
  97. H. Arnold, I. Moroz, and T. Palmer, Stochastic parametrizations and model uncertainty in the Lorenz ’96 system, Phil. Trans. R. Soc. A. 371, 20110479 (2013).
  98. A. Dawson and T. Palmer, Simulating weather regimes: Impact of model resolution and stochastic parameterization, Clim Dyn 44, 2177 (2015).
  99. J. Berner et al., Stochastic parameterization: Toward a new view of weather and climate models, Bull. Am. Meteorol. Soc. 98, 565 (2017).
  100. F. Cecconi, G. Costantini, C. Guardiani, M. Baldovin, and A. Vulpiani, Correlation, response and entropy approaches to allosteric behaviors: A critical comparison on the ubiquitin case, Phys. Biol. 20, 056002 (2020).
  101. B. Meyssignac, R. Guillaume-Castel, and R. Roca, Revisiting the global energy budget dynamics with a multivariate earth energy balance model to account for the warming pattern effect, J. Clim. 36, 8113 (2023).
  102. J. J. Barsugli and P. D. Sardeshmukh, Global atmospheric sensitivity to tropical sst anomalies throughout the Indo-Pacific basin, J. Clim. 15, 3427 (2002).
  103. C. Zhou, M. D. Zelinka, and S. A. Klein, Analyzing the dependence of global cloud feedback on the spatial pattern of sea surface temperature change with a Green's function approach, J. Adv. Model. Earth Syst. 9, 2174 (2017).
  104. B. Zhang, M. Zhao, and Z. Tan, Using a Green's function approach to diagnose the pattern effect in GFDL AM4 and CM4, J. Clim. 36, 1105 (2023).
  105. Y. Dong, K. C. Armour, M. D. Zelinka, C. Proistosescu, D. S. Battisti, C. Zhou, and T. Andrews, Intermodel spread in the pattern effect and its contribution to climate sensitivity in CMIP5 and CMIP6 Models, J. Clim. 33, 7755 (2020).
  106. M. J. Alessi and M. A. Rugenstein, Surface temperature pattern scenarios suggest higher warming rates than current projections, Geophys. Res. Lett. 50, e2023GL105795 (2023).
  107. A. I. Williams, N. Jeevanjee, and J. Bloch-Johnson, Circus tents, convective thresholds, and the non-linear climate response to tropical ssts, Geophys. Res. Lett. 50, e2022GL101499 (2023).
  108. F. Falasca, A. Basinski, L. Zanna, and M. Zhao, A fluctuation-dissipation theorem perspective on radiative responses to temperature perturbations, J. Clim. 38, 3239 (2025).
  109. I. M. Held, H. Guo, A. Adcroft, J. P. Dunne, L. W. Horowitz, J. Krasting, E. Shevliakova, M. Winton, M. Zhao, M. Bushuk, A. T. Wittenberg, B. Wyman, B. Xiang, R. Zhang, W. Anderson, V. Balaji, L. Donner, K. Dunne, J. Durachta, P. P. G. Gauthier et al., Structure and performance of GFDL's CM4.0 climate model, J. Adv. Model. Earth Syst. 11, 3691 (2019).
  110. A. Adcroft, W. Anderson, V. Balaji, C. Blanton, M. Bushuk, C. O. Dufour et al., The GFDL global ocean and sea ice model OM4.0: Model description and simulation features, J. Adv. Model. Earth Syst. 11, 3167 (2019).
  111. M. Zhao et al., The GFDL global atmosphere and land model AM4.0/LM4.0: 1. Simulation characteristics with prescribed ssts, J. Adv. Model. Earth Syst. 10, 691 (2018).
  112. M. Zhao et al., The GFDL global atmosphere and land model am4.0/LM4.0: 2. Model description, sensitivity studies, and tuning strategies, J. Adv. Model. Earth Syst. 10, 735 (2018).
  113. R. T. Pierrehumbert, Principles of Planetary Climate (Cambridge University Press, Cambridge, 2010).
  114. D. M. Romps, J. T. Seeley, and J. P. Edman, Why the forcing from carbon dioxide scales as the logarithm of its concentration, J. Clim. 35, 4027 (2022).
  115. M. Zhao, An investigation of the effective climate sensitivity in gfdl's new climate models CM4.0 and SPEAR, J. Clim. 35, 5637 (2022).
  116. S. M. Kang, P. Ceppi, Y. Yu, and I.-S. Kang, Recent global climate feedback controlled by southern ocean cooling, Nat. Geosci. 16, 775 (2023).
  117. S. Van Loon, M. Rugenstein, and E. A. Barnes, Observation-based estimate of Earth's effective radiative forcing, Proc. Natl. Acad. Sci. USA 122, e2425445122 (2025).
  118. L. Fredericks, M. Rugenstein, D. W. J. Thompson, S. Van Loon, F. Falasca, R. Basinski-Ferris, P. Ceppi, Q. Wu, J. Bloch-Johnson, M. Alessi, S. M. Kang, Quantifying the radiative response to surface temperature variability: A critical comparison of current methods, arXiv:2511.00731.
  119. E. Guilyardi, P. Braconnot, F.-F. Jin, S. T. Kim, M. Kolasinski, T. Lee, and I. Musat, Atmosphere feedbacks during ENSO in a coupled GCM with a modified atmospheric convection scheme, J. Clim. 22, 5698 (2009).
  120. L. A. Smith, Disentangling uncertainty and error: On the predictability of nonlinear systems, in Nonlinear D ynamics and Statistics, edited by A. I. Mees (Birkhäuser Boston, Boston, MA, 2001), pp. 31–64.
  121. P. Sardeshmukh, C. Penland, and G. Compo, Learning ENSO dynamics from data, J. Clim. 38, 7437 (2025).
  122. S. Lang et al., AIFS–ecmwf's data-driven forecasting system, arXiv:2406.01465.
  123. B. Lütjens, R. Ferrari, D. Watson‐Parris, and N. E. Selin, The impact of internal variability on benchmarking deep learning climate emulators, J. Adv. Model. Earth Syst. 17, e2024MS004619 (2025).
  124. G. Geogdzhayev, A. N. Souza, G. R. Flierl, and R. Ferrari, An eof-based emulator of means and covariances of monthly climate fields, EGUsphere [preprint], https://egusphere.copernicus.org/preprints/2025/egusphere-2025-3768/.
  125. S. Bouabid, A. Souza, and R. Ferrari, Score-based generative emulation of impact-relevant earth system model outputs, arXiv:2510.04358.
  126. T. Palmer, A nonlinear dynamical perspective on model error: A proposal for non-local stochastic-dynamic parametrization in weather and climate prediction models, Proc. Natl. Acad. Sci. USA 127, 279 (2001).
  127. F. Alet et al., Skillful joint probabilistic weather forecasting from marginals, arXiv:2506.10772.
  128. Y. Chen, M. Goldstein, M. Hua, M. Albergo, N. Boffi, and E. Vanden-Eijnden, Probabilistic forecasting with stochastic interpolants and föllmer processes, arXiv:2403.13724.
  129. A. Zhou, A. Wikner, A. Lancelin, P. Hassanzadeh, and A. B. Farimani, Reframing generative models for physical systems using stochastic interpolants, arXiv:2509.26282.
  130. N. Mücke and B. Sanderse, Physics-aware generative models for turbulent fluid flows through energy-consistent stochastic interpolants, arXiv:2504.05852.
  131. E. Lorenz, Predictability: A problem partly solved, in Seminar on Predictability, 4–8 September 1995 (ECMWF, 1995).
  132. A. Gritsun and G. Branstator, Climate response using a three-dimensional operator based on the fluctuation–dissipation theorem, J. Atmos. Sci. 64 2558 (2007).
  133. L. Giorgini, F. Falasca, and A. Souza, Predicting forced responses of probability distributions via the fluctuation-dissipation theorem and generative modeling, Proc. Natl. Acad. Sci. USA 122, e2509578122 (2025).
  134. L. Onsager and S. Machlup, Fluctuations and irreversible processes, Phys. Rev. 91, 1505 (1953).
  135. D. Lucente, A. Baldassarri, A. Puglisi, A. Vulpiani, and M. Viale, Inference of time irreversibility from incomplete information: Linear systems and its pitfalls, Phys. Rev. Res. 4, 043103 (2022).
  136. A general framework for linking free and forced fluctuations via Koopmanism, Chaos, Solitons Fractals 202, 117540 (2026).
  137. M. J. Kusner, J. Loftus, C. Russell, and R. Silva, Counterfactual fairness, in Advances in Neural Information Processing Systems, edited by I. Guyon, U. V. Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan, and R. Garnett (Curran Associates, Inc., Long Beach, CA, USA, 2017), Vol. 30.
  138. J. Runge, V. Petoukhov, J. F. Donges, J. Hlinka, N. Jajcay, M. Vejmelka, D. Hartman, N. Marwan, M. Paluš, and J. Kurths, Identifying causal gateways and mediators in complex spatio-temporal systems, Nat. Commun. 6, 8502 (2015).
  139. N. Ay and D. Polani, Information flows in causal networks, Adv. Complex Syst. 11, 17 (2008).
  140. M. Falcioni and A. Vulpiani, The relevance of chaos for the linear response theory, Phys. A 215, 481 (1995).
  141. M. Falcioni, S. Isola, and A. Vulpiani, Correlation functions and relaxation properties in chaotic dynamics and statistical mechanics, Phys. Lett. A 144, 341 (1990).
  142. C. E. Leith, Climate response and fluctuation dissipation, J. Atmos. Sci. 32, 2022 (1975).
  143. V. Lucarini, Interpretable and equation-free response theory for complex systems, arXiv:2502.07908.
  144. G. Di Antonio and G. V. Vinci, Nonlinear correlations underlie linear response and causality, Phys. Rev. Res. 7, L042029 (2025).
  145. E. Lorenz, Deterministic nonperiodic flow, J. Atmos. Sci. 20, 130 (1963).
  146. J. Charney and J. DeVore, Multiple flow equilibria in the atmosphere and blocking, J. Atmos. Sci. 36, 1205 (1979).
  147. A. J. Roberts, Resolving the multitude of microscale interactions accurately models stochastic partial differential equations, LMS J. Comput. Math. 9, 193 (2006).
  148. A. Roberts, Normal form transforms separate slow and fast modes in stochastic dynamical systems, Phys. A 387, 12 (2008).
  149. S. Kravtsov, M. Ghil, and D. Kondrashov, Empirical model reduction and the modeling hierarchy in climate dynamics and the geosciences, in Stochastic Physics and Climate Modeling, edited by T. Palmer and P. Williams (Cambridge University Press, Cambridge, 2009), pp. 35–72.
  150. F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel, P. Prettenhofer, R. Weiss, V. Dubourg, J. Vanderplas, A. Passos, D. Cournapeau, M. Brucher, M. Perrot, and E. Duchesnay, Scikit-learn: Machine learning in Python, J. Mach. Learn. Res. 12, 2825 (2011).
  151. D. P. Kingma and J. Ba, Adam: A method for stochastic optimization, arXiv:1412.6980.
  152. J. Adler and O. Öktem, Deep Bayesian inversion, Data-driven Models in Inverse Problems, edited by T. A. Bubba (De Gruyter, Berlin, Boston, 2025), pp. 359–412.
  153. F. Takens, Detecting strange attractors in turbulence, in Dynamical Systems and Turbulence, Lecture N Otes in Mathematics, edited by D. Rand and L. Young (Springer, Berlin, 1981), Vol. 898, p. 21.

Outline

Information

Phys. Rev. Research 7, 043314– Published 19 December, 2025
  • Received 14 July 2025
  • Accepted 13 November 2025
  • Corrected 20 January 2026
  • Corrected 5 March 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation