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  • Open Access

Targeted Calibration to Adjust Stability Biases in Complex Dynamical System Models

Daniel Pals1,‡,§, Sebastian Bathiany2,3,*,‡,§, Joel Kuettel2,4, Richard A. Wood5, and Niklas Boers2,3,6,†

  • *Contact author: sebastian.bathiany@tum.de
  • †Contact author: n.boers@tum.de
  • ‡Also at Technical University of Munich, Germany; Munich Climate Center; TUM School of Engineering and Design, Department of Aerospace and Geodesy, Earth System Modelling Group.
  • §These authors contributed equally to this work.
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Phys. Rev. X 16, 021007 – Published 7 April, 2026

DOI: https://doi.org/10.1103/624z-1p1s

Abstract

Models of complex dynamical systems like the Earth’s climate often involve large numbers of uncertain parameters. Comprehensive exploration of the parameter space is typically prohibitive due to excessive computational costs, and systematic gradient-based parameter optimization is not feasible because such models are typically not differentiable. This is especially problematic in cases where the models intend to describe highly nonlinear and possibly abrupt dynamics, where sensitivity to parameter changes is high. Components of Earth’s climate system, such as the North Atlantic Overturning Circulation, the polar ice sheets, or the Amazon rainforest, are at risk of undergoing critical transitions in response to anthropogenic climate change. However, estimates of the critical forcing thresholds are highly uncertain because the parameter spaces of complex climate models cannot be fully explored. Concerns have been raised that the above Earth system components are too stable in state-of-the-art models. Here, we introduce a method for efficient, systematic, and objective calibration of dynamical complex system models, targeted at adjusting system stability. Given a number of physical or observational constraints, our method moves the system in a direction where the system loses or gains stability, guided by indicators of “critical slowing down”. In contrast to a brute force approach, where the computational cost would exponentially increase with the number of parameters, our method scales polynomially and thus evades the curse of dimensionality. We successfully apply our method to a conceptual double-fold bifurcation model and a physically plausible reduced-order model of the global ocean circulation. Our method can efficiently adjust stability biases across a range of complex system models, helping to reveal hidden instabilities and resulting state transitions they induce. In particular, it can help us to explore and improve the representation of key multistable components of the Earth system in climate models.

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Popular Summary

Numerical models of complex systems like the Earth system are expensive to run and involve many uncertain and typically hand-tuned parameters. New methods for calibrating such complex models efficiently, objectively, and systematically are needed. In the context of anthropogenic climate change, there is particular concern that specific tipping elements, like the Atlantic meridional overturning circulation, might be overly stable in models due to imperfect parameter choices. Here, we present a methodology for systematically calibrating numerical models toward parameter configurations that yield more or less stable dynamics. We apply our method to a simple bistable model and a conceptual ocean circulation model, demonstrating that our method can help find hidden tipping points and can calibrate complex models under user-defined constraints.

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Phys. Rev. X 16, 021007– Published 7 April, 2026
  • Received 14 October 2024
  • Accepted 16 March 2026
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