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Feedback control systems

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Related Experiment Videos

Adversarial dynamical systems characterize when data-driven learning succeeds or fails.

Matthew J Colbrook1, Igor Mezić2, Alexei Stepanenko3

  • 1DAMTP, University of Cambridge, Cambridge, UK. m.colbrook@damtp.cam.ac.uk.

Nature Communications
|July 14, 2026
PubMed
Summary

Data-driven spectral learning can reliably infer system dynamics, but only under specific conditions. Adversarial systems define the boundary between learning success and failure, offering guarantees for physical systems.

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Area of Science:

  • Dynamical systems theory
  • Machine learning
  • Spectral analysis

Background:

  • Many complex systems resist traditional analytical modeling, necessitating data-driven approaches for dynamic inference.
  • Existing data-driven methods face challenges in convergence and generalization, raising questions about the reliability of learned system behaviors.
  • Koopman operator learning offers a framework for analyzing nonlinear dynamics via linear spectral representations.

Purpose of the Study:

  • To establish the conditions under which data-driven spectral learning of dynamical systems is reliable.
  • To develop optimal data-driven spectral algorithms with convergence and certification guarantees.
  • To delineate the boundary between accessible and inaccessible learning regimes using adversarial dynamical systems.

Main Methods:

  • Development of optimal data-driven spectral algorithms within the Koopman operator learning framework.
  • Construction of adversarial dynamical systems to prove impossibility results for learning.
  • Validation of the theoretical framework on physical systems, including oscillators, chaotic fluid flows, and Arctic sea ice concentration.

Main Results:

  • Convergence theory for Koopman operator approximations, resolving a key problem in spectral analysis.
  • Matching impossibility results demonstrating that learning guarantees fail without specific conditions, irrespective of data quality.
  • Successful application to Arctic sea ice forecasting, uncovering hidden decline modes and outperforming existing models.

Conclusions:

  • The study precisely characterizes the conditions for successful data-driven spectral learning in dynamical systems.
  • The developed framework provides guarantees for physical systems and enables reliable long-range forecasting with error bounds.
  • The approach offers a computationally efficient alternative to state-of-the-art deep learning models for complex system analysis.