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  3. Interpretable And Equation-free Response Theory For Complex Systems.
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  3. Interpretable And Equation-free Response Theory For Complex Systems.

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Interpretable and equation-free response theory for complex systems.

Valerio Lucarini1

  • 1School of Mathematical, Physical and Computational Sciences, College of Science and Engineering, University of Leicester , Leicester, UK.

Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences
|June 18, 2026

View abstract on PubMed

Summary
This summary is machine-generated.

We developed a new response theory for Markov chains, enabling predictions of system sensitivity to perturbations. This data-driven approach works even without knowing the underlying equations, offering interpretable insights into complex systems.

Keywords:
KoopmanismMarkov chainsMarkov state modellingProny methodmodel reductionmultiple time scalesresponse theory

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Area of Science:

  • Complex systems
  • Statistical mechanics
  • Dynamical systems theory

Background:

  • Response theory analyzes system sensitivity to perturbations.
  • Existing methods struggle with complex, multiscale systems, lacking interpretability and data-driven flexibility.
  • Markov state modelling (MSM) offers a framework for analyzing complex systems.

Purpose of the Study:

  • To develop interpretable, data-driven response theory formulas for Markov chains.
  • To predict responses of observables and higher-order correlations in complex systems.
  • To provide a foundation for equation-agnostic analysis of system dynamics.

Main Methods:

  • Linear and nonlinear response formulas derived for Markov chains.
  • Algebraic expansions inspired by Koopman analysis to identify time scales and modes.
  • Application to a simple metastable system for illustration.

    • Main Results:

    • Simple, implementable expressions for predicting system responses.
    • Methodology is amenable to purely data-driven implementation, even with unknown dynamics.
    • Explicit and interpretable expressions for Green's functions at all orders were obtained.

    Conclusions:

    • The proposed framework offers a powerful, interpretable, and data-driven approach to response theory for complex systems.
    • Provides a dynamical foundation for methods like the Prony method in signal analysis.
    • Enhances understanding of critical transitions and control in complex systems.