Optimal parameterizing manifolds for anticipating tipping points and higher-order critical transitions

Area of Science:

• Dynamical Systems Theory
• Computational Modeling
• Data-Driven Science

Background:

• Classical reduced-order modeling relies on invariant or slow manifolds.
• The

Purpose of the Study:

• To present a general, variational approach for deriving low-order reduced models from non-autonomous systems.
• To introduce the optimal parameterizing manifold (OPM) as a successor to invariant/slow manifolds when slaving breaks down.
• To enable accurate prediction of critical transitions and tipping phenomena using reduced models.

Main Methods:

• The optimal parameterizing manifold (OPM) concept is introduced, defining a manifold that optimally averages unresolved variables conditioned on resolved ones.
• Parameterizations are derived from continuous deformations of those valid near instability onset, using auxiliary backward-forward systems.
• Analytic parameterization formulas are obtained by optimizing backward integration time, guided by data-informed minimization of parameterization defects.

Main Results:

• The OPM approach successfully derives reduced systems capable of predicting higher-order critical transitions and catastrophic tipping phenomena.
• Optimizing backward integration time per scale/variable allows for accurate parameterization, even in chaotic regimes where classical methods fail.
• Reduced models trained on pre-transition regimes demonstrate enhanced predictive accuracy for subsequent critical events.

Conclusions:

• The OPM framework offers a robust method for developing accurate reduced-order models from complex, non-autonomous systems.
• This variational approach overcomes limitations of traditional manifold concepts, particularly when "slaving" breaks down.
• The derived reduced systems provide valuable tools for understanding and predicting abrupt changes in dynamical systems.