Ruelle--Pollicott Theory for Metastable Systems: A Unified Framework for Tipping Transitions

What the math gives to ML

The paper's transferable asset is a residue-resolved spectral sensitivity principle: a small relaxation eigenvalue creates large response only when the observable and perturbation direction actually couple to that spectral block. Projectors and nilpotent Jordan components explain amplification that eigenvalue-gap monitoring alone misses, including strong transient response from non-normal dynamics with a frozen spectrum. This suggests replacing crude spectral-radius or smallest-gap regularization in recurrent and continuous-time networks with a mode-selective penalty based on observable–perturbation residues. The most direct first target is a neural ODE or state-space model, where local Jacobian modes, input perturbations, and readout sensitivities can be estimated with Arnoldi or Schur methods.