Ornstein's d-bar distance as a dynamical-fidelity metric for surrogate models of chaotic systems

Learned surrogates of chaotic systems cannot be judged by pointwise trajectory error,because nearby trajectories separate exponentially. The de facto replacement is to compareinvariant measures - where the system spends its time - via Wasserstein-type distances.But the invariant measure is a static object: two systems can share it exactly while disagreeing about every dynamical property (mixing, entropy production, temporal ordering). Weimport Ornstein’s ¯d-distance from ergodic theory - a metric on stationary processes thatmetrizes dynamical, not merely distributional, equivalence - and turn it into a practicalevaluation metric for chaotic surrogates. Our estimator computes exact optimal transport between empirical n-block distributions of symbolized trajectories under Hamming cost; everycomputed value is a statistically certified lower bound on ¯d, is reported against a same-processnoise floor, and is complemented by an independent entropy-rate certificate obtained froma Fano-type inequality. On Lorenz-63 and the Kuramoto–Sivashinsky equation, surrogatesconstructed to match the invariant measure - IAAFT surrogates (exact marginal, matchedspectrum) and time-rescaled flows (exactly the same invariant measure) - are invisible tomeasure- and spectrum-based baselines yet are separated from truth by ¯dat more than 20×the noise floor over 8 seeds (and remain detectable with as few as 104 samples). On a family of trained echo-state-network surrogates, ¯d resolves dynamical failures that state-spaceWasserstein cannot. We also report what the metric does not see (time reversal, at binarypartitions), and release code.