A Response Theory Probe for Learned Stochastic AI Simulators, Tested on Lorenz-63

Abstract:Machine-learning emulators of chaotic and stochastic systems are usually validated on forecast skill and long-run statistics. Neither certifies that an emulator responds correctly to forcing, the property that projection and attribution studies rely on. Linear response theory makes this testable: the forced response follows from unperturbed correlations through a generalized fluctuation-dissipation relation, and decomposes over the stochastic Ruelle-Pollicott resonances of the Koopman generator. Building on the Koopmanism Response framework, we turn this into a calibrated, mode-resolved test for learned surrogates: each surrogate rollout passes or fails each check, and failure rates are compared with those of independent realizations of the true system. On stochastic Lorenz-63, a three-variable toy model, we evaluate SINDy, an MLP, a reservoir computer, a neural ODE and a neural SDE with learned diffusion, over up to 80 rollouts each. A sparse-regression model with the correct library passes every check at rates consistent with the true system. Invariant-statistics fidelity and response fidelity dissociate in both directions: a quarter of reservoir-computer rollouts pass every invariant-statistics check and match the static susceptibility $\chi(0)$, yet misrepresent the slow relaxation modes, while the neural ODE and SDE rarely meet the invariant-statistics floor but recover those modes in three quarters of rollouts. As expected of a time-integrated quantity dominated here by fast relaxation, $\chi(0)$ does not separate these cases. For a fixed network, the training formulation (one-step drift, flow map, or multi-step through the integrator) decides which of these properties it gets right.

16 pages, 2 figures, 10 tables. Accepted at the NeurIPS 2026 workshop "AI for Stochastic Dynamics"

Dynamical Systems (math.DS); Machine Learning (cs.LG); Chaotic Dynamics (nlin.CD)

arXiv:2610.06798 [math.DS]

(or arXiv:2610.06798v1 [math.DS] for this version)

https://doi.org/10.48550/arXiv.2610.06798 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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