arXiv：cs.LG· Jo\~ao B\"oger, Simon Driscoll, Niccol\`o Zagli, Valerio Lucarini, Francisco Camara Pereira·· 4 小时前AI 评分38
面向学习型随机 AI 模拟器的响应理论探针：在 Lorenz-63 上的测试
A Response Theory Probe for Learned Stochastic AI Simulators, Tested on Lorenz-63
AI 导读
研究者提出一种基于 Koopmanism Response 框架的校准式、模式分辨测试，用于检验学习型 AI 模拟器能否正确响应外力。
正文
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.
| Comments: | 16 pages, 2 figures, 10 tables. Extended version of the short paper accepted at the NeurIPS 2026 workshop "AI for Stochastic Dynamics" |
| Subjects: | Dynamical Systems (math.DS); Machine Learning (cs.LG); Chaotic Dynamics (nlin.CD) |
| Cite as: | arXiv:2610.06798 [math.DS] |
| (or arXiv:2610.06798v2 [math.DS] for this version) | |
| https://doi.org/10.48550/arXiv.2610.06798 arXiv-issued DOI via DataCite |
Submission history
From: João Paulo De Souza Böger [view email]
[v1]
Mon, 5 Oct 2026 17:51:23 UTC (143 KB)
[v2]
Tue, 6 Oct 2026 12:18:00 UTC (148 KB)
来源：arXiv：cs.LG · arxiv.org