发表机构
Technical University of Denmark; University of Cambridge; University of Leicester; Great Bay University（丹麦技术大学; 剑桥大学; 莱斯特大学; 大湾区大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究提出基于线性响应理论的检验框架，用于验证学习型随机模拟器对强迫的响应保真度，并在Lorenz-63模型上测试多种方法，发现不同模型在统计与响应保真度上存在分离，训练方式决定其表现。
AI 中文摘要
混沌与随机系统的机器学习仿真器通常通过预报技巧和长期统计量进行验证，但这两者均不能证明仿真器对强迫的响应是否正确，而后者正是投影与归因研究所依赖的性质。线性响应理论使这一验证成为可能：受迫响应通过广义涨落-耗散关系由未受扰动的相关性决定，并可分解为Koopman生成元的随机Ruelle-Pollicott共振模式。基于Koopmanism Response框架，我们将其转化为针对学习型替代模型的校准的、模式分辨的检验：每次替代模型 rollout 通过或失败各项检验，并将失败率与真实系统的独立实现进行比较。在随机Lorenz-63（一个三变量玩具模型）上，我们评估了SINDy、MLP、储备池计算机、神经ODE以及具有学习扩散项的神经SDE，每种模型最多进行80次rollout。具有正确函数库的稀疏回归模型以与真实系统一致的比率通过所有检验。不变统计保真度与响应保真度在两个方向上均发生分离：四分之一的储备池计算机rollout通过所有不变统计检验并匹配静态磁化率χ(0)，但错误表征了慢弛豫模式；而神经ODE和SDE很少达到不变统计基线，但在四分之三的rollout中恢复了这些模式。正如对由快弛豫主导的时间积分量所预期的那样，χ(0)无法区分这些情况。对于固定网络，训练公式（单步漂移、流映射或多步积分器）决定了其获得这些性质中的哪些。
英文摘要
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 $χ(0)$, 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, $χ(0)$ 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.
Comments16 pages, 2 figures, 10 tables. Extended version of the short paper accepted at the NeurIPS 2026 workshop "AI for Stochastic Dynamics"
