Noise poses a challenge for learning dynamical-system models because already small variations can distort the dynamics described by trajectory data. This work builds on operator inference from scientific machine learning to infer low-dimensional models from high-dimensional state trajectories polluted with noise. The presented analysis shows that, under certain conditions, the inferred operators are unbiased estimators of the well-studied projection-based reduced operators from traditional model reduction. Furthermore, the connection between operator inference and projection-based model reduction enables bounding the mean-squared errors of predictions made with the learned models with respect to traditional reduced models. The analysis also motivates an active operator inference approach that judiciously samples high-dimensional trajectories with the aim of achieving a low mean-squared error by reducing the effect of noise. Numerical experiments with high-dimensional linear and nonlinear state dynamics demonstrate that predictions obtained with active operator inference have orders of magnitude lower mean-squared errors than operator inference with traditional, equidistantly sampled trajectory data.

翻译：对学习动态系统模型来说,噪音是一个挑战,因为已经小的变异会扭曲轨迹数据描述的动态。这项工作建立在操作者从科学机器学习到从受噪音污染的高维状态轨迹推断低维模型的推论上。介绍的分析表明,在某些情况下,推断的操作者是经过仔细研究的基于预测的减少的操作者从传统的模型减少中公正估计出一个低度错误。此外,操作者推论和基于投影的模型减少之间的关联使得能够约束与所学模型一起对传统降幅模型所作的预测的中度错误。分析还激励了一个主动操作者推论方法,该方法通过减少噪音的影响,明智地采样高维度轨迹,以达到一个低度的中度错误。具有高度线性和非线性状态动态的数值实验表明,与主动操作者推论获得的预测,其平均值的误差幅度比操作者根据传统、等近的抽样轨迹数据推断的平均数要低。