This work introduces a non-intrusive model reduction approach for learning reduced models from partially observed state trajectories of high-dimensional dynamical systems. The proposed approach compensates for the loss of information due to the partially observed states by constructing non-Markovian reduced models that make future-state predictions based on a history of reduced states, in contrast to traditional Markovian reduced models that rely on the current reduced state alone to predict the next state. The core contributions of this work are a data sampling scheme to sample partially observed states from high-dimensional dynamical systems and a formulation of a regression problem to fit the non-Markovian reduced terms to the sampled states. Under certain conditions, the proposed approach recovers from data the very same non-Markovian terms that one obtains with intrusive methods that require the governing equations and discrete operators of the high-dimensional dynamical system. Numerical results demonstrate that the proposed approach leads to non-Markovian reduced models that are predictive far beyond the training regime. Additionally, in the numerical experiments, the proposed approach learns non-Markovian reduced models from trajectories with only 20% observed state components that are about as accurate as traditional Markovian reduced models fitted to trajectories with 99% observed components.

这项工作引入了一种非侵入式模型简化方法，用于从部分观察到的高维动态系统的状态轨迹中学习简化模型。与依赖当前简化状态的传统马尔可夫简化模型相比，所提出的方法通过构建基于简化状态历史进行未来状态预测的非马尔可夫简化模型来补偿由于部分观察到的状态而导致的信息丢失单独预测下一个状态。这项工作的核心贡献是一个数据采样方案，用于从高维动力系统中采样部分观察到的状态，以及一个回归问题的公式，以将非马尔可夫约简项拟合到采样状态。在一定条件下，所提出的方法从数据中恢复了与使用侵入式方法获得的非常相同的非马尔可夫项，这些方法需要高维动力系统的控制方程和离散算子。数值结果表明，所提出的方法导致非马尔可夫简化模型的预测远远超出训练制度。此外，在数值实验中，所提出的方法从只有 20% 观察到的状态分量的轨迹中学习非马尔可夫简化模型，这些模型与适合具有 99% 观察到的分量的轨迹的传统马尔可夫简化模型一样准确。数值结果表明，所提出的方法导致非马尔可夫简化模型的预测远远超出训练制度。此外，在数值实验中，所提出的方法从只有 20% 观察到的状态分量的轨迹中学习非马尔可夫简化模型，这些模型与适合具有 99% 观察到的分量的轨迹的传统马尔可夫简化模型一样准确。数值结果表明，所提出的方法导致非马尔可夫简化模型的预测远远超出训练制度。此外，在数值实验中，所提出的方法从只有 20% 观察到的状态分量的轨迹中学习非马尔可夫简化模型，这些模型与适合具有 99% 观察到的分量的轨迹的传统马尔可夫简化模型一样准确。