Data-driven modeling is useful for reconstructing nonlinear dynamical systems when the underlying process is unknown or too expensive to compute. Having reliable uncertainty assessment of the forecast enables tools to be deployed to predict new scenarios unobserved before. In this work, we first extend parallel partial Gaussian processes for predicting the vector-valued transition function that links the observations between the current and next time points, and quantify the uncertainty of predictions by posterior sampling. Second, we show the equivalence between the dynamic mode decomposition and the maximum likelihood estimator of the linear mapping matrix in the linear state space model. The connection provides a data generating model of dynamic mode decomposition and thus, uncertainty of predictions can be obtained. Furthermore, we draw close connections between different data-driven models for approximating nonlinear dynamics, through a unified view of data generating models. We study two numerical examples, where the inputs of the dynamics are assumed to be known in the first example and the inputs are unknown in the second example. The examples indicate that uncertainty of forecast can be properly quantified, whereas model or input misspecification can degrade the accuracy of uncertainty quantification.

翻译：数据驱动建模在底层过程未知或计算成本过高时，对于重构非线性动力系统具有重要价值。可靠的预报不确定性评估能够使相关工具应用于预测前所未见的新场景。本研究首先扩展了并行偏高斯过程，用于预测连接当前与下一时间点观测值的向量值转移函数，并通过后验抽样量化预测的不确定性。其次，我们证明了动态模态分解与线性状态空间模型中线性映射矩阵的最大似然估计之间的等价性。这一联系为动态模态分解提供了数据生成模型，从而能够获得预测的不确定性。此外，通过数据生成模型的统一视角，我们建立了近似非线性动力学的不同数据驱动模型之间的紧密联系。通过两个数值算例展开研究：第一个算例假设动力系统输入已知，第二个算例则假设输入未知。算例表明，预报不确定性能够得到合理量化，但模型或输入的错误设定会降低不确定性量化的准确性。