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 {probabilistic generative} 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 generative 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：数据驱动建模可在底层过程未知或计算成本过高时有效重构非线性动力系统。可靠的预测不确定性评估使相关工具能够应用于预测未曾观测的新场景。本文首先扩展并行局部高斯过程，用于预测连接当前与下一时刻观测值的向量值转移函数，并通过后验采样量化预测的不确定性。其次，我们证明了动态模式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量之间的等价性。该关联为动态模式分解提供了概率生成模型，从而能够获得预测的不确定性。此外，我们通过生成模型的统一视角，揭示了不同数据驱动模型在逼近非线性动力学时的密切关联。我们研究了两个数值算例：第一个算例假设动力输入已知，第二个算例中输入未知。算例表明，预测不确定性可被合理量化，但模型或输入设定错误会降低不确定性量化的准确性。