Data-driven modeling is useful for reconstructing nonlinear dynamical systems when the true data generating mechanism is unknown or too expensive to compute. Having reliable uncertainty assessment of the forecast enables tools to be deployed to predict new scenarios that haven't been observed before. In this work, we derive internal uncertainty assessments from a few models for probabilistic forecasts. First, we extend the parallel partial Gaussian processes for predicting the one-step-ahead 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 maximum likelihood estimator of a linear mapping matrix in a linear state space model. This connection provides data generating models of dynamic mode decomposition and thus, the uncertainty of the predictions can be obtained. Third, we draw close connections between data-driven models of nonlinear dynamical systems, such as proper orthogonal decomposition, dynamic mode decomposition and parallel partial Gaussian processes, 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：数据驱动建模在真实数据生成机制未知或计算成本过高时，对于重构非线性动力学系统具有重要价值。对预测结果进行可靠的不确定性评估，能够使相关工具应用于预测此前未观测到的新场景。本研究基于多种模型推导了概率预测的内部不确定性评估方法。首先，我们扩展了并行偏高斯过程，用于预测连接当前与下一个时间点观测的一步超前向量值转移函数，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之间的等价性。这种关联为动态模式分解提供了数据生成模型，从而能够获得预测的不确定性。第三，通过数据生成模型的统一视角，我们揭示了非线性动力学系统数据驱动模型（如本征正交分解、动态模式分解及并行偏高斯过程）之间的密切联系。我们研究了两个数值算例：第一个算例假设动力学输入已知，第二个算例假设输入未知。算例结果表明，预测的不确定性能够得到合理量化，而模型或输入的误设会降低不确定性量化的精度。