Dynamical systems arise in a wide variety of mathematical models from science and engineering. A common challenge is to quantify uncertainties on model inputs (parameters) that correspond to a quantitative characterization of uncertainties on observable Quantities of Interest (QoI). To this end, we consider a stochastic inverse problem (SIP) with a solution described by a pullback probability measure. We call this an observation-consistent solution, as its subsequent push-forward through the QoI map matches the observed probability distribution on model outputs. A distinction is made between QoI useful for solving the SIP and arbitrary model output data. In dynamical systems, model output data are often given as a series of state variable responses recorded over a particular time window. Consequently, the dimension of output data can easily exceed $\mathcal{O}(1E4)$ or more due to the frequency of observations, and the correct choice or construction of a QoI from this data is not self-evident. We present a new framework, Learning Uncertain Quantities (LUQ), that facilitates the tractable solution of SIPs for dynamical systems. Given ensembles of predicted (simulated) time series and (noisy) observed data, LUQ provides routines for filtering data, unsupervised learning of the underlying dynamics, classifying observations, and feature extraction to learn the QoI map. Subsequently, time series data are transformed into samples of the underlying predicted and observed distributions associated with the QoI so that solutions to the SIP are computable. Following the introduction and demonstration of LUQ, numerical results from several SIPs are presented for a variety of dynamical systems arising in the life and physical sciences. For scientific reproducibility, we provide links to our Python implementation of LUQ and to all data and scripts required to reproduce the results in this manuscript.

中文翻译：

---

从动态系统中学习兴趣量以进行观测一致反演

动力系统出现在来自科学和工程的各种数学模型中。一个常见的挑战是量化模型输入（参数）的不确定性，这些不确定性对应于可观察的感兴趣数量 (QoI) 的不确定性的定量表征。为此，我们考虑一个随机逆问题 (SIP)，其解决方案由回调概率度量描述。我们称其为观察一致的解决方案，因为它随后通过 QoI 映射的推进与模型输出上观察到的概率分布相匹配。区分 QoI 可用于解决 SIP 和任意模型输出数据。在动力系统中，模型输出数据通常作为在特定时间窗口内记录的一系列状态变量响应给出。最后，由于观察频率的原因，输出数据的维度很容易超过 $\mathcal{O}(1E4)$ 或更多，并且从这些数据中正确选择或构建 QoI 并不是不言而喻的。我们提出了一个新的框架，学习不确定量（LUQ），它促进了动态系统的 SIP 的易处理解决方案。给定预测（模拟）时间序列和（嘈杂）观察数据的集合，LUQ 提供了用于过滤数据、底层动态的无监督学习、对观察进行分类和特征提取以学习 QoI 图的例程。随后，时间序列数据被转换为与 QoI 相关的潜在预测和观察分布的样本，以便 SIP 的解决方案是可计算的。随着LUQ的介绍和演示，针对生命和物理科学中出现的各种动力系统，展示了来自多个 SIP 的数值结果。为了科学的可重复性，我们提供了指向我们的 LUQ 的 Python 实现以及重现本手稿中结果所需的所有数据和脚本的链接。