Mechanical and Industrial Engineering, Northeastern University, Boston, MA, 02215, USA.
Electrical and Systems Engineering, Washington University in St. Louis, St. Louis, MO, 63130, USA.
Sci Rep. 2022 Jul 27;12(1):12807. doi: 10.1038/s41598-022-15669-7.
Dynamical systems pervasively seen in most real-life applications are complex and behave by following certain evolution rules or dynamical patterns, which are linear, non-linear, or stochastic. The underlying dynamics (or evolution rule) of such complex systems, if found, can be used for understanding the system behavior, and furthermore for system prediction and control. It is common to analyze the system's dynamics through observations in different modality approaches. For instance, to recognize patient deterioration in acute care, it usually relies on monitoring and analyzing vital signs and other observations, such as blood pressure, heart rate, respiration, and electroencephalography. These observations convey the information describing the same target system, but the dynamics is not able to be directly characterized due to high complexity of individual modality and maybe time-delay interactions among modalities. In this work, we suppose that the state behavior of a dynamical system follows an intrinsic dynamics shared among these modalities. We specifically propose a new deep auto-encoder framework using the Koopman operator theory to derive the joint linear dynamics for a target system in a space spanned by the intrinsic coordinates. The proposed method aims to reconstruct the original system states by learning the information provided among multiple modalities. Furthermore, with the derived intrinsic dynamics, our method is capable of restoring the missing observations within and across modalities, and used for predicting the future states of the system that follows the same evolution rule.
在大多数实际应用中广泛存在的动态系统是复杂的，并按照特定的演化规则或动态模式表现出来，这些规则或模式可以是线性的、非线性的或随机的。如果找到此类复杂系统的潜在动态（或演化规则），就可以用于理解系统行为，并且可以进一步用于系统预测和控制。通过不同的模态方法进行观察来分析系统的动态是很常见的。例如，要识别急性护理中的患者恶化情况，通常依赖于监测和分析生命体征和其他观察结果，例如血压、心率、呼吸和脑电图。这些观察结果传达了描述同一目标系统的信息，但由于单个模态的复杂性以及模态之间可能存在的时滞相互作用，动态无法直接描述。在这项工作中，我们假设动态系统的状态行为遵循这些模态之间共享的内在动态。我们特别提出了一种新的深度自动编码器框架，使用 Koopman 算子理论来推导目标系统在由内在坐标构成的空间中的联合线性动力学。所提出的方法旨在通过学习多个模态之间提供的信息来重建原始系统状态。此外，通过推导出的内在动力学，我们的方法能够恢复模态内和模态间的缺失观察结果，并用于预测遵循相同演化规则的系统的未来状态。
