Deep multi-modal learning for joint linear representation of nonlinear dynamical systems

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.