Physically-Relevant Information Learning in High-Dimensional Time-Derivatives Spaces

Understanding the physics of many-body complex dynamical systems may be a non-trivial task. High-dimensional analysis approaches are often deemed necessary to prevent losing important information. Typically, these use order parameters or descriptors capturing information related to, e.g., relative positions, symmetries, etc., of the units in the studied system. However, in many cases, gaining information related to the relative positions of the constitutive units (or their velocities) alone may be insufficient, and to reach a more complete physical knowledge, one should ideally learn and correlate with each other both structure and dynamics. Here we demonstrate how to achieve such a goal efficiently by building and navigating high-dimensional Time-Derivatives (TiDe) spaces. A TiDe space can be generated for virtually any type of system/phenomenon from the time-series data collected along its observation over time. Each TiDe's dimension corresponds to a growing-order time-derivative of the extracted data, thus containing information related to different physical phenomena/events, which can be easily extracted via unsupervised approaches. We demonstrate how, by definition, TiDes can be directly analyzed without a need for prior dimensionality reduction, providing results that are intrinsically intuitive to interpret. We show the potential of the method by analyzing two prototypical example datasets extracted from molecular dynamics simulations or experimental tracking of different types of complex dynamical systems. Our results demonstrate how efficiently one can navigate and learn in information-rich TiDe spaces, which provide a robust general framework for data analysis and for studying complex dynamical systems from the data collected along their observation over time.