Understanding the physics of many-body complex dynamical systems is typically non-trivial. 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 (or velocities) of the constitutive units 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 efficiently achieve such a goal by building and navigating high-dimensional Time-Derivatives (TiDe) space. A TiDe space can be easily generated for virtually any type of system/phenomenon under study 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 types of physical phenomena/events that 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 complex dynamical systems. Our results demonstrate how efficiently one can navigate and learn in such information-rich TiDe spaces, which provide robust general frameworks for data analysis and for studying complex dynamical systems from the data collected along their observation over time.
Simone Martino, Domiziano Doria, Chiara Lionello et al.
Reconstructing the physical complexity of many-body dynamical systems can be challenging. Starting from the trajectories of their constitutive units (raw data), typical approaches require selecting appropriate descriptors to convert them into time-series, which are then analyzed to extract interpretable information. However, identifying the most effective descriptor is often non-trivial. Here, we report a data-driven approach to compare the efficiency of various descriptors in extracting information from noisy trajectories and translating it into physically relevant insights. As a prototypical system with non-trivial internal complexity, we analyze molecular dynamics trajectories of an atomistic system where ice and water coexist in equilibrium near the solid/liquid transition temperature. We compare general and specific descriptors often used in aqueous systems: number of neighbors, molecular velocities, Smooth Overlap of Atomic Positions (SOAP), Local Environments and Neighbors Shuffling (LENS), Orientational Tetrahedral Order, and distance from the fifth neighbor ($d_5$). Using Onion Clustering -- an efficient unsupervised method for single-point time-series analysis -- we assess the maximum extractable information for each descriptor and rank them via a high-dimensional metric. Our results show that advanced descriptors like SOAP and LENS outperform classical ones due to higher signal-to-noise ratios. Nonetheless, even simple descriptors can rival or exceed advanced ones after local signal denoising. For example, $d_5$, initially among the weakest, becomes the most effective at resolving the system's non-local dynamical complexity after denoising. This work highlights the critical role of noise in information extraction from molecular trajectories and offers a data-driven approach to identify optimal descriptors for systems with characteristic internal complexity.