Physically-Relevant Information Learning in High-Dimensional Time-Derivatives Spaces
Authors: Domiziano Doria, Matteo Becchi, Giovanni M. Pavan
Organizations: Department of Applied Science and Technology, Politecnico di Torino, Torino 10129, Italy
Abstract
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.
Identifying the dynamical state variables of a system from high-dimensional observations is a central problem across physical sciences. The challenge is that the state variables are not directly observable and must be inferred from raw high-dimensional data without supervision. Here we introduce DySIB (Dynamical Symmetric Information Bottleneck) as a method to learn low-dimensional representations of time-series data by maximizing predictive mutual information between past and future observation windows while penalizing representation complexity. This objective operates entirely in latent space and avoids reconstruction of the observations. We apply DySIB to an experimental video dataset of a physical pendulum, where the underlying state space is known. The method, with hyperparameters of the learning architecture set self-consistently by the data, recovers a two-dimensional representation that matches the dimensionality, topology, and geometry of the pendulum phase space, with the learned coordinates aligning smoothly with the canonical angle and angular velocity. These results demonstrate, on a well-characterized experimental system, that predictive information in latent space can be used to recover interpretable dynamical coordinates directly from high-dimensional data.
K. Michael Martini, Eslam Abdelaleem, Paarth Gulati +1
This work addresses the problem of learning the dynamics of high-dimensional probability densities over time using unlabeled samples, without assuming access to trajectory information. We introduce two-parameter flows that learn only sampling-time transports from a base distribution to each marginal and then extract a physics-time velocity by regressing on coupled synthetic trajectories. We prove that the resulting physics-time dynamics are unique and inherit regularity from the sampling-time transports. Because we can build on standard, well-developed conditional flow matching techniques for learning the base-to-marginal transports, our approach scales to high dimensions and avoids per-step optimal-transport couplings, while allowing admissible non-gradient dynamics that can naturally explain rotational or circulating physics phenomena.
Paul Schwerdtner, Tobias Blickhan, Benjamin Peherstorfer
In recent years, weak-form methods have made significant advances in data-driven discovery of dynamical systems. However, in high-dimensional settings, current techniques can prove expensive in both computation and memory. In this work, we introduce TT-WSINDy, which combines techniques of the Multidimensional Approximation of Nonlinear Dynamics (MANDy) and Weak Sparse Identification of Nonlinear Dynamics (WSINDy) methods, implementing requisite computations in the tensor-train (TT) format. We demonstrate that this method is able to search an exponentially-growing space of candidate functions -- performing weak-form transformation, regression, and sparsification -- without suffering from the curse of dimensionality.