Leveraging Geometric Structure in Learning Dynamical Systems

Fluid-structural interactions arise in many engineering applications, such as large-scale offshore wind turbines and urban air mobility vehicles. One of the key challenges in modeling such dynamical interactions is to identify the appropriate mathematical representation that respects the physical constraints that are encoded in the observed data. The goal of this project is to model such dynamical interactions with a novel machine learning algorithm that incorporates the geometrical constraints that can be revealed from the data. The key scientific product of this research is a stable and accurate Artificial Intelligence (AI) model for long-time dynamical predictions under external disturbances. In the urban air mobility vehicles application, for example, one is interested in predicting the structural deformation with the strong coupling to the unsteady aerodynamics. Another important byproduct of this research is the reduction of the high energy consumption in training and prediction using AI models. Besides the engineering applications, this project is likely to generate new mathematical questions, fostering interactions between computational mathematics and data-enabled science. This project will contribute to the NSF's mission of advancing STEM through the training of graduate students in an interdisciplinary research training environment to be proficient in mathematical analysis, applied differential geometry, statistical learning, and scientific computing. The goal of the project is to reveal geometric structures encoded in the measured time series of the dynamics, i.e., the data manifold, to enable compact reduced-order modeling for scalable numerical simulations. The proposed approach is to develop provably convergent computational algorithms to identify vector fields of dynamical systems under the data manifold. The focus is on two hypotheses: under the manifold assumption and under a more refined structural constraint, such as Lie groups. The proposed project is to: 1) Develop an operator-valued kernel that ensures the vector fields lie on the tangent bundle in the limit of large data. The underlying vector field will be identified through standard regression algorithms with the novel operator-valued kernel and extend it to noisy measurements that perturb the time series to be slightly off the manifolds. 2) Devise a normal correction to the standard Euler scheme that approximates the exponential maps of the vector fields induced by the dynamics to ensure the robustness of the proposed model on nontrivial geometry. 3) Devise a data-driven Lie group analysis, in the form of a constrained optimization, to identify potential Lie group structure from the data. 4) Extract scaling laws from the identified Lie groups, which represent the equivariance among dynamics at different operating conditions. With such a scaling law structure, the reduced order model is represented by a manifold structure that can be decomposed into a product of fiber bundles and base space. Here, the fiber bundle represents the scaling laws, while the reduced-order model can be identified as a vector field on the base space. The effectiveness of these approaches will be demonstrated numerically on a series of dynamics with increasing complexity, including the academic test examples with inertial manifold structure (e.g., Kuramoto-Sivashinsky and Korteweg-de Vries equations), to real-world fluid-structural interaction problems with limit cycle oscillations and nontrivial scaling law structures. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.