Shane A. McQuarrie

Gavin Paxton, Seunghee Cheon, Rudy Geelen et al.

We present a dynamic subspace approach for efficiently approximating large-scale systems by learning time-continuous trajectories on the Grassmannian manifold. By parameterizing a low-dimensional basis as a geodesic path, the method allows for adaptive tracking of evolving physics. Our approach decouples the geometric drift of the subspace from the intrinsic state evolution. This avoids the typical rank inflation required by static low-dimensional approximation methods to maintain accuracy, effectively breaking the Kolmogorov barrier in transport-dominated phenomena. To ensure scalability for high-dimensional data, the optimization is performed in a reduced feature space, rendering the computational cost independent of the large original state dimension. Numerical results for a 1D transport equation and a large-scale turbulent airfoil wake demonstrate that this dynamic subspace approach achieves higher accuracy than static linear approximations at equivalent ranks, positioning it as a robust and scalable method for the low-rank modeling of complex, non-stationary dynamical systems.

Jack DeChant, Rudy Geelen, Shane A. McQuarrie et al.

We present a dynamic subspace approach for efficiently approximating large-scale systems by learning time-continuous trajectories on the Grassmannian manifold. By parameterizing a low-dimensional basis as a geodesic path, the method allows for adaptive tracking of evolving physics. Our approach decouples the geometric drift of the subspace from the intrinsic state evolution. This avoids the typical rank inflation required by static low-dimensional approximation methods to maintain accuracy, effectively breaking the Kolmogorov barrier in transport-dominated phenomena. To ensure scalability for high-dimensional data, the optimization is performed in a reduced feature space, rendering the computational cost independent of the large original state dimension. Numerical results for a 1D transport equation and a large-scale turbulent airfoil wake demonstrate that this dynamic subspace approach achieves higher accuracy than static linear approximations at equivalent ranks, positioning it as a robust and scalable method for the low-rank modeling of complex, non-stationary dynamical systems.

Shane A. McQuarrie, Mengwu Guo, Anirban Chaudhuri

This work develops an active learning framework to intelligently enrich data-driven reduced-order models (ROMs) of parametric dynamical systems, which can serve as the foundation of virtual assets in a digital twin. Data-driven ROMs are explainable, computationally efficient scientific machine learning models that aim to preserve the underlying physics of complex dynamical simulations. Since the quality of data-driven ROMs is sensitive to the quality of the limited training data, we seek to identify training parameters for which using the associated training data results in the best possible parametric ROM. Our approach uses the operator inference methodology, a regression-based strategy which can be tailored to particular parametric structure for a large class of problems. We establish a probabilistic version of parametric operator inference, casting the learning problem as a Bayesian linear regression. Prediction uncertainties stemming from the resulting probabilistic ROM solutions are used to design a sequential adaptive sampling scheme to select new training parameter vectors that promote ROM stability and accuracy globally in the parameter domain. We conduct numerical experiments for several nonlinear parametric systems of partial differential equations and compare the results to ROMs trained on random parameter samples. The results demonstrate that the proposed adaptive sampling strategy consistently yields more stable and accurate ROMs than random sampling does under the same computational budget.