Rudy Geelen

Rudy Geelen, Laura Balzano, Stephen Wright et al.

We present a novel method for learning reduced-order models of dynamical systems using nonlinear manifolds. First, we learn the manifold by identifying nonlinear structure in the data through a general representation learning problem. The proposed approach is driven by embeddings of low-order polynomial form. A projection onto the nonlinear manifold reveals the algebraic structure of the reduced-space system that governs the problem of interest. The matrix operators of the reduced-order model are then inferred from the data using operator inference. Numerical experiments on a number of nonlinear problems demonstrate the generalizability of the methodology and the increase in accuracy that can be obtained over reduced-order modeling methods that employ a linear subspace approximation.

Rudy Geelen, Laura Balzano, Karen Willcox

We present a novel framework for learning cost-efficient latent representations in problems with high-dimensional state spaces through nonlinear dimension reduction. By enriching linear state approximations with low-order polynomial terms we account for key nonlinear interactions existing in the data thereby reducing the problem's intrinsic dimensionality. Two methods are introduced for learning the representation of such low-dimensional, polynomial manifolds for embedding the data. The manifold parametrization coefficients can be obtained by regression via either a proper orthogonal decomposition or an alternating minimization based approach. Our numerical results focus on the one-dimensional Korteweg-de Vries equation where accounting for nonlinear correlations in the data was found to lower the representation error by up to two orders of magnitude compared to linear dimension reduction techniques.

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.

Jin Yi Yong, Rudy Geelen, Johann Guilleminot

This paper presents a probabilistic approach to represent and quantify model-form uncertainties in the reduced-order modeling of complex systems using operator inference techniques. Such uncertainties can arise in the selection of an appropriate state-space representation, in the projection step that underlies many reduced-order modeling methods, or as a byproduct of considerations made during training, to name a few. Following previous works in the literature, the proposed method captures these uncertainties by expanding the approximation space through the randomization of the projection matrix. This is achieved by combining Riemannian projection and retraction operators - acting on a subset of the Stiefel manifold - with an information-theoretic formulation. The efficacy of the approach is assessed on canonical problems in fluid mechanics by identifying and quantifying the impact of model-form uncertainties on the inferred operators.

Harsh Sharma, Hongliang Mu, Patrick Buchfink et al.

This work presents two novel approaches for the symplectic model reduction of high-dimensional Hamiltonian systems using data-driven quadratic manifolds. Classical symplectic model reduction approaches employ linear symplectic subspaces for representing the high-dimensional system states in a reduced-dimensional coordinate system. While these approximations respect the symplectic nature of Hamiltonian systems, linear basis approximations can suffer from slowly decaying Kolmogorov $N$-width, especially in wave-type problems, which then requires a large basis size. We propose two different model reduction methods based on recently developed quadratic manifolds, each presenting its own advantages and limitations. The addition of quadratic terms to the state approximation, which sits at the heart of the proposed methodologies, enables us to better represent intrinsic low-dimensionality in the problem at hand. Both approaches are effective for issuing predictions in settings well outside the range of their training data while providing more accurate solutions than the linear symplectic reduced-order models.