Report on Current Developments in Reduced-Order Modeling (ROM) for Physical Systems
General Direction of the Field
The field of reduced-order modeling (ROM) for physical systems is currently witnessing a significant shift towards more robust, non-intrusive, and structure-preserving methods. This shift is driven by the need to address the limitations of traditional ROM techniques, particularly in the context of advection-dominated flows and nonlinear systems. The recent advancements are characterized by a focus on local and data-driven approaches that enhance the generalization capabilities of ROMs, improve computational efficiency, and ensure the preservation of key physical properties such as energy conservation and stability.
One of the primary trends is the development of space-local ROMs, which aim to overcome the challenges posed by the slow decay of singular values in advection-dominated flows. These methods partition the domain into subdomains and apply local basis functions, thereby achieving a sparse representation that generalizes better outside the training regime. This approach not only reduces computational costs but also enhances the stability and accuracy of the ROMs.
Another notable trend is the emergence of non-intrusive ROM techniques that leverage spectral submanifolds (SSMs) and machine learning methods. These approaches are particularly valuable for systems where the explicit knowledge of nonlinear coefficients is unavailable, such as in generic finite-element solvers. By treating the system nonlinearity as a black box, these methods enable the construction of low-dimensional ROMs that are both rigorous and computationally efficient.
Structure-preserving learning methods are also gaining traction, especially for multi-symplectic partial differential equations (PDEs). These methods ensure that the reduced-order models inherit the energy conservation and multi-symplectic properties of the full-order models, even when the fully discrete operators are inaccessible. This is achieved through data-driven approaches that infer the dynamics of the PDEs without relying on intrusive modifications to the solver.
Lastly, there is a growing emphasis on cost-informed dimensionality reduction techniques for structural digital twin technologies. These methods aim to minimize misclassification costs by optimizing the dimensionality of the input feature space, thereby mitigating the curse of dimensionality and improving the predictive performance of classification models.
Noteworthy Papers
Space-Local Reduced-Order Models for Advection-Dominated Flows: Introduces a novel space-local POD approach that significantly improves generalization and stability in advection-dominated flows.
Non-intrusive Model Reduction via Spectral Submanifolds: Proposes a non-intrusive algorithm for computing SSMs, enabling ROM construction for generic finite-element models with up to cubic-order nonlinearities.
Energy-Preserving Machine Learning for Multi-Symplectic PDEs: Develops a non-intrusive, data-driven method that preserves energy and multi-symplectic properties, tested on complex PDEs.
Cost-Informed Dimensionality Reduction for Structural Digital Twins: Formulates a decision-theoretic approach to dimensionality reduction, focusing on minimizing misclassification costs in asset management.
