Interpolated Adaptive Linear Reduced Order Modeling for Deformation Dynamics

2025-09-29

AI Summary

Traditional linear reduced-order models (ROMs) rely on fixed subspaces, rendering them inadequate for capturing nonlinear geometric deformations under large displacements, thereby limiting accuracy. This paper proposes an adaptive linear ROM framework that dynamically updates the reduced-order mapping via an online error-driven adaptation mechanism. By leveraging Grassmann manifold interpolation, the method enables robust interpolation and update of the reduced basis over the subspace manifold spanned by historical displacement snapshots. This approach explicitly relaxes the static subspace assumption inherent in conventional ROMs, achieving a favorable balance between computational efficiency and approximation fidelity. Experimental results demonstrate that, at comparable computational cost, the proposed method significantly reduces simulation error relative to PCA-based linear ROMs, markedly improving both accuracy and numerical stability in large-deformation scenarios.

Technology Category

• Machine Learning: Learning with Manifolds
• Intelligent Robots: State Estimation
• Cognitive Modeling & Cognitive Systems: Adaptive Behavior

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Linear reduced-order modeling (ROM) is widely used for efficient simulation of deformation dynamics, but its accuracy is often limited by the fixed linearization of the reduced mapping. We propose a new adaptive strategy for linear ROM that allows the reduced mapping to vary dynamically in response to the evolving deformation state, significantly improving accuracy over traditional linear approaches. To further handle large deformations, we introduce a historical displacement basis combined with Grassmann interpolation, enabling the system to recover robustly even in challenging scenarios. We evaluate our method through quantitative online-error analysis and qualitative comparisons with principal component analysis (PCA)-based linear ROM simulations, demonstrating substantial accuracy gains while preserving comparable computational costs.

Problem

• Adaptive linear ROM for dynamic deformation mapping
• Handling large deformations with historical displacement basis
• Improving accuracy while maintaining computational efficiency

Innovation

• Adaptive linear ROM with dynamic reduced mapping
• Historical displacement basis with Grassmann interpolation
• Maintains computational cost while improving accuracy