Low-Rank Koopman Deformables with Log-Linear Time Integration

2026-02-07

AI Summary

This work addresses the high computational cost and limited flexibility of traditional deformable body subspace simulation, which relies on sequential time integration and struggles to support optimization tasks under geometric variations. The authors propose a novel approach based on Dynamic Mode Decomposition (DMD) to construct a low-rank Koopman operator that enables direct future-state prediction via matrix operations, yielding an efficient model for deformable dynamics. The key innovation lies in achieving, for the first time, shared Koopman dynamics across varying shapes and mesh resolutions—overcoming the limitation of existing DMD methods that are confined to fixed geometry and discretization. The resulting framework supports trajectory prediction with logarithmic-linear time complexity, allowing large numbers of time steps to be skipped while preserving accuracy, thereby significantly accelerating simulation and enabling efficient control, initial condition estimation, and shape optimization.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Machine Learning: Learning with Manifolds
• Computer Vision: Diffusion Models for Vision

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
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

We present a low-rank Koopman operator formulation for accelerating deformable subspace simulation. Using a Dynamic Mode Decomposition (DMD) parameterization of the Koopman operator, our method learns the temporal evolution of deformable dynamics and predicts future states through efficient matrix evaluations instead of sequential time integration. This yields log-linear scaling in the number of time steps and allows large portions of the trajectory to be skipped while retaining accuracy. The resulting temporal efficiency is especially advantageous for optimization tasks such as control and initial-state estimation, where the objective often depends largely on the final configuration. To broaden the scope of Koopman-based reduced-order models in graphics, we introduce a discretization-agnostic extension that learns shared dynamic behavior across multiple shapes and mesh resolutions. Prior DMD-based approaches have been restricted to a single shape and discretization, which limits their usefulness for tasks involving geometry variation. Our formulation generalizes across both shape and discretization, which enables fast shape optimization that was previously impractical for DMD models. This expanded capability highlights the potential of Koopman operator learning as a practical tool for efficient deformable simulation and design.

Problem

• Koopman operator
• deformable simulation
• shape variation
• reduced-order modeling
• time integration

Innovation

• Low-Rank Koopman Operator
• Dynamic Mode Decomposition
• Log-Linear Time Integration
• Discretization-Agnostic Learning
• Deformable Simulation