Low-Rank Koopman Deformables with Log-Linear Time Integration
For computer graphics and simulation researchers, this work provides a more efficient and generalizable reduced-order model for deformable objects, though it is an incremental improvement over existing DMD-based methods.
The paper introduces a low-rank Koopman operator formulation that accelerates deformable subspace simulation by learning temporal dynamics via DMD, achieving log-linear time scaling and enabling large trajectory skips. It also extends Koopman models to be discretization-agnostic, allowing generalization across shapes and mesh resolutions for tasks like shape optimization.
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.