Low-Rank Koopman Deformables with Log-Linear Time Integration

TL;DR

This paper introduces a low-rank Koopman operator approach for deformable simulation that enables fast, accurate predictions and broad applicability across different shapes and discretizations, significantly improving efficiency in graphics tasks.

Contribution

It develops a discretization-agnostic, low-rank Koopman model that accelerates deformable simulations and enables shape optimization across multiple geometries.

Findings

Achieves log-linear scaling in simulation time steps.

Allows skipping large trajectory portions with maintained accuracy.

Enables fast shape optimization across different geometries.

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.