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.

翻译：本文提出了一种低秩Koopman算子框架，用于加速可变形体子空间模拟。通过采用动态模态分解对Koopman算子进行参数化，本方法能够学习可变形动力学的时序演化规律，并通过高效的矩阵运算而非顺序时间积分来预测未来状态。该方法实现了时间步数上的对数线性复杂度，允许在保持精度的前提下跳过轨迹的大部分区段。由此获得的时间效率在控制和初始状态估计等优化任务中尤为有利，因为这类任务的目标函数通常高度依赖于最终构型。为拓展基于Koopman的降阶模型在图形学中的应用范围，我们提出了一种与离散化方式无关的扩展方法，能够学习跨多种形状和网格分辨率的共享动态行为。先前基于DMD的方法受限于单一形状和离散化方案，这限制了其在涉及几何变化任务中的实用性。我们的框架实现了形状与离散化方案的双重泛化，使得快速形状优化——这对传统DMD模型而言原本不切实际——成为可能。这一拓展能力凸显了Koopman算子学习作为高效可变形体模拟与设计实用工具的潜力。