Low-Rank Koopman Deformables with Log-Linear Time Integration

TL;DR

The paper tackles the challenge of real-time deformable simulation across varying geometries by learning a low-rank Koopman operator via Dynamic Mode Decomposition. It introduces a discretization-agnostic neural extension that conditions basis functions and eigenvalues on geometry, enabling fast long-horizon predictions through matrix exponentiation and log-linear scaling. Key contributions include incorporating momentum in the lifted state, handling external forces through a DMDc-like framework, and achieving generalization across shapes and discretizations with robust real-time performance for interaction, control, and optimization. This approach offers a practical pathway to fast, geometry-general deformable simulation and design, with potential extensions to real-world data and physics-informed training.

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.

Figures (16)

• Figure 1: Behavior under larger time steps. We compare deformation and kinetic energy for implicit Euler and our method (both numerical and neural) at two time step sizes. Because our dynamics are represented by a linear operator, increasing the time step corresponds to exponentiating this operator, yielding nearly identical behavior across step sizes (middle and right, yellow). This computation scales only logarithmically with the step size. In contrast, linear model reduction integrated with implicit Euler, requiring Newton solves at each step, exhibits significantly different and strongly damped behavior as the time step increases (left).
• Figure 2: Generalization across discretizations. Our neural formulation provides a discretization-agnostic extension of the numerical DMD model, allowing a single model to represent the dynamics of meshes with widely varying resolutions (2.5k, 10k, and 500k vertices).
• Figure 3: Ablation on the momentum term. Unlike prior reduced-order models for deformable objects that represent the state using displacement alone, accurate operator approximation requires both displacement and momentum. We ablate the state representation accordingly: using displacement only leads to unstable and physically implausible behavior (top), while incorporating the momentum term yields realistic and stable dynamics (bottom).
• Figure 4: Generalization to force magnitude. Our model is trained on a single force magnitude and evaluated on unseen forces scaled by $0.5\times$ and $2\times$. Despite these changes, the predicted deformations remain visually consistent with the ground truth and incur low reconstruction error, demonstrating that the learned Koopman dynamics generalize to variations in force magnitude beyond the training set.
• Figure 5: Training pipeline. We represent the Koopman basis functions using neural fields, allowing them to be evaluated continuously over the reference domain and shared across different discretizations. Both the basis functions and the eigenvalues are further conditioned on a geometry code to support generalization across shapes. During training, the current state is projected into the Koopman subspace, advanced in time using the learned eigenvalues, and lifted back to predict the next state. The model is trained by minimizing the difference between this predicted next state and the ground-truth simulation.