Interpolated Adaptive Linear Reduced Order Modeling for Deformation Dynamics

TL;DR

The paper tackles the limitations of fixed linear reduced-order models for deformable objects under large deformations by proposing an adaptive linear ROM whose basis $U(t)$ is dynamically updated from the decoder Jacobian, aligning with the local tangent of a learned nonlinear deformation manifold. It further enhances robustness by incorporating a historical displacement basis $\Phi$ and blending it with the current basis via Grassmann interpolation parameterized by $\lambda$, enabling reliable recovery to the undeformed state. Latent dynamics are evolved in the reduced space by solving a minimized energy $E(u(q))$ with implicit integration, using Newton-type updates in the reduced coordinates, while preserving the efficiency of a linear ROM. Empirically, the approach yields 40–70% lower online error than PCA-based ROM at 1.3–1.5× the cost, across diverse objects and challenging scenarios, suggesting strong potential for real-time deformation dynamics in graphics and robotics where large strains occur.

Abstract

Linear reduced-order modeling (ROM) is widely used for efficient simulation of deformation dynamics, but its accuracy is often limited by the fixed linearization of the reduced mapping. We propose a new adaptive strategy for linear ROM that allows the reduced mapping to vary dynamically in response to the evolving deformation state, significantly improving accuracy over traditional linear approaches. To further handle large deformations, we introduce a historical displacement basis combined with Grassmann interpolation, enabling the system to recover robustly even in challenging scenarios. We evaluate our method through quantitative online-error analysis and qualitative comparisons with principal component analysis (PCA)-based linear ROM simulations, demonstrating substantial accuracy gains while preserving comparable computational costs.

Figures (6)

• Figure 1: Bunny compression under large deformation: PCA (middle) exhibits noticeable artifacts compared to FOM and ours.
• Figure 2: Continuous reduced-order modeling (CROM). The encoder $e_{\theta_e}$ maps a full displacement field $u(t, \mathbf{x})$ to latent coordinates $q$, and the decoder $g_{\theta_g}$ reconstructs continuous displacements from $q$ and spatial position $x_i$.
• Figure 3: Comparison of adaptive linear model reduction under large deformation. Left: Two identical objects before compression. Right: Results after compression—without historical interpolation and with historical interpolation. The interpolation enables perfect recovery to the original state, whereas the standard one fails under large deformation.
• Figure 4: Ablation study scenarios: Each subfigure shows three armadillos—FOM, PCA, and ours. As highlighted by the red circle, ours exhibits greater similarity to FOM.
• Figure 5: Effect of basis size $r$ on relative online error for the two scenarios. Increasing $r$ improves accuracy for both methods and reduces the performance gap, but our approach remains consistently more accurate than PCA.