Accelerate Neural Subspace-Based Reduced-Order Solver of Deformable Simulation by Lipschitz Optimization

TL;DR

This paper introduces a Lipschitz-aware framework to accelerate neural subspace reduced-order solvers for deformable simulations. By minimizing a second-order Lipschitz energy of the elasticity term and incorporating cubature during training, the authors reshape the objective landscape within a nonlinear subspace to speed up Newton-type convergence, while preserving the coverage of the configuration manifold. The approach supports both supervised and unsupervised subspace learning and demonstrates up to 6.83× speedups across dynamic and quasi-static tests with complex deformations, though at the cost of longer training times and with current focus on the elastic term. The method provides a practical augmentation to existing neural reduced-order solvers, enabling faster, accurate simulations for large-DOF deformable objects and offering a path for further acceleration in dynamic scenarios.

Abstract

Reduced-order simulation is an emerging method for accelerating physical simulations with high DOFs, and recently developed neural-network-based methods with nonlinear subspaces have been proven effective in diverse applications as more concise subspaces can be detected. However, the complexity and landscape of simulation objectives within the subspace have not been optimized, which leaves room for enhancement of the convergence speed. This work focuses on this point by proposing a general method for finding optimized subspace mappings, enabling further acceleration of neural reduced-order simulations while capturing comprehensive representations of the configuration manifolds. We achieve this by optimizing the Lipschitz energy of the elasticity term in the simulation objective, and incorporating the cubature approximation into the training process to manage the high memory and time demands associated with optimizing the newly introduced energy. Our method is versatile and applicable to both supervised and unsupervised settings for optimizing the parameterizations of the configuration manifolds. We demonstrate the effectiveness of our approach through general cases in both quasi-static and dynamics simulations. Our method achieves acceleration factors of up to 6.83 while consistently preserving comparable simulation accuracy in various cases, including large twisting, bending, and rotational deformations with collision handling. This novel approach offers significant potential for accelerating physical simulations, and can be a good add-on to existing neural-network-based solutions in modeling complex deformable objects.

Figures (10)

• Figure 1: Network training settings for effective neural subspace construction. (a) The supervised setting. (b) The unsupervised setting. Conventional methods only consider the loss shown in blue but do not optimize the Lipschitz loss (shown in orange) to control the landscape of simulation objective in the subspace.
• Figure 2: Comparison of the relationship between subspace dimension and training cost without and with the cubature approximation. The data points are collected on the bunny problem with $44k$ DOFs and $53k$ tetrahedrons using a RTX3090. $300$ cubatures are used in this example.
• Figure 3: Performance on a compressed and twisted bar. The simulation results using subspaces are rendered in blue, while full-space simulation results are overlaid in purple as reference shapes. (a) Results of vanilla neural subspace. (b) Results of our method with optimized Lipschitz energy. (c) Comparison of the convergence speed in the simulation when converging to a similar termination energy. (d) Simulation error distribution with full-space simulation as the reference. The solid lines are Kernel Density Estimation (KDE) plots that visualize the estimated probability density curves of the simulation error.
• Figure 4: Our method and the vanilla subspace construction produce complex deformations on the dinosaur mesh by applying interactions. We also report the simulation time cost distribution of the two methods.
• Figure 5: For the elephant example, we project full-space simulation's state samples to the configuration manifold induced by different subspace constructions. The projected shapes are rendered in blue and the source shapes are rendered in purple. We also report the projection error and simulation time distribution.