[论文解读] High-order Differentiable Autoencoder for Nonlinear Model Reduction

Siyuan Shen, Yin Yang+5 more|arXiv (Cornell University)|Feb 19, 2021

Model Reduction and Neural Networks被引用 4

一句话总结

本文提出一种高阶可微分深度自编码器（DAE），用于基于物理模拟中的非线性模型降维，通过复步有限差分（CSFD）与反向自动微分相结合，高效计算二阶和三阶导数，实现实时弹性形变。该方法在GPU上实现了深度、高维DAE（10+层）的实时性能，相较于线性降维，在捕捉复杂非线性动力学方面表现更优。

ABSTRACT

This paper provides a new avenue for exploiting deep neural networks to improve physics-based simulation. Specifically, we integrate the classic Lagrangian mechanics with a deep autoencoder to accelerate elastic simulation of deformable solids. Due to the inertia effect, the dynamic equilibrium cannot be established without evaluating the second-order derivatives of the deep autoencoder network. This is beyond the capability of off-the-shelf automatic differentiation packages and algorithms, which mainly focus on the gradient evaluation. Solving the nonlinear force equilibrium is even more challenging if the standard Newton's method is to be used. This is because we need to compute a third-order derivative of the network to obtain the variational Hessian. We attack those difficulties by exploiting complex-step finite difference, coupled with reverse automatic differentiation. This strategy allows us to enjoy the convenience and accuracy of complex-step finite difference and in the meantime, to deploy complex-value perturbations as collectively as possible to save excessive network passes. With a GPU-based implementation, we are able to wield deep autoencoders (e.g., $10+$ layers) with a relatively high-dimension latent space in real-time. Along this pipeline, we also design a sampling network and a weighting network to enable \emph{weight-varying} Cubature integration in order to incorporate nonlinearity in the model reduction. We believe this work will inspire and benefit future research efforts in nonlinearly reduced physical simulation problems.

研究动机与目标

• 为解决线性模型降维在捕捉可变形固体复杂非线性动力学方面的局限性。
• 通过克服高阶导数计算的计算挑战，实现使用深度自编码器进行精确、实时的物理模拟。
• 开发一种支持牛顿动力学子空间模拟所需二阶和三阶导数的可微分DAE框架。
• 结合神经积分法与区域分解，实现在复杂几何体中的高效、局部化非线性降维。
• 展示高阶可微分DAE在实时动画与模拟中的可行性与性能。

提出的方法

• 通过复步有限差分（CSFD）增强深度自编码器，以高效计算高阶导数，实现对时变切空间的精确投影。
• 将CSFD与反向自动微分结合，仅通过O(n)次网络前向传播即可评估DAE的一阶、二阶和三阶导数。
• 设计具有可学习权重的神经积分网络，以实现在降维空间中的自适应、非均匀采样与精确积分。
• 采用基于形变子结构的区域分解方法，应用局部非线性子空间，提升复杂模型中的效率与真实感。
• 使用广义牛顿-欧拉方程将局部形变与全局刚体运动耦合，实现自由浮动模拟。
• 部署GPU加速流水线，实现在深度、高维DAE（如10+层，潜在空间>50）下的实时模拟。

Figure 1. In this paper, we exploit deep autoencoder (DAE) networks to accelerate physics-based simulation. In order to model nonlinear subspace dynamics accurately, second- and high-order derivatives of the deep decoder net must be efficiently evaluated to match the subspace simulation frame rate.

实验结果

研究问题

• RQ1 能否高效计算深度自编码器的高阶导数，以实现在基于物理模拟中的精确非线性模型降维？
• RQ2 如何将复步有限差分适配至深度神经网络，以支持牛顿动力学所需的二阶与三阶导数评估？
• RQ3 能否将可微分DAE集成至拉格朗日力学中，实现实时弹性形变模拟？
• RQ4 与线性模型降维相比，该方法在捕捉复杂非线性形变行为方面表现如何？
• RQ5 区域分解与神经积分法能否提升大规模、几何复杂模型中非线性降维的可扩展性与准确性？

主要发现

• 所提出的CSFD增强DAE实现了使用深度自编码器（10+层）和高维潜在空间对非线性可变形固体的实时模拟。
• 该方法在复杂模型（如320根绳索的气球）和具有30+自由度的下落兔子上实现了实时性能（FPS），优于单核全尺寸模拟。
• 采用具有可学习权重的神经积分法显著提升了非线性降维空间中的积分精度与采样效率。
• 结合局部非线性子空间（如每区域n_p=10, n_q=6）的区域分解，实现了对仙人掌等复杂模型的可扩展且细节丰富的模拟。
• 该框架通过广义牛顿-欧拉方程支持自由浮动物体的鲁棒模拟，碰撞处理采用隐式惩罚力。
• 该方法表明，通过CSFD实现的高阶可微性能够准确建模非线性子空间动力学中的惯性效应，而标准自动微分工具无法支持此功能。

Figure 2. Increased nonlinearity better fits training poses but also makes the network more wiggling.