用神经网络提升隐式曲面弹性模拟的精度与可微性
Neurally Integrated Finite Elements for Differentiable Elasticity on Evolving Domains
- 用小神经网络优化隐式网格单元的积分点，提升数值稳定性
- 实现形状与材料参数的全程可微，支持物理优化
- 适合需要可微物理模拟的3D重建与编辑任务
我们提出一种针对随时间演变的隐式函数定义域的弹性模拟器，具备高效、鲁棒且对形状和材料参数完全可微的特点。该模拟器源于3D重建应用：如今从图像中恢复几何形状常以隐式函数形式表示，但物理模拟需准确预测此类形状在形变下的响应，这一直具有挑战性。核心技术创新在于训练一个小型神经网络，用于拟合隐式网格单元上的积分点，实现稳健的数值积分。结合混合有限元方法，构建出从隐式表面演化到弹性响应的平滑、全可微的模拟模型。我们在前向模拟隐式曲面、编辑过程中的3D形状直接模拟，以及结合可微渲染的新型基于物理的形状与拓扑优化任务上验证了方法的有效性。
原文摘要 · Abstract (English)
We present an elastic simulator for domains defined as evolving implicit functions, which is efficient, robust, and differentiable with respect to both shape and material. This simulator is motivated by applications in 3D reconstruction: it is increasingly effective to recover geometry from observed images as implicit functions, but physical applications require accurately simulating and optimizing-for the behavior of such shapes under deformation, which has remained challenging. Our key technical innovation is to train a small neural network to fit quadrature points for robust numerical integration on implicit grid cells. When coupled with a Mixed Finite Element formulation, this yields a smooth, fully differentiable simulation model connecting the evolution of the underlying implicit surface to its elastic response. We demonstrate the efficacy of our approach on forward simulation of implicits, direct simulation of 3D shapes during editing, and novel physics-based shape and topology optimizations in conjunction with differentiable rendering.
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