NNWarp: Neural Network-Based Nonlinear Deformation

NNWarp is a highly re-usable and efficient neural network (NN) based nonlinear deformable simulation framework. Unlike other machine learning applications such as image recognition, where different inputs have a uniform and consistent format (e.g., an array of all the pixels in an image), the input for deformable simulation is quite variable, high-dimensional, and parametrization-unfriendly. Consequently, even though the neural network is known for its rich expressivity of nonlinear functions, directly using an NN to reconstruct the force-displacement relation for general deformable simulation is nearly impossible. NNWarp obviates this difficulty by partially restoring the force-displacement relation via warping the nodal displacement simulated using a simplistic constitutive model-the linear elasticity. In other words, NNWarp yields an incremental displacement fix per mesh node based on a simplified (therefore incorrect) simulation result other than synthesizing the unknown displacement directly. We introduce a compact yet effective feature vector including geodesic, potential and digression to sort training pairs of per-node linear and nonlinear displacement. NNWarp is robust under different model shapes and tessellations. With the assistance of deformation substructuring, one NN training is able to handle a wide range of 3D models of various geometries. Thanks to the linear elasticity and its constant system matrix, the underlying simulator only needs to perform one pre-factorized matrix solve at each time step, which allows NNWarp to simulate large models in real time.

1 Introduction

Nonlinear shape deformation is ubiquitous in our every day life and simulating deformable objects has long been considered as an important yet challenging task for computer graphics and animation. In the past ten years, the finite element method (FEM) based frameworks [1] become more and more popular due to its versatility of encoding various material behaviors. With the prescribed external force $\mathbf {f}_\mathtt {ext}$, the dynamic equilibrium is forwarded by solving a high-dimensional nonlinear system at each time step. Most nonlinear solvers start with an initial guess of the unknown displacement $\mathbf {u}$ and iteratively refine the result until the system converges in order to calculate the deformed model shape. While conceptually straightforward, the requirement of repetitive evaluations of the nonlinear internal force $\mathbf {f}_\mathtt {int}$ or/and its gradient $\partial \mathbf {f}_\mathtt {int}/\partial \mathbf {u}$ makes the simulation rather computational expensive.