We propose a hybrid neural network (NN) and PDE approach for learning generalizable PDE dynamics from motion observations. Many NN approaches learn an end-to-end model that implicitly models both the governing PDE and constitutive models (or material models). Without explicit PDE knowledge, these approaches cannot guarantee physical correctness and have limited generalizability. We argue that the governing PDEs are often well-known and should be explicitly enforced rather than learned. Instead, constitutive models are particularly suitable for learning due to their data-fitting nature. To this end, we introduce a new framework termed "Neural Constitutive Laws" (NCLaw), which utilizes a network architecture that strictly guarantees standard constitutive priors, including rotation equivariance and undeformed state equilibrium. We embed this network inside a differentiable simulation and train the model by minimizing a loss function based on the difference between the simulation and the motion observation. We validate NCLaw on various large-deformation dynamical systems, ranging from solids to fluids. After training on a single motion trajectory, our method generalizes to new geometries, initial/boundary conditions, temporal ranges, and even multi-physics systems. On these extremely out-of-distribution generalization tasks, NCLaw is orders-of-magnitude more accurate than previous NN approaches. Real-world experiments demonstrate our method's ability to learn constitutive laws from videos.

翻译：我们提出了一种结合神经网络（NN）与偏微分方程（PDE）的混合方法，用于从运动观测中学习可泛化的PDE动力学。许多神经网络方法采用端到端模型，隐式同时建模了控制PDE与本构模型（或材料模型）。由于缺乏显式PDE知识，这些方法无法保证物理正确性且泛化能力有限。我们认为，控制PDE通常是已知的，应显式地强制实施而非学习。相反，本构模型因其数据拟合特性而特别适合学习。为此，我们提出了一种名为“神经本构定律”（NCLaw）的新框架，该框架利用一种网络架构，严格保证了标准本构先验知识，包括旋转等变性和未变形状态平衡。我们将此网络嵌入可微仿真中，并通过最小化基于仿真与运动观测差异的损失函数来训练模型。我们在从固体到流体的各类大变形动力系统中验证了NCLaw。在单一运动轨迹上训练后，我们的方法可泛化至新几何结构、初始/边界条件、时间范围，甚至多物理系统。在这些极端分布外泛化任务中，NCLaw的精度比先前的神经网络方法高数个数量级。真实世界实验证明了我们的方法能够从视频中学习本构定律。