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：我们提出一种混合神经网络与偏微分方程方法，用于从运动观测中学习通用的偏微分方程动力学。许多神经网络方法学习端到端模型，隐式地同时拟合控制偏微分方程与本构模型（或材料模型）。由于缺乏显式的偏微分方程知识，这些方法无法保证物理正确性且泛化能力有限。我们认为控制偏微分方程通常是已知的，应被显式强化而非学习；而本构模型因其数据拟合特性，特别适合通过学习获得。为此，我们提出名为“神经本构定律”的新框架，采用能严格保证标准本构先验（包括旋转等变性与未变形状态平衡性）的网络架构。我们将该网络嵌入可微分仿真流程，通过最小化仿真结果与运动观测差异的损失函数训练模型。我们在从固体到流体的多种大变形动力学系统上验证了NCLaw。在单一运动轨迹上训练后，我们的方法可泛化至新几何形状、初始/边界条件、时间尺度甚至多物理系统。在这些极端分布外泛化任务中，NCLaw的精度较此前神经网络方法提升数个数量级。真实世界实验证明了该方法从视频中学习本构定律的能力。