  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.
