  This work proposes an approach for latent-dynamics learning that exactly
enforces physical conservation laws. The method comprises two steps. First, the
method computes a low-dimensional embedding of the high-dimensional
dynamical-system state using deep convolutional autoencoders. This defines a
low-dimensional nonlinear manifold on which the state is subsequently enforced
to evolve. Second, the method defines a latent-dynamics model that associates
with the solution to a constrained optimization problem. Here, the objective
function is defined as the sum of squares of conservation-law violations over
control volumes within a finite-volume discretization of the problem; nonlinear
equality constraints explicitly enforce conservation over prescribed subdomains
of the problem. Under modest conditions, the resulting dynamics model
guarantees that the time-evolution of the latent state exactly satisfies
conservation laws over the prescribed subdomains.
