  We propose a novel neural network architecture (TSympOCNet) to address
high--dimensional optimal control problems with linear and nonlinear dynamics.
An important application of this method is to solve the path planning problem
of multi-agent vehicles in real time. The new method extends our previous
SympOCNet framework by introducing a time-dependent symplectic network into the
architecture. In addition, we propose a more general latent representation,
which greatly improves model expressivity based on the universal approximation
theorem. We demonstrate the efficacy of TSympOCNet in path planning problems
with obstacle and collision avoidance, including systems with Newtonian
dynamics and non-convex environments, up to dimension 512. Our method shows
significant promise in handling efficiently both complex dynamics and
constraints.
