  This work extends the paradigm of evolutional deep neural networks (EDNNs) to
solving parametric time-dependent partial differential equations (PDEs) on
domains with geometric structure. By introducing positional embeddings based on
eigenfunctions of the Laplace-Beltrami operator, geometric properties are
encoded intrinsically and Dirichlet, Neumann and periodic boundary conditions
of the PDE solution are enforced directly through the neural network
architecture. The proposed embeddings lead to improved error convergence for
static PDEs and extend EDNNs towards computational domains of realistic
complexity. Several steps are taken to improve performance of EDNNs: Solving
the EDNN update equation with a Krylov solver avoids the explicit assembly of
Jacobians and enables scaling to larger neural networks. Computational
efficiency is further improved by an ad-hoc active sampling scheme that uses
the PDE dynamics to effectively sample collocation points. A modified linearly
implicit Rosenbrock method is proposed to alleviate the time step requirements
of stiff PDEs. Lastly, a completely training-free approach, which automatically
enforces initial conditions and only requires time integration, is compared
against EDNNs that are trained on the initial conditions. We report results for
the Korteweg-de Vries equation, a nonlinear heat equation and (nonlinear)
advection-diffusion problems on domains with and without holes and various
boundary conditions, to demonstrate the effectiveness of the method. The
numerical results highlight EDNNs as a promising surrogate model for
parametrized PDEs with slow decaying Kolmogorov n-width.
