Abstract
We propose a Viscosity-enhanced Generative Pre-Trained Physics-Informed
Neural Network with a transform layer (VGPT-PINN) for solving parameterized
nonlinear conservation laws. The VGPT-PINN extends the traditional
physics-informed neural networks and its recently proposed generative
pre-trained strategy for linear model reduction to nonlinear model reduction
and shock-capturing domains. By utilizing an adaptive meta-network, a
simultaneously trained transform layer, viscosity enhancement strategies,
implementable shock interaction analysis, and a separable training process, the
VGPT-PINN efficiently captures complex parameter-dependent shock formations and
interactions. Numerical results of VGPT-PINN applied to the families of
inviscid Burgers' equation and the Euler equations, parameterized by their
initial conditions, demonstrate the robustness and accuracy of the proposed
technique. It accurately solves for the viscosity solution via very few neurons
without leveraging any {\it a priori} knowledge of the equations or its initial
condition.