EGPT-PINN: Entropy-enhanced Generative Pre-Trained Physics Informed Neural Networks for parameterized nonlinear conservation laws

We propose an entropy-enhanced Generative Pre-Trained Physics-Informed Neural Network with a transform layer (EGPT-PINN) for solving parameterized nonlinear conservation laws. The EGPT-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, entropy enhancement strategies, implementable shock interaction analysis, and a separable training process, the EGPT-PINN efficiently captures complex parameter-dependent shock formations and interactions. Numerical results of EGPT-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 the viscosity solution via very few neurons without leveraging any {\it a priori} knowledge of the equations or its initial condition. Moreover, via a simple augmentation of the loss function by model-data mismatch, we demonstrate the robustness of EGPT-PINN in solving inverse problems more accurately than the vanilla and entropy-enhanced versions of PINN.