  Neural networks, while powerful, often lack interpretability.
Physics-Informed Neural Networks (PINNs) address this limitation by
incorporating physics laws into the loss function, making them applicable to
solving Ordinary Differential Equations (ODEs) and Partial Differential
Equations (PDEs). The recently introduced PINC framework extends PINNs to
control applications, allowing for open-ended long-range prediction and control
of dynamic systems. In this work, we enhance PINC for modeling highly nonlinear
systems such as gas-lifted oil wells. By introducing skip connections in the
PINC network and refining certain terms in the ODE, we achieve more accurate
gradients during training, resulting in an effective modeling process for the
oil well system. Our proposed improved PINC demonstrates superior performance,
reducing the validation prediction error by an average of 67% in the oil well
application and significantly enhancing gradient flow through the network
layers, increasing its magnitude by four orders of magnitude compared to the
original PINC. Furthermore, experiments showcase the efficacy of Model
Predictive Control (MPC) in regulating the bottom-hole pressure of the oil well
using the improved PINC model, even in the presence of noisy measurements.
