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.

通过在 PINC 网络中引入跳跃连接以及改进 ODE 的某些项，我们在训练过程中实现了更准确的梯度，从而为油井系统的建模过程提供了有效的解决方案。我们提出的改进的 PINC 展现出卓越的性能，将油井应用中的验证预测误差平均降低了 67％，并显著增强了网络层之间的梯度流动，使其幅度相对于原始 PINC 增加了四个数量级。此外，实验展示了通过改进的 PINC 模型，在存在噪声测量的情况下，使用模型预测控制 (MPC) 来调节油井的底孔压力的有效性。