Physics-Informed Neural Networks with Skip Connections for Modeling and Control of Gas-Lifted Oil Wells

TL;DR

This work tackles the challenge of modeling and controlling nonlinear gas‑lifted oil wells using Physics‑Informed Neural Nets for Control (PINC). It introduces skip connections and surrogate ODE term adjustments, plus a hierarchical network to predict algebraic variables, yielding dramatically improved gradient flow and a 67% reduction in validation prediction error on the oil‑well task. The enhanced PINC enables reliable long‑range predictions and effective nonlinear MPC, maintaining performance under measurement noise. Together, these innovations expand PINC’s applicability to complex industrial systems and open avenues for adaptive loss weighting and PDE extensions in future work.

Abstract

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.

Figures (15)

• Figure 1: Schematics of a gas-lifted oil well well_model_paper:2012. Liquid and gas from the reservoir enter the well at the bottom of the tubing, referred to as the bottom hole. From there, the gas and liquid flow up through the tubing until they exit through the production choke. The tubing is encased by a larger tube called the annulus. At the top of the annulus, gas is injected through the gas-lift choke. This gas flows down the annulus and enters the production tubing through a check valve (directional/one-way valve) near the annulus's bottom. This gas helps "lift" the reservoir flow up the tubing, increasing the production of the well. This kind of artificial lift is commonly used when the pressure in the reservoir is insufficient to sustain the flow from the reservoir to the top side.
• Figure 2: PINN structure for solving an IVP. The neural network maps the time $t$ to the state at this time: $\mathbf{y}(t)$.
• Figure 3: Physics-informed Neural Networks for Control (PINC). Left: PINC network with time $t$, initial condition $\mathbf{y}(0)$, and control input $\mathbf{u}$ as inputs. This neural network can make predictions of the state at time $t$ given any initial condition and control input. Right: PINC network in self-loop mode, allowing for long-range simulations. The output state prediction at the time $t=T$ is fed back as the initial condition of the PINC to make a new prediction, progressing $T$ seconds every iteration. A different control input may be applied at every iteration. For the first iteration, the initial condition must come from the outside, possibly measured from the actual system.
• Figure 4: Illustration of the improved neural network architecture for PINNs, as proposed by improved_NN_struct. $\mathbf{X}$ is the input to the neural network, which is projected into a higher dimensional space through the two encoder layers, resulting in the higher dimensional vectors $\mathbf{U}$ and $\mathbf{V}$. For each layer of the main network, we first calculate the intermediate activation $\mathbf{Z}$ utilizing the weight matrix and bias vector of this layer, then $\mathbf{Z}$ is weighted by the encoder outputs $\mathbf{U}$ and $\mathbf{V}$ to calculate the final activation $\mathbf{A}$ of the layer.
• Figure 5: Hierarchical architecture for computing algebraic variables from the states. An additional, independently trained neural network is trained to predict the algebraic variables of the oil well, for instance, the bottom-hole pressure. The inputs to this supporting network are the state predictions from the PINC and the control input.