Physics-informed neural nets for control of dynamical systems

Abstract

Physics-informed neural networks (PINNs) incorporate established physical principles into the training of deep neural networks, ensuring that they adhere to the underlying physics of the process while reducing the need for labeled data, since the desired output is not a prerequisite for physics-informed training. For modeling systems described by Ordinary Differential Equations (ODEs), traditional PINNs typically take continuous time as an input variable and produce the solution to the corresponding ODE. However, in their original form, PINNs neither accommodate control inputs nor do they effectively simulate variable long-range intervals without experiencing a significant decline in prediction accuracy. In this context, this work introduces a novel framework known as “Physics-Informed Neural Nets for Control” (PINC). PINC presents an innovative PINN-based architecture tailored to control problems, capable of simulating longer-range time horizons that are not predetermined during training. This increased flexibility sets it apart from traditional PINNs. The variable simulation time is achieved by adding inputs to the PINC network that convey the initial condition and the control signal for a particular time interval. Simulating variable long-range intervals involves running the PINC net across a sequence of shorter intervals. In this autoregressive process, the network predictions are linked in a self-feedback mode, with the initial state (input) of the next interval set to the last predicted state (network output) of the previous interval. We showcase the effectiveness of our proposal in identifying and controlling three nonlinear dynamic systems: the Van der Pol oscillator, the four-tank system, and an electric submersible pump. Crucially, these experiments demonstrate that learning the dynamics of these systems can be achieved without relying on any sample collected from the actual process, and it offers faster inference speed compared to numerical simulations.