Gray-box Modeling for Interacting Active Brownian Particle Control

Active matter systems offer exciting opportunities for applications such as directed self-assembly and microactuation. However, controlling these systems at the population level presents challenges, particularly due to the absence of accurate dynamic models for interacting active particles. Model Predictive Control (MPC) has proven effective in diverse fields like chemical processing and manufacturing, but it typically relies on accurate system dynamics, which are unavailable for interacting active matter systems. In this work, we propose a data-driven approach that combines MPC with gray-box modeling to address this challenge. Gray-box models incorporate partially known dynamics derived from physics, while approximating difficult-to-model terms using a neural network. Specifically, we employ an advection-diffusion equation as the gray-box model, where the advection term is learned to capture the emergent behavior of interacting active hard disks. We demonstrate the effectiveness of this approach by using the gray-box model within the MPC framework to control the particle densities of interacting active particles under an actuated magnetic-like field that orients them. Our simulations show that the gray-box model achieves low prediction error for number density across a sequence of test inputs. Finally, we apply the method to two control tasks: splitting a population of active hard disks into two groups and generating the fastest traveling wave.