Learning continuum-level closures for control of interacting active particles

Active matter swarms-collectives of self-propelled particles that can self-assemble, ferry microscopic cargo, or endow materials with dynamic properties-remain hard to steer. In crowded systems, tracking or controlling individual agents becomes challenging, so strategies must operate on macroscopic fields like particle density. Yet predicting how density evolves is difficult because of inter-agent interactions. For model-based feedback control methods-such as Model Predictive Control (MPC)-fast, accurate, and differentiable models are crucial. Detailed agent-based simulations are too slow, necessitating coarse-grained continuum models. However, constructing accurate closures-approximations that express the effects of unresolved microscopic states (e.g., agent positions) on continuum dynamics in terms of the modeled continuum fields (e.g., density)-is challenging for active matter swarms. We present a learning-for-control framework that learns continuum closures from agent simulations, demonstrated with active Brownian particles under a controllable external field. Our Universal Differential Equation (UDE) framework represents the continuum as an advection-diffusion equation. A neural operator learns the advection term, providing closure relations for microscopic effects such as self-propulsion, interactions, and external-field responses. This UDE approach, embedding universal function approximators in differential equations, ensures adherence to physical laws (e.g., conservation) while learning complex dynamics directly from data. We embed this learned continuum model into MPC for precise agent-simulation control. We demonstrate this framework's capabilities by dynamically exchanging particle densities between two groups and by simultaneously controlling particle density and mean flux to follow a prescribed sinusoidal profile. These results highlight the framework's potential to control complex active-matter dynamics, foundational for programmable materials.