  Flow control is key to maximize energy efficiency in a wide range of
applications. However, traditional flow-control methods face significant
challenges in addressing non-linear systems and high-dimensional data, limiting
their application in realistic energy systems. This study advances
deep-reinforcement-learning (DRL) methods for flow control, particularly
focusing on integrating group-invariant networks and positional encoding into
DRL architectures. Our methods leverage multi-agent reinforcement learning
(MARL) to exploit policy invariance in space, in combination with
group-invariant networks to ensure local symmetry invariance. Additionally, a
positional encoding inspired by the transformer architecture is incorporated to
provide location information to the agents, mitigating action constraints from
strict invariance. The proposed methods are verified using a case study of
Rayleigh-B\'enard convection, where the goal is to minimize the Nusselt number
Nu. The group-invariant neural networks (GI-NNs) show faster convergence
compared to the base MARL, achieving better average policy performance. The
GI-NNs not only cut DRL training time in half but also notably enhance learning
reproducibility. Positional encoding further enhances these results,
effectively reducing the minimum Nu and stabilizing convergence. Interestingly,
group invariant networks specialize in improving learning speed and positional
encoding specializes in improving learning quality. These results demonstrate
that choosing a suitable feature-representation method according to the purpose
as well as the characteristics of each control problem is essential. We believe
that the results of this study will not only inspire novel DRL methods with
invariant and unique representations, but also provide useful insights for
industrial applications.
