# Advanced deep-reinforcement-learning methods for flow control: group-invariant and positional-encoding networks improve learning speed and quality

arXiv: https://arxiv.org/abs/2407.17822 · HTML: https://mlsanity.com/paper/2407.17822/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 49 references that are in the map.
- First posted on arXiv 25 Jul 2024.

## Builds on (most cited first, 22 of 22)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [Trust Region Policy Optimization](https://mlsanity.com/paper/1502.05477/) (2015)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Learning Invariant Representations for Reinforcement Learning without Reconstruction](https://mlsanity.com/paper/2006.10742/) (2020)
- [A Survey of Convolutional Neural Networks: Analysis, Applications, and Prospects](https://mlsanity.com/paper/2004.02806/) (2020)
- [Exploration in Deep Reinforcement Learning: A Survey](https://mlsanity.com/paper/2205.00824/) (2022)
- [Recent Advances in Reinforcement Learning in Finance](https://mlsanity.com/paper/2112.04553/) (2021)
- [SO(2)-Equivariant Reinforcement Learning](https://mlsanity.com/paper/2203.04439/) (2022)
- [Rotation-invariant convolutional neural networks for galaxy morphology prediction](https://mlsanity.com/paper/1503.07077/) (2015)
- [A review on Deep Reinforcement Learning for Fluid Mechanics](https://mlsanity.com/paper/1908.04127/) (2019)
- [Roto-Translation Equivariant Convolutional Networks: Application to Histopathology Image Analysis](https://mlsanity.com/paper/2002.08725/) (2020)
- [Effective control of two-dimensional Rayleigh--B'enard convection: invariant multi-agent reinforcement learning is all you need](https://mlsanity.com/paper/2304.02370/) (2023)
- [SINDy-RL: Interpretable and Efficient Model-Based Reinforcement Learning](https://mlsanity.com/paper/2403.09110/) (2024)
- [Distributed Control of Partial Differential Equations Using Convolutional Reinforcement Learning](https://mlsanity.com/paper/2301.10737/) (2023)
- [Towards Applicable Reinforcement Learning: Improving the Generalization and Sample Efficiency with Policy Ensemble](https://mlsanity.com/paper/2205.09284/) (2022)
- [Deep Reinforcement Learning for Turbulence Modeling in Large Eddy Simulations](https://mlsanity.com/paper/2206.11038/) (2022)
- [Residual-based physics-informed transfer learning: A hybrid method for accelerating long-term CFD simulations via deep learning](https://mlsanity.com/paper/2206.06817/) (2022)
- [Finite volume method network for acceleration of unsteady computational fluid dynamics: non-reacting and reacting flows](https://mlsanity.com/paper/2105.03332/) (2021)
- [Symmetry reduction for deep reinforcement learning active control of chaotic spatiotemporal dynamics](https://mlsanity.com/paper/2104.05437/) (2021)
- [LIFT: Reinforcement Learning in Computer Systems by Learning From Demonstrations](https://mlsanity.com/paper/1808.07903/) (2018)
- [Active flow control for three-dimensional cylinders through deep reinforcement learning](https://mlsanity.com/paper/2309.02462/) (2023)

## Built on by (most cited first, 5 of 5)

- [Real-time optimal control of high-dimensional parametrized systems by deep learning-based reduced order models](https://mlsanity.com/paper/2409.05709/) (2024)
- [Surrogate Modeling of 3D Rayleigh-Benard Convection with Equivariant Autoencoders](https://mlsanity.com/paper/2505.13569/) (2025)
- [Control of Rayleigh-B'enard Convection: Effectiveness of Reinforcement Learning in the Turbulent Regime](https://mlsanity.com/paper/2504.12000/) (2025)
- [Invariant Control Strategies for Active Flow Control using Graph Neural Networks](https://mlsanity.com/paper/2503.22775/) (2025)
- [Latent feedback control of distributed systems in multiple scenarios through deep learning-based reduced order models](https://mlsanity.com/paper/2412.09942/) (2024)

## Abstract (arXiv)

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.