pith. sign in

arxiv: 2606.30238 · v1 · pith:A7YGB2A2new · submitted 2026-06-29 · 💻 cs.MA

Sparse Sensor Placement in Multi-Agent Reinforcement Learning Control of Rayleigh-B\'enard Convection

Jan Stenner , Hans Harder , Sebastian Peitz This is my paper

Pith reviewed 2026-06-30 03:54 UTC · model grok-4.3

classification 💻 cs.MA
keywords sparse sensor placementmulti-agent reinforcement learningRayleigh-Bénard convectiongrouped regularizationpolicy distillationmulti-agent transformerfluid flow control
0
0 comments X

The pith

Sparse apprentice policies distilled from dense multi-agent RL experts achieve comparable control of Rayleigh-Bénard convection at maximal or near-maximal sparsity levels.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that dense expert policies trained with multi-agent reinforcement learning on windowed observations can be distilled into sparse apprentices using grouped regularization, while retaining similar control performance over the fluid system. This matters because real-world control often faces limits on sensor numbers due to cost or placement constraints, and the method identifies which spatial regions and state components matter most for effective actuation. The approach relies on a grouping scheme that keeps pruning decisions consistent across overlapping observation windows, paired with ordered non-convex and iterative reweighted regularization to protect control-relevant signals. Experiments confirm that the resulting sparse policies match expert behavior closely, with especially strong sparsity results under fixed initial conditions and stable training when using multi-agent transformers over standard PPO methods. A further test demonstrates that policies can even be trained from scratch on the learned minimal sensor sets, cutting per-agent observations from 360 down to 12 without breaking the overall learning trend.

Core claim

Dense expert policies are trained with windowed observations in a multi-agent RL setting for Rayleigh-Bénard convection control; these are then distilled into sparse apprentice policies via supervised learning that applies grouped regularization to the encoder input weights. The distillation uses a grouping construction to enforce consistent pruning across overlapping windows together with ordered non-convex grouped regularization and iterative reweighted grouped regularization. Multi-agent transformer policies prove more stable to train than PPO baselines, and the resulting sparse apprentices retain control behavior comparable to the dense experts across both fixed and varying initial-condi

What carries the argument

Grouped regularization applied to encoder input weights, with a construction that enforces consistent pruning decisions across overlapping observation windows, combined with ordered non-convex and iterative reweighted grouped regularization.

If this is right

  • Sparse apprentices retain control behavior comparable to dense experts across the tested settings.
  • Maximal sparsity is reached in all fixed-initial-condition variants and maximal or near-maximal sparsity in varying-initial-condition variants.
  • Training from the learned minimal sensor sets reduces per-agent observation size from 360 to 12 while preserving the overall training trend and lowering data throughput.
  • Multi-agent transformer policies train more stably than proximal policy optimization baselines.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The identified control-relevant spatial regions and state components could guide sensor placement decisions in other spatially distributed fluid or chaotic systems.
  • The large reduction in required observations suggests the method may scale to higher-resolution simulations or physical hardware with strict bandwidth limits.
  • The consistent pruning across sliding windows may extend to other reinforcement learning tasks that use overlapping temporal or spatial observation buffers.

Load-bearing premise

The grouping construction that enforces consistent pruning across overlapping observation windows, combined with ordered non-convex and iterative reweighted grouped regularization, preserves sufficient control-relevant information during distillation from dense to sparse policies.

What would settle it

A direct side-by-side simulation run in which a sparse apprentice policy produces measurably larger deviations in the controlled temperature or velocity fields than its dense expert counterpart under identical initial conditions and actuation limits would falsify the claim of retained control behavior.

Figures

Figures reproduced from arXiv: 2606.30238 by Hans Harder, Jan Stenner, Sebastian Peitz.

Figure 1
Figure 1. Figure 1: Illustration for the Rayleigh-Bénard convection control setup referenced in Section 2.1. [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: (a) Schematic view of different agents and their observation windows of sensor sets for [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Architecture of the Multi-Agent Transformer (MAT) network used as an apprentice model [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Overlapping windows with sensors in one dimension. The window of one agent is high [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: MAT-versus-PPO training curves for both initial-condition settings. Panel (a) shows fixed [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Global Nusselt-number trajectories over time for the fixed-initial-condition scenario, com [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Varying-IC performance comparison. Panel (a) shows cumulative-reward box plots on the [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Fixed-IC window sparsity plots for the GO/GR regularizer and channel-grouping combina [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Varying-IC window sparsity plots for the GO/GR regularizer and channel-grouping com [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: MAT training with full versus minimal sensor sets. Both panels show mean and standard [PITH_FULL_IMAGE:figures/full_fig_p015_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Illustration of the overlap construction in two modes. One mode is an overlap mode, [PITH_FULL_IMAGE:figures/full_fig_p021_11.png] view at source ↗
read the original abstract

This paper studies sparse sensor placement for control of Rayleigh-B\'enard convection with multi-agent reinforcement learning. We train dense expert policies with windowed observations and distill sparse apprentice policies by supervised learning with grouped regularization on encoder input weights. The framework combines ordered non-convex grouped regularization and iterative reweighted grouped regularization, and uses a grouping construction that enforces consistent pruning across overlapping observation windows. Experiments with fixed and varying initial conditions show that Multi-Agent Transformer policies train more stably than proximal policy optimization baselines, while sparse apprentices retain control behavior comparable to dense experts. Sparsity results are strong for the proposed grouped methods across settings, including maximal sparsity in all fixed-initial-condition setting variants and maximal or near-maximal sparsity in varying-initial-condition setting variants. As an additional proof of concept, training from learned minimal sensor sets reduces per-agent observation size from 360 to 12 and preserves the overall training trend in simulation while reducing data throughput. The results provide both an interpretable basis for identifying control-relevant spatial regions and state components, and a practical pathway toward sensor-efficient control under realistic hardware constraints.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper proposes a sparse sensor placement framework for multi-agent RL control of Rayleigh-Bénard convection. Dense expert policies are trained on windowed observations and distilled into sparse apprentice policies via supervised learning that applies ordered non-convex grouped regularization combined with iterative reweighted grouped regularization; a grouping construction enforces consistent pruning across overlapping windows. Experiments in fixed- and varying-initial-condition regimes show that Multi-Agent Transformer policies train more stably than PPO baselines, that sparse apprentices retain comparable closed-loop control behavior, and that the grouped methods achieve maximal or near-maximal sparsity; a proof-of-concept further reduces per-agent observation size from 360 to 12 while preserving training trends.

Significance. If the quantitative results hold, the work supplies both an interpretable method for identifying control-relevant spatial regions and state components and a practical route to sensor-efficient MARL policies under hardware constraints. The grouped-regularization construction that maintains consistency across time windows, together with explicit ablations of the non-convex and reweighted penalties, constitutes a clear technical contribution; the stability advantage of MAT over PPO in this fluid-control setting is also of interest to the community.

minor comments (2)
  1. [Abstract] Abstract: the claims of 'comparable performance' and 'maximal sparsity' are stated without any numerical values, error metrics, or baseline comparisons; including at least one quantitative summary statistic per setting would make the abstract self-contained.
  2. The manuscript would benefit from an explicit statement of the precise performance metric (e.g., Nusselt-number deviation or integrated control cost) used to declare 'comparable behavior' in the results tables.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. The feedback affirms the technical contributions of the grouped regularization approach and the stability observations for Multi-Agent Transformer policies. Since no specific major comments or requested changes were provided in the report, we have no point-by-point revisions to address at this stage.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The manuscript describes an empirical pipeline: dense expert policies are trained on windowed observations, then sparse apprentices are obtained via supervised distillation with grouped regularization on encoder weights. Sparsity and closed-loop performance are reported as experimental outcomes across fixed- and varying-initial-condition regimes, supported by ablations of the regularization scheme. No load-bearing step reduces by the paper's own equations or self-citation to a fitted quantity or definitional identity; the grouping construction and regularization are methodological choices whose effectiveness is evaluated externally against control metrics. The derivation chain is therefore self-contained against the reported benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no explicit free parameters, axioms, or invented entities; all statements are high-level empirical claims.

pith-pipeline@v0.9.1-grok · 5722 in / 986 out tokens · 23037 ms · 2026-06-30T03:54:52.599682+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

36 extracted references · 23 canonical work pages · 6 internal anchors

  1. [1]

    A review on sparse solutions in optimal control of partial differential equations

    Eduardo Casas. A review on sparse solutions in optimal control of partial differential equations. SeMA Journal, 74, 04 2017. doi: 10.1007/s40324-017-0121-5

    work page doi:10.1007/s40324-017-0121-5 2017

  2. [2]

    B. W. Brunton, S. L. Brunton, J. L. Proctor, and J. N. Kutz. Sparse sensor placement opti- mization for classification.SIAM Journal on Applied Mathematics, 76(5):2099–2122, 2016. doi: 10.1137/15M1036713. URLhttps://doi.org/10.1137/15M1036713

    work page doi:10.1137/15m1036713 2099

  3. [3]

    Brunton, J

    Krithika Manohar, Bingni W. Brunton, J. Nathan Kutz, and Steven L. Brunton. Data-Driven Sparse Sensor Placement for Reconstruction: Demonstrating the Benefits of Exploiting Known Patterns.IEEE Control Systems, 38(3):63–86, January 2018. doi: 10.1109/MCS.2018.2810460

    work page doi:10.1109/mcs.2018.2810460 2018

  4. [4]

    Brunton, and Bernd R

    Thomas Duriez, Steven L. Brunton, and Bernd R. Noack.Taming Nonlinear Dynamics with MLC, pages 93–120. Springer International Publishing, Cham, 2017. ISBN 978-3-319-40624-4. doi: 10.1007/978-3-319-40624-4_5. URLhttps://doi.org/10.1007/978-3-319-40624-4_5

    work page doi:10.1007/978-3-319-40624-4_5 2017

  5. [5]

    M. A. Bucci, O. Semeraro, A. Allauzen, G. Wisniewski, L. Cordier, and L. Mathelin. Control of chaotic systems by deep reinforcement learning.Proceedings of the Royal Society A: Mathe- matical, Physical and Engineering Sciences, 475(2231):20190351, 11 2019. ISSN 1364-5021. doi: 10.1098/rspa.2019.0351. URLhttps://doi.org/10.1098/rspa.2019.0351

    work page doi:10.1098/rspa.2019.0351 2019

  6. [6]

    Artificial neural networks trained through deep reinforcement learning discover control strategies for active flow control.Journal of Fluid Mechanics, 865:281–302, February 2019

    Jean Rabault, Miroslav Kuchta, Atle Jensen, Ulysse Réglade, and Nicolas Cerardi. Artificial neural networks trained through deep reinforcement learning discover control strategies for active flow control.Journal of Fluid Mechanics, 865:281–302, February 2019. ISSN 1469-7645. doi: 10.1017/jfm.2019.62. URLhttp://dx.doi.org/10.1017/jfm.2019.62

    work page doi:10.1017/jfm.2019.62 2019

  7. [7]

    Reinforcement learning of chaoticsystemscontrolinpartiallyobservableenvironments.Flow, Turbulence and Combustion, 115:1357–1378, 01 2025

    Max Weissenbacher, Anastasia Borovykh, and Georgios Rigas. Reinforcement learning of chaoticsystemscontrolinpartiallyobservableenvironments.Flow, Turbulence and Combustion, 115:1357–1378, 01 2025. doi: 10.1007/s10494-024-00632-5

    work page doi:10.1007/s10494-024-00632-5 2025

  8. [8]

    Brunton, and Kunihiko Taira

    Sebastian Peitz, Jan Stenner, Vikas Chidananda, Oliver Wallscheid, Steven L. Brunton, and Kunihiko Taira. Distributed control of partial differential equations using convolutional rein- forcement learning, 2024. ISSN 0167-2789. URLhttps://www.sciencedirect.com/science/ article/pii/S0167278924000472

    2024

  9. [9]

    Vignon, J

    Colin Vignon, Jean Rabault, Joel Vasanth, Francisco Alcántara-Ávila, Mikael Mortensen, and Ricardo Vinuesa. Effective control of two-dimensional rayleigh–bénard convection: Invariant multi-agent reinforcement learning is all you need.Physics of Fluids, 35(6):065146, 06 2023. ISSN 1070-6631. doi: 10.1063/5.0153181. URLhttps://doi.org/10.1063/5.0153181

    work page doi:10.1063/5.0153181 2023

  10. [10]

    Advanced deep-reinforcement-learning methods for flow control: group-invariant and positional-encoding networks improve learning speed and quality, 2024

    JoongooJeon, JeanRabault, JoelVasanth, FranciscoAlcántara-Ávila, ShilajBaral, andRicardo Vinuesa. Advanced deep-reinforcement-learning methods for flow control: group-invariant and positional-encoding networks improve learning speed and quality, 2024. URLhttps://arxiv. org/abs/2407.17822

    work page arXiv 2024

  11. [11]

    Controlofrayleigh- bénard convection: Effectiveness of reinforcement learning in the turbulent regime, 2025

    ThorbenMarkmann, MichielStraat, SebastianPeitz, andBarbaraHammer. Controlofrayleigh- bénard convection: Effectiveness of reinforcement learning in the turbulent regime, 2025. URL https://arxiv.org/abs/2504.12000

    work page arXiv 2025

  12. [12]

    Song Han, Huizi Mao, and William J. Dally. Deep compression: Compressing deep neural networks with pruning, trained quantization and huffman coding, 2016. URLhttps://arxiv. org/abs/1510.00149. 17

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  13. [13]

    In value-based deep reinforcement learning, a pruned network is a good network, 21–27 Jul 2024

    Johan Samir Obando Ceron, Aaron Courville, and Pablo Samuel Castro. In value-based deep reinforcement learning, a pruned network is a good network, 21–27 Jul 2024. URLhttps: //proceedings.mlr.press/v235/obando-ceron24a.html

    2024

  14. [14]

    Christos Louizos, Max Welling, and Diederik P. Kingma. Learning sparse neural networks throughL 0 regularization. InInternational Conference on Learning Representations, 2018. URLhttps://openreview.net/forum?id=H1Y8hhg0b

    2018

  15. [15]

    Parametric pde control with deep reinforcement learning and differentiableL 0-sparse polynomial policies, 2024

    Nicolò Botteghi and Urban Fasel. Parametric pde control with deep reinforcement learning and differentiableL 0-sparse polynomial policies, 2024. URLhttps://arxiv.org/abs/2403.15267

    work page arXiv 2024

  16. [16]

    Representational similarity learning with application to brain networks

    Urvashi Oswal, Christopher Cox, Matthew Lambon-Ralph, Timothy Rogers, and Robert Nowak. Representational similarity learning with application to brain networks. In Maria Flo- rina Balcan and Kilian Q. Weinberger, editors,Proceedings of The 33rd International Con- ference on Machine Learning, volume 48 ofProceedings of Machine Learning Research, pages 1041...

    2016

  17. [17]

    Learning to share: Simul- taneous parameter tying and sparsification in deep learning

    Dejiao Zhang, Haozhu Wang, Mario Figueiredo, and Laura Balzano. Learning to share: Simul- taneous parameter tying and sparsification in deep learning. InInternational Conference on Learning Representations, 2018. URLhttps://openreview.net/forum?id=rypT3fb0b

    2018

  18. [18]

    S. S. Hotegni, M. Berkemeier, and S. Peitz. Multi-objective optimization for sparse deep multi- task learning, 2024. URLhttps://arxiv.org/abs/2308.12243

    work page arXiv 2024

  19. [19]

    A reduction of imitation learning and structured prediction to no-regret online learning

    Stephane Ross, Geoffrey Gordon, and Drew Bagnell. A reduction of imitation learning and structured prediction to no-regret online learning. In Geoffrey Gordon, David Dunson, and Miroslav Dudík, editors,Proceedings of the Fourteenth International Conference on Artificial Intelligence and Statistics, volume 15 ofProceedings of Machine Learning Research, pag...

    2011

  20. [20]

    Policy Distillation

    Andrei A. Rusu, Sergio Gomez Colmenarejo, Caglar Gulcehre, Guillaume Desjardins, James Kirkpatrick, Razvan Pascanu, Volodymyr Mnih, Koray Kavukcuoglu, and Raia Hadsell. Policy distillation, 2016. URLhttps://arxiv.org/abs/1511.06295

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  21. [21]

    Progressive reinforce- ment learning with distillation for multi-skilled motion control

    Glen Berseth, Cheng Xie, Paul Cernek, and Michiel Van de Panne. Progressive reinforce- ment learning with distillation for multi-skilled motion control. InInternational Conference on Learning Representations, 2018. URLhttps://openreview.net/forum?id=B13njo1R-

    2018

  22. [22]

    Pops: Policy pruning and shrinking for deep reinforcement learn- ing.IEEE Journal of Selected Topics in Signal Processing, 14(4):789–801, May 2020

    Dor Livne and Kobi Cohen. Pops: Policy pruning and shrinking for deep reinforcement learn- ing.IEEE Journal of Selected Topics in Signal Processing, 14(4):789–801, May 2020. ISSN 1941-0484. doi: 10.1109/jstsp.2020.2967566. URLhttp://dx.doi.org/10.1109/JSTSP.2020. 2967566

    work page doi:10.1109/jstsp.2020.2967566 2020

  23. [23]

    Multi-agent reinforcement learning is a sequence modeling problem

    Muning Wen, Jakub Grudzien Kuba, Runji Lin, Weinan Zhang, Ying Wen, Jun Wang, and Yaodong Yang. Multi-agent reinforcement learning is a sequence modeling problem. In Alice H. Oh, Alekh Agarwal, Danielle Belgrave, and Kyunghyun Cho, editors,Advances in Neural In- formation Processing Systems, 2022. URLhttps://openreview.net/forum?id=1W8UwXAQubL

    2022

  24. [24]

    Candès, Michael B

    Emmanuel J. Candès, Michael B. Wakin, and Stephen P. Boyd. Enhancing sparsity by reweightedℓ1 minimization, October 2008. ISSN 1531-5851. URLhttp://dx.doi.org/10. 1007/s00041-008-9045-x. 18

    2008

  25. [25]

    Plug-and-Play Benchmarking of Reinforcement Learning Algorithms for Large-Scale Flow Control

    Jannis Becktepe, Aleksandra Franz, Nils Thuerey, and Sebastian Peitz. Plug-and-play bench- marking of reinforcement learning algorithms for large-scale flow control, 2026. URLhttps: //arxiv.org/abs/2601.15015

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  26. [26]

    Sutton and Andrew G

    Richard S. Sutton and Andrew G. Barto.Reinforcement Learning: An Introduction. The MIT Press, second edition, 2018. URLhttp://incompleteideas.net/book/the-book-2nd.html

    2018

  27. [27]

    Proximal Policy Optimization Algorithms

    John Schulman, Filip Wolski, Prafulla Dhariwal, Alec Radford, and Oleg Klimov. Proximal policy optimization algorithms.CoRR, abs/1707.06347, 2017. URLhttp://arxiv.org/abs/ 1707.06347

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  28. [28]

    Matthijs T. J. Spaan.Partially Observable Markov Decision Processes, pages 387–414. Springer Berlin Heidelberg, Berlin, Heidelberg, 2012. ISBN 978-3-642-27645-3. doi: 10.1007/ 978-3-642-27645-3_12. URLhttps://doi.org/10.1007/978-3-642-27645-3_12

    work page doi:10.1007/978-3-642-27645-3_12 2012

  29. [29]

    Springer Berlin Heidelberg, Berlin, Heidelberg, 2010

    Lucian Buşoniu, Robert Babuška, and Bart De Schutter.Multi-agent Reinforcement Learning: An Overview, pages 183–221. Springer Berlin Heidelberg, Berlin, Heidelberg, 2010. ISBN 978-3-642-14435-6. doi: 10.1007/978-3-642-14435-6_7. URLhttps://doi.org/10.1007/ 978-3-642-14435-6_7

    work page doi:10.1007/978-3-642-14435-6_7 2010

  30. [30]

    Ordered weighted l1 regularized regression with strongly correlated covariates: Theoretical aspects

    Mario Figueiredo and Robert Nowak. Ordered weighted l1 regularized regression with strongly correlated covariates: Theoretical aspects. In Arthur Gretton and Christian C. Robert, editors, Proceedings of the 19th International Conference on Artificial Intelligence and Statistics, vol- ume 51 ofProceedings of Machine Learning Research, pages 930–938, Cadiz,...

  31. [31]

    URLhttps://proceedings.mlr.press/v51/figueiredo16.html

    PMLR. URLhttps://proceedings.mlr.press/v51/figueiredo16.html

  32. [32]

    Xiangrong Zeng and Mário A. T. Figueiredo. The ordered weightedℓ1 norm: Atomic formula- tion, projections, and algorithms, 2015. URLhttps://arxiv.org/abs/1409.4271

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  33. [33]

    Nonconvex sortedℓ1 minimization for sparse ap- proximation.Journal of the Operations Research Society of China, 3(2):207–229, February

    Xiao-Lin Huang, Lei Shi, and Ming Yan. Nonconvex sortedℓ1 minimization for sparse ap- proximation.Journal of the Operations Research Society of China, 3(2):207–229, February

  34. [34]

    doi: 10.1007/s40305-014-0069-4

    ISSN 2194-6698. doi: 10.1007/s40305-014-0069-4. URLhttp://dx.doi.org/10.1007/ s40305-014-0069-4

    work page doi:10.1007/s40305-014-0069-4

  35. [35]

    Attention Is All You Need

    Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N. Gomez, Lukasz Kaiser, and Illia Polosukhin. Attention is all you need, 2023. URLhttps://arxiv. org/abs/1706.03762

    work page internal anchor Pith review Pith/arXiv arXiv 2023

  36. [36]

    Wagner, Christopher Hill, Matin Raayai Ardakani, Johannes Blaschke, Jean-Michel Campin, Valentin Churavy, Navid C

    Simone Silvestri, Gregory L. Wagner, Christopher Hill, Matin Raayai Ardakani, Johannes Blaschke, Jean-Michel Campin, Valentin Churavy, Navid C. Constantinou, Alan Edelman, John Marshall, Ali Ramadhan, Andre Souza, and Raffaele Ferrari. Oceananigans.jl: A julia library that achieves breakthrough resolution, memory and energy efficiency in global ocean simu...

    work page arXiv 2024