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REVIEW 5 major objections 5 minor 72 references

A brief history of sparse sensor readings is sufficient to produce optimal control actions in real time for high-dimensional, parameter-dependent systems, without knowing the parameters or solving the governing equations online.

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2026-08-01 12:48 UTC pith:AODDZVG4

load-bearing objection Useful sensor-only control extension of SHRED-ROM with clean experiments, but the zero-padded 'instantaneous control' claim is ill-posed for the first L steps and the paper needs error bars, timings, and a baseline. the 5 major comments →

arxiv 2607.19302 v1 pith:AODDZVG4 submitted 2026-07-21 cs.LG math.OC

Real-time optimal control with shallow recurrent decoder networks

Matteo Tomasetto , Francesco Braghin , J. Nathan Kutz , Andrea Manzoni This is my paper
classification cs.LG math.OC MSC 68T0749K2093B52
keywords real-time optimal controlshallow recurrent decoder networksreduced-order modelingsensor-based feedbackparametric PDE-constrained optimizationimitation learninglatent sensor forecasterfluid flow control
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper aims to establish that a shallow recurrent decoder network, trained on a handful of expert-optimal trajectories, can replace the expensive online solution of PDE-constrained optimal control problems. The learned policy maps a short window of sparse sensor readings directly to distributed control actions, and generalizes to parameter regimes never seen during training. If correct, this makes closed-loop real-time control feasible for large-scale systems where traditional optimization-based control is too slow. The same architecture also predicts the controlled state and the scenario parameters, and a latent sensor forecaster keeps the loop closed when sensors fail or data arrive late.

Core claim

The central claim is that a map û_k^µ = f_X(f_T(s_{k-L:k}^µ)), composed of an LSTM encoder and a shallow decoder, trained in a supervised way on expert demonstrations, reproduces the optimal control of high-dimensional parametric PDEs from limited sensor data. The decoder outputs coefficients in a POD basis of the control space (and optionally of the state and parameter space), so training and inference stay cheap. Reported test errors on three benchmark problems (density advection-diffusion in a fluidic pinball, unsteady flow past a cylinder, and double-gyre tracking) are a few percent, with the policy suppressing vortex shedding and tracking a reference flow. A second LSTM, the latent sens

What carries the argument

The load-bearing object is the shallow recurrent decoder (SHRED) as used in SHRED-ROM: a long short-term memory network reads a time-window of sparse sensor values and produces a latent vector, and a shallow decoder network lifts that latent to POD coefficients of the requested output field (control, state, or parameters). Time-lagged sensors are justified by Takens embedding, which says the history of measurements encodes the underlying attractor; the paper extends this reasoning to non-autonomous forced systems for the forecaster. The latent sensor forecaster is a second LSTM that maps past sensor values and past SHRED latents to the next sensor value, and can be run autoregressively durin

Load-bearing premise

A fixed-length window of sparse, noise-free sensor readings uniquely determines the expert's optimal control for every parameter value in the range considered.

What would settle it

Construct two distinct parameter configurations (or two initial states) that produce identical sensor readings over the full lag window L yet call for different optimal controls; if the policy outputs a single control for that window, it is wrong for at least one of them. A concrete search: simulate several parameter pairs in the double-gyre or cylinder-wake test cases, record their sensor trajectories, and look for a window collision with divergent target controls.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Real-time closed-loop control of high-dimensional, parametric systems becomes possible with only a short sensor history and no parameter knowledge at inference time.
  • The same trained policy can output the corresponding optimal state trajectory and the scenario parameter alongside the control, which is useful for monitoring and estimation.
  • Sensor failures or communication delays can be bridged by the latent forecaster, which keeps control nearly optimal even with 50% per-step failure probability.
  • Because control inference is a single network forward pass, it satisfies latency constraints that rule out repeated high-fidelity optimization.
  • The method is data-hungry only in the offline phase: it needs a handful of expert optimal trajectories, then generalizes across the parameter range.

Where Pith is reading between the lines

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

  • An implication the authors leave implicit is that the policy's ceiling is set by the expert solver: training on suboptimal or non-unique demonstrations would compress an average behavior, so the quality of the expert data is part of the method's assumptions.
  • The Takens-style justification covers a single attractor; the parametric case requires that sensor windows also disambiguate the parameter and the required control. A testable extension is to construct two parameter configurations whose L-sensor histories coincide on the training set but demand different controls, and check whether the policy fails at that crossing.
  • The forecaster opens a route to event-triggered control: instead of querying sensors every step, the controller could rely on predictions and only wake the sensors when the forecast uncertainty grows, saving communication energy in wireless control networks.
  • Because the decoder is shallow and the latent dimension is small, the learned policy could be distilled into a lookup or a linear time-varying law per operating region, which might further reduce hardware requirements.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

5 major / 5 minor

Summary. The paper proposes a data-driven, sensor-only feedback control strategy for high-dimensional parametric PDE systems. A SHRED-ROM model maps a sliding window of sparse state sensor readings to distributed control actions, trained by imitation learning on expert optimal-control trajectories. A latent sensor forecaster is introduced to maintain control when sensor measurements are missing. The method is tested on three problems: fluidic-pinball density control, unsteady flow control around a cylinder, and double-gyre flow tracking. The reported results show small mean relative errors on held-out parameter scenarios and effective closed-loop loss reduction compared with uncontrolled dynamics.

Significance. If the stated claims hold, the work is a valuable contribution: it offers a parameter-agnostic, computationally cheap policy for distributed optimal control, addressing a long-standing latency bottleneck. The paper includes public code and data, uses challenging high-dimensional benchmarks, and explicitly treats sensor failure via a latent forecaster. The main strengths are the breadth of numerical experiments and the reproducibility infrastructure. However, the central claim that 'limited state sensor readings are all you need' is weakened by the double-gyre experiment's use of reference-flow sensors, and several load-bearing aspects -- the initial-transient behavior, closed-loop distribution shift, autoregressive forecaster accuracy, and real-time feasibility -- are not quantitatively supported.

major comments (5)
  1. [Section III, policy definition and pre-padding] The pre-padding convention makes the sensor-to-control map ill-posed at early times. Section III states that s_k=0 for k<=L, so for the first L time steps the input window to f_X(f_T(s_{k-L:k})) is identical for every parameter mu. Yet the expert controls at those times are generally mu-dependent: in the double-gyre case the explicit formula contains d(vbar)/dt at t=0, which depends on eps and omega, and in the fluidic-pinball case the adjoint-based control depends on the future target trajectory. A deterministic network can only fit a compromise for these inputs. The paper reports only aggregate mean relative errors and never checks per-step errors for the first L controls or the variance of the expert initial controls. Please either remove/warm-start these steps, show that first-L controls are effectively parameter-independent, or quantify the resulting error. As written, the claim of
  2. [Section IV, closed-loop deployment (Figures 4, 6, 11)] The policy is trained on sensor windows from expert optimal trajectories, but during closed-loop deployment the sensor windows come from the controlled trajectory under the learned policy. This is classic covariate shift in behavioral cloning and can lead to compounding errors. The paper does not discuss this distribution shift or report how much the deployed sensor/state trajectories deviate from the expert trajectories. The loss histories in Figures 4 and 11 show only the final closed-loop cost, not the input-distribution mismatch. Please add an explicit comparison of open-loop versus closed-loop sensor trajectories, or at least a discussion of why compounding is not a concern here.
  3. [Section III, latent feedback loop; Section IV forecaster errors] The sensor forecaster phi_z is trained on ground-truth sensor histories and on latent variables z_k computed from those ground-truth histories. In the failure mode, however, phi_z is run autoregressively, feeding its own predictions back into both f_T and phi_z. The reported test errors (0.19% and 0.69%) are one-step-ahead errors on true inputs; they do not quantify error accumulation over multi-step outages. Figures 7 and 10 show aggregate control performance but not sensor-forecast error as a function of outage length. Since the robustness claim of the paper relies on this forecaster, please report multi-step/autoregressive errors or compare against a no-forecaster baseline for the same outage durations.
  4. [Section IV, Double gyre flow tracking] In this test case, half of the sensors measure the horizontal velocity of the reference flow, i.e. the target trajectory, rather than the state of the controlled system. The text acknowledges that these sensors are 'needed to specify which instance of double gyre flow to track.' This means the controller is not purely state-based in this benchmark, and the abstract's claim of relying 'solely on limited state sensor readings' is overstated. Please clarify that the reference signal is assumed measurable/available, or provide a variant using only state sensors.
  5. [Sections I--IV, real-time claim] The title and introduction emphasize 'real-time' control, but the paper reports no wall-clock inference times anywhere in Section IV. The relevant comparison is the per-step inference latency against the control interval (1 s in the fluidic pinball case, 0.1 s in the flow-control cases). Without timing measurements, the central real-time claim is unsupported. Please report inference time per control step, and ideally training time and hardware used.
minor comments (5)
  1. [Section II vs Section III, pre-padding notation] The pre-padding convention is inconsistent: Section II says s(t_{k-L})=0 for k<=L, while Section III says s_k=0 for k<=L. Please clarify which entries of the window are zero-padded.
  2. [Section III, latent forecaster terminology] The text refers to z_k as 'latent control variables,' but z_k = f_T(s_{k-L:k}) are sensor embeddings, not control variables. This terminology is confusing and should be changed to 'latent sensor variables' or 'latent state variables.'
  3. [Section IV, unsteady flow control, Figure 5 caption] The caption says 'two different test scenarios' but the figure has three columns and the text preceding it says 'three test scenarios.' The caption should be corrected.
  4. [Conclusions, sensor-placement independence] The conclusion that the method is 'independent on sensor placement' is stronger than the evidence in Section IV, which only tests random placements. Please soften the claim or add a sensor-placement sensitivity study.
  5. [Section III, latent forecaster equation] Minor typo: 'even through different values are also possible' should read 'even though.'

Circularity Check

0 steps flagged

No circular derivation: the sensor-to-control map is fit to expert demonstrations and tested on held-out parameters; self-citations provide scaffolding, not the result itself.

full rationale

The paper's central claim is that a policy û_k^μ = f_X(f_T(s_{k-L:k}^μ)), trained by supervised regression on expert optimal trajectories, can produce real-time controls for unseen parameter scenarios from sparse sensor windows. This is a standard supervised learning setup: the loss L(θ_T, θ_X) directly compares predicted controls to expert controls on training scenarios, and the numerical sections evaluate on held-out test parameters (e.g., 'mean relative error in test scenarios equal to 2.68%' for the unsteady flow case). The sensor-to-control map is not defined in terms of the predicted quantity, and no fitted parameter is renamed as a prediction. The invocation of Takens embedding is a theoretical justification for why sensor histories may encode state information, not a hidden input to the derivation. The SHRED-ROM architecture and hyperparameters are taken from the authors' prior work [62], but the generalization claims are independently tested on new PDE control problems and compared to full-order adjoint/expert solutions, so the self-citations are supporting infrastructure rather than a load-bearing uniqueness argument. The latent sensor forecaster φ_z is a separately trained supervised model; using it in the feedback loop applies an independently fitted map rather than assuming the target. The acknowledged need for expert demonstration data ('the need for controlled snapshots may represent a limiting factor') is a data-requirement limitation, not circularity. The zero-padding issue raised for k≤L is a possible well-posedness or transient-accuracy concern, but it is not a circular reduction: it questions whether the regression target is determinate from the input, not whether the prediction is equivalent to the input by construction. Overall, the derivation is self-contained and externally falsifiable, with only a modest self-referential burden from building directly on the authors' earlier SHRED and latent-feedback work.

Axiom & Free-Parameter Ledger

5 free parameters · 6 axioms · 0 invented entities

Central claim rests on supervised learning from expert demonstrations plus a handful of manually chosen hyperparameters and POD ranks; the main theoretical crutch is Takens embedding, which is invoked beyond its usual scope.

free parameters (5)
  • SHRED-ROM hyperparameters (LSTM layers/width, SDN layers/width, dropout, optimizer, epochs) = LSTM 2x64, SDN 2x350/400, dropout 0.1, Adam, 200 epochs, batch 64
    Taken as defaults from [62]; no ablation or sensitivity analysis, yet central results depend on them.
  • POD reduced dimensions per test case = 300 (pinball); 200 velocity + 20 control (unsteady flow); 15 per component, 60 total (double gyre)
    Chosen by hand to balance compression and accuracy; no convergence study reported.
  • Sensor lag L = 10 (pinball), 20 (unsteady flow), 25 (double gyre)
    Manually chosen; the paper notes L=25 covers two periods at minimum double-gyre frequency, but no sensitivity analysis.
  • Sensor count and placement = 1 mobile (pinball); 3 fixed (unsteady flow); 6 fixed (double gyre)
    Chosen ad hoc; the paper claims robustness to placement but only tests random placement in one case.
  • Cost functional weights beta and beta_g in fluidic pinball = beta=1e-4, beta_g=10
    Define the expert's optimality criterion; not varied.
axioms (6)
  • standard math Takens embedding theorem
    Invoked in Section II to justify that time-lagged sensor readings reconstruct the high-dimensional state; implies the sensor-to-control map is learnable from windows of length L.
  • standard math Forced delay embedding (Stark 1999)
    Invoked in Section III for the latent sensor forecaster, requiring both sensor and control history to reconstruct state in non-autonomous systems.
  • standard math Existence/solvability of the discrete KKT optimality system
    Section III assumes the adjoint-based solutions exist and are unique, providing expert labels.
  • domain assumption Optimality of the expert feedback law for double gyre
    The control formula from Strazzullo et al. is asserted to be optimal after discretization; no derivation in this paper.
  • domain assumption The learned map generalizes across the parameter distribution
    The entire method assumes i.i.d. parameter sampling and that interpolation in parameter space is as easy as learning from sparse sensors; not theoretically justified.
  • domain assumption Sensor readings are noise-free and the model/plant are the same in training and deployment
    Noiseless deterministic simulation; no sensor noise, model mismatch, or disturbances are modeled, despite the discussion of sensor failures.

pith-pipeline@v1.3.0-alltime-deepseek · 17965 in / 11493 out tokens · 110293 ms · 2026-08-01T12:48:02.573735+00:00 · methodology

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read the original abstract

Controlling dynamical systems in real-time across multiple scenarios is critical to enabling adaptive control strategies, ensuring stability and efficiency. However, to tailor control actions in response to varying scenarios, traditional optimal control problems typically require several system simulations, which are often computationally demanding due to the high-dimensionality of the underlying spatio-temporal dynamics. In this work, we exploit SHallow REcurrent Decoder networks-based Reduced Order Modeling (SHRED-ROM) to synthesize a real-time closed-loop controller for high-dimensional and parametric dynamics, relying solely on limited state sensor readings. After training the model on a few optimal examples given by an expert demonstrator, SHRED-ROM mimics the expert behavior with effective distributed control actions in new scenarios, alleviating the curse of dimensionality. Moreover, a sensor forecaster is synthesized and used to close the loop at the latent level, thus efficiently mitigating possible sensor failures or delays. The performance of the proposed optimal control strategy is finally assessed on three challenging high-dimensional cases dealing with either parametric density control or fluid flow control.

Figures

Figures reproduced from arXiv: 2607.19302 by Andrea Manzoni, Francesco Braghin, J. Nathan Kutz, Matteo Tomasetto.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
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Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
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Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
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Figure 9. Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p013_9.png] view at source ↗
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Figure 10. Figure 10: FIG. 10 [PITH_FULL_IMAGE:figures/full_fig_p014_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11 [PITH_FULL_IMAGE:figures/full_fig_p015_11.png] view at source ↗

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Reference graph

Works this paper leans on

72 extracted references · 2 linked inside Pith

  1. [1]

    Aggelogiannaki and H

    E. Aggelogiannaki and H. Sarimveis. Nonlinear model predictive control for distributed parameter systems us- ing data driven artificial neural network models.Comput- ers and Chemical Engineering, 32(6):1225–1237, 2008

    2008

  2. [2]

    A. Alla, G. Bertaglia, and E. Calzola. A pinn ap- proach for the online identification and control of un- known pdes.Journal of Optimization Theory and Appli- cations, 206(1), 2025

    2025

  3. [3]

    Reduced Order Mod- eling and Deep Learning for the real-time approximation of PDEs (DREAM)

    – numerical schemes suffer from the curse of dimen- sionality and become computationally intractable as the state and control dimensions increase, thus limiting their applicability in real-time control scenarios. Moreover, projection-based ROMs require complete knowledge of the underlying physics and are limited by the linearity assumption, lacking accura...

  4. [4]

    A. Alla, B. Haasdonk, and A. Schmidt. Feedback con- trol of parametrized PDEs via model order reduction and dynamic programming principle.Advances in Computa- tional Mathematics, 46(9), 2020

    2020

  5. [5]

    Alla and M

    A. Alla and M. Hinze. HJB-POD feedback control for navier-stokes equations. InProgress in Industrial Math- ematics at ECMI 2014, pages 861–868, Cham, 2016. Springer International Publishing

    2014

  6. [6]

    Alla and S

    A. Alla and S. Volkwein. Asymptotic stability of POD based model predictive control for a semilinear parabolic PDE.Advances in Computational Mathemat- ics, 41(5):1073–1102, 2015

    2015

  7. [7]

    Amsallem, M

    D. Amsallem, M. Zahr, Y. Choi, and C. Farhat. Design optimization using hyper-reduced-order models.Struc- tural and Multidisciplinary Optimization, 51(4):919–940, 2015

    2015

  8. [8]

    E. A. Antonelo, E. Camponogara, L. O. Seman, J. P. Jordanou, E. R. de Souza, and J. F. H¨ ubner. Physics- informed neural nets for control of dynamical systems. Neurocomputing, 579:127419, 2024

    2024

  9. [9]

    Bao and J

    Y. Bao and J. N. Kutz. Data assimilation and discrep- ancy modeling with shallow recurrent decoders, 2025. 16

    2025

  10. [10]

    Bardi and I

    M. Bardi and I. Capuzzo-Dolcetta.Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equa- tions. Birkh¨ auser Boston, MA, 2009

    2009

  11. [11]

    Barry-Straume, A

    J. Barry-Straume, A. Sarshar, A. A. Popov, and A. Sandu. Physics-informed neural networks for PDE- constrained optimization and control. arXiv:2205.03377, 2022

    Pith/arXiv arXiv 2022

  12. [12]

    Benner, S

    P. Benner, S. Gugercin, and K. Willcox. A survey of projection-based model reduction methods for paramet- ric dynamical systems.SIAM review, 57(4):483–531, 2015

    2015

  13. [13]

    Benner, E

    P. Benner, E. Sachs, and S. Volkwein. Model order re- duction for PDE constrained optimization.International Series of Numerical Mathematics, 165:303–326, 2014

    2014

  14. [14]

    Bieker, S

    K. Bieker, S. Peitz, S. L. Brunton, J. N. Kutz, and M. Dellnitz. Deep model predictive flow control with limited sensor data and online learning.Theoretical and Computational Fluid Dynamics, 34(4):577–591, 2020

    2020

  15. [15]

    Camacho and C

    E. Camacho and C. Bordons.Model Predictive Control. Springer London, 2004

    2004

  16. [16]

    Courant and D

    R. Courant and D. Hilbert.Methods of mathematical physics: partial differential equations. John Wiley & Sons, 2008

    2008

  17. [17]

    N. Demo, M. Strazzullo, and G. Rozza. An extended physics informed neural network for preliminary analysis of parametric optimal control problems.Computers and Mathematics with Applications, 143(May):383–396, 2023

    2023

  18. [18]

    Draeger, S

    A. Draeger, S. Engell, and H. Ranke. Model predictive control using neural networks.IEEE Control Systems Magazine, 15(5):61–66, 1995

    1995

  19. [19]

    M. R. Ebers, J. P. Williams, K. M. Steele, and J. Nathan Kutz. Leveraging arbitrary mobile sensor trajectories with shallow recurrent decoder networks for full-state reconstruction.IEEE Access, 12:97428–97439, 2024

    2024

  20. [20]

    N. B. Erichson, L. Mathelin, Z. Yao, S. L. Brunton, M. W. Mahoney, and J. N. Kutz. Shallow neural net- works for fluid flow reconstruction with limited sensors. Proceedings of the Royal Society A: Mathematical, Physi- cal and Engineering Sciences, 476(2238):20200097, 2020

    2020

  21. [21]

    Fresca, L

    S. Fresca, L. Dede’, and A. Manzoni. A Comprehensive Deep Learning-Based Approach to Reduced Order Mod- eling of Nonlinear Time-Dependent Parametrized PDEs. Journal of Scientific Computing, 87(2):1–36, 2021

    2021

  22. [22]

    M. L. Gao, Y. Bao, A. S. Rude, X. Shen, and J. N. Kutz. Uq-shred: uncertainty quantification of shallow recurrent decoder networks for sparse sensing via engression, 2026

    2026

  23. [23]

    M. L. Gao, J. P. Williams, and J. N. Kutz. Sparse iden- tification of nonlinear dynamics and koopman operators with shallow recurrent decoder networks.Proceedings of the National Academy of Sciences, 123(16):e2508144123, 2026

    2026

  24. [24]

    Ghiglieri and S

    J. Ghiglieri and S. Ulbrich. Optimal flow control based on POD and MPC and an application to the cancellation of Tollmien–Schlichting waves.Optimization Methods and Software, 29(5):1042–1074, 2014

    2014

  25. [25]

    M. D. Gunzburger, L. Hou, and T. P. Svobodny. Anal- ysis and finite element approximation of optimal control problems for the stationary navier-stokes equations with distributed and neumann controls.Mathematics of Com- putation, 57(195):123–151, 1991

    1991

  26. [26]

    Hesthaven and S

    J. Hesthaven and S. Ubbiali. Non-intrusive reduced order modeling of nonlinear problems using neural networks. Journal of Computational Physics, 363:55–78, 2018

    2018

  27. [27]

    M. K. Hickner, U. Fasel, A. G. Nair, B. W. Brunton, and S. L. Brunton. Data-Driven Unsteady Aeroelastic Mod- eling for Control.AIAA Journal, 61(2):780–792, 2023

    2023

  28. [28]

    Hochreiter and J

    S. Hochreiter and J. Schmidhuber. Long short-term memory.Neural computation, 9(8):1735–1780, 1997

    1997

  29. [29]

    Hussein, M

    A. Hussein, M. M. Gaber, E. Elyan, and C. Jayne. Im- itation learning: A survey of learning methods.ACM Comput. Surv., 50(2), 2017

    2017

  30. [30]

    Hwang, J

    R. Hwang, J. Y. Lee, J. Y. Shin, and H. J. Hwang. Solving PDE-Constrained Control Problems Using Oper- ator Learning.Proceedings of the 36th AAAI Conference on Artificial Intelligence, AAAI 2022, 36(Phase 1):4504– 4512, 2022

    2022

  31. [31]

    Kaiser, J

    E. Kaiser, J. N. Kutz, and S. L. Brunton. Sparse identi- fication of nonlinear dynamics for model predictive con- trol in the low-data limit.Proceedings of the Royal Soci- ety A: Mathematical, Physical and Engineering Sciences, 474(2219), 2018

    2018

  32. [32]

    G. E. Karniadakis, I. G. Kevrekidis, L. Lu, P. Perdikaris, S. Wang, and L. Yang. Physics-informed machine learn- ing.Nature Reviews Physics, 3(6):422–440, May 2021

    2021

  33. [33]

    D. E. Kirk.Optimal control theory: an introduction. Courier Corporation, 2004

    2004

  34. [34]

    S. Klus, F. N¨ uske, S. Peitz, J. H. Niemann, C. Clementi, and C. Sch¨ utte. Data-driven approximation of the Koop- man generator: Model reduction, system identification, and control.Phys. D Nonlinear Phenom., 406:1–32, 2020

    2020

  35. [35]

    Korda and I

    M. Korda and I. Mezi´ c. Linear predictors for nonlinear dynamical systems: Koopman operator meets model pre- dictive control.Automatica, 93:149–160, 2018

    2018

  36. [36]

    Kunisch and S

    K. Kunisch and S. Volkwein. Control of the Burgers Equation by a Reduced-Order Approach Using Proper Orthogonal Decomposition.Journal of Optimization Theory and Applications, 102(2):345–371, 1999

    1999

  37. [37]

    Kunisch and S

    K. Kunisch and S. Volkwein. Proper orthogonal decom- position for optimality systems.ESAIM: Mod´ elisation math´ ematique et analyse num´ erique, 42(1):1–23, 2008

    2008

  38. [38]

    Kunisch, S

    K. Kunisch, S. Volkwein, and L. Xie. HJB-POD-Based Feedback Design for the Optimal Control of Evolution Problems.SIAM Journal on Applied Dynamical Systems, 3(4):701–722, 2004

    2004

  39. [39]

    J. N. Kutz.Data-driven modeling & scientific computa- tion: methods for complex systems & big data. Oxford University Press, 2013

    2013

  40. [40]

    J. N. Kutz and S. L. Brunton. Parsimony as the ultimate regularizer for physics-informed machine learning.Non- linear Dynamics, 107(3):1801–1817, 2022

    2022

  41. [41]

    J. N. Kutz, M. Reza, F. Faraji, and A. Knoll. Shallow recurrent decoder for reduced order modeling of plasma dynamics, 2024

    2024

  42. [42]

    Leibfritz and S

    F. Leibfritz and S. Volkwein. Reduced order output feed- back control design for PDE systems using proper orthog- onal decomposition and nonlinear semidefinite program- ming.Linear Algebra and its Applications, 415(2):542– 575, 2006. Special Issue on Order Reduction of Large- Scale Systems

    2006

  43. [43]

    Manzoni, A

    A. Manzoni, A. Quarteroni, and S. Salsa.Optimal control of partial differential equations. Springer, 2021

    2021

  44. [44]

    S. K. Mitusch, S. W. Funke, and J. S. Dokken. dolfin- adjoint 2018.1: automated adjoints for fenics and fire- drake.Journal of Open Source Software, 4(38):1292, 2019

    2018

  45. [45]

    Mowlavi and S

    S. Mowlavi and S. Nabi. Optimal control of PDEs using physics-informed neural networks.Journal of Computa- 17 tional Physics, 473:111731, 2023

    2023

  46. [46]

    Negri, A

    F. Negri, A. Manzoni, and G. Rozza. Reduced basis ap- proximation of parametrized optimal flow control prob- lems for the stokes equations.Computers & Mathematics with Applications, 69(4):319–336, 2015

    2015

  47. [47]

    Negri, G

    F. Negri, G. Rozza, A. Manzoni, and A. Quarteroni. Re- duced basis method for parametrized elliptic optimal con- trol problems.SIAM Journal on Scientific Computing, 35(5):A2316–A2340, 2013

    2013

  48. [48]

    Peitz and K

    S. Peitz and K. Bieker. On the universal transformation of data-driven models to control systems.Automatica, 149:110840, 2023

    2023

  49. [49]

    Peitz and S

    S. Peitz and S. Klus. Koopman operator-based model reduction for switched-system control of PDEs.Auto- matica, 106:184–191, 2019

    2019

  50. [50]

    Peitz and S

    S. Peitz and S. Klus.Feedback Control of Nonlinear PDEs Using Data-Efficient Reduced Order Models Based on the Koopman Operator, pages 257–282. Springer Interna- tional Publishing, Cham, 2020

    2020

  51. [51]

    Quarteroni.Numerical Models for Differential Prob- lems

    A. Quarteroni.Numerical Models for Differential Prob- lems. Springer Cham, 2017

    2017

  52. [52]

    Quarteroni, A

    A. Quarteroni, A. Manzoni, and F. Negri.Reduced basis methods for partial differential equations: an introduc- tion, volume 92. Springer, 2015

    2015

  53. [53]

    S. Riva, C. Introini, A. Cammi, and J. N. Kutz. Robust state estimation from partial out-core measurements with shallow recurrent decoder for nuclear reactors.Progress in Nuclear Energy, 189:105928, 2025

    2025

  54. [54]

    Sauer, J

    T. Sauer, J. Yorke, and M. Casdagli. Embedol- ogy.Journal of Statistical Physics, 65:579–616, 1991. https://doi.org/10.1007/BF01053745

    doi:10.1007/bf01053745 1991

  55. [55]

    Schmidt and B

    A. Schmidt and B. Haasdonk. Data-driven surrogates of value functions and applications to feedback control for dynamical systems.IF AC-PapersOnLine, 51(2):307–312,

  56. [56]

    J. Stark. Delay embeddings for forced systems. i. deterministic forcing.Journal of Nonlinear Science, 9(3):255–332, 1999

    1999

  57. [57]

    S. C. Shadden, F. Lekien, and J. E. Marsden. Defi- nition and properties of lagrangian coherent structures from finite-time lyapunov exponents in two-dimensional aperiodic flows.Physica D: Nonlinear Phenomena, 212(3):271–304, 2005

    2005

  58. [58]

    Strazzullo, F

    M. Strazzullo, F. Ballarin, R. Mosetti, and G. Rozza. Model Reduction for Parametrized Optimal Control problems in Environmental Marine Science and En- gineering.SIAM Journal on Scientific Computing, 40(4):B1055–B1079, 2018

    2018

  59. [59]

    Strazzullo, F

    M. Strazzullo, F. Ballarin, T. Iliescu, and C. Canuto. New feedback control and adaptive evolve-filter-relax regularization for the navier-stokes equations in the convection-dominated regime, 2025

    2025

  60. [60]

    Tomasetto, F

    M. Tomasetto, F. Braghin, and A. Manzoni. Latent feed- back control of distributed systems in multiple scenar- ios through deep learning-based reduced order models. Computer Methods in Applied Mechanics and Engineer- ing, 442:118030, 2025

    2025

  61. [61]

    F. Takens. Detecting strange attractors in turbu- lence. InDynamical Systems and Turbulence, War- wick 1980, Lecture Notes in Mathematics, volume 898, pages 366–381. Springer Berlin Heidelberg, 1981. https://doi.org/10.1007/BFb0091924

    doi:10.1007/bfb0091924 1980

  62. [62]

    Tomasetto, J

    M. Tomasetto, J. P. Williams, F. Braghin, A. Manzoni, and J. N. Kutz. Reduced order modeling with shallow recurrent decoder networks.Nature Communications, 16(1), 2025

    2025

  63. [63]

    Tomasetto, A

    M. Tomasetto, A. Manzoni, and F. Braghin. Real-time optimal control of high-dimensional parametrized sys- tems by deep learning-based reduced order models.Inter- national Journal for Numerical Methods in Engineering, 127(1):e70237, 2026

    2026

  64. [64]

    S. Wang, M. A. Bhouri, and P. Perdikaris. Fast PDE-constrained optimization via self-supervised oper- ator learning. arXiv:2110.13297, 2021

    Pith/arXiv arXiv 2021

  65. [65]

    Tr¨ oltzsch.Optimal control of partial differential equa- tions: theory, methods, and applications, volume 112

    F. Tr¨ oltzsch.Optimal control of partial differential equa- tions: theory, methods, and applications, volume 112. American Mathematical Soc., 2010

    2010

  66. [66]

    J. P. Williams, O. Zahn, and J. N. Kutz. Sensing with shallow recurrent decoder networks.Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 480(2298):20240054, 2024

    2024

  67. [67]

    J. P. Williams, J. N. Kutz, and K. Manohar. Reservoir computing for system identification and model predictive control.Neural Networks, 202:109031, 2026

    2026

  68. [68]

    Yermakov, D

    A. Yermakov, D. Zoro, L. M. Gao, and J. N. Kutz. T-shred: symbolic regression for regularization and model discovery with transformer shallow recurrent de- coders.Philosophical Transactions of the Royal Soci- ety A: Mathematical, Physical and Engineering Sciences, 384(2317):20240586, 2026

    2026

  69. [69]

    D. Ye, J. Williams, M. Gao, S. Riva, M. Tomasetto, D. Zoro, and J. N. Kutz. Pyshred: A python package for shallow recurrent decoding for sparse sensing, model reduction and scientific discovery, 2025

    2025

  70. [70]

    Zhang, S

    Z. Zhang, S. Liu, A. Alla, J. Darbon, and G. E. Karni- adakis. Pinns in pde constrained optimal control prob- lems: Direct vs indirect methods, 2026

    2026

  71. [71]

    P. Yin, G. Xiao, K. Tang, and C. Yang. Aonn: An adjoint-oriented neural network method for all-at- once solutions of parametric optimal control problems. SIAM Journal on Scientific Computing, 46(1):C127– C153, 2024

    2024

  72. [2018]

    9th Vienna International Conference on Mathe- matical Modelling