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REVIEW 3 major objections 5 minor 46 references

This paper claims that a local neural operator trained only on short-horizon data can replace a first-principles timestepper in equation-free control, demonstrated by stabilizing an unstable Liouville–Bratu steady state.

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2026-08-04 14:37 UTC pith:WUTLF43A

load-bearing objection Useful extension of equation-free control to neural-operator surrogates, but the central claim that the controller works on the true system is unsupported: it is only tested on the same approximate surrogate used to design it. the 3 major comments →

arxiv 2509.23975 v2 pith:WUTLF43A submitted 2025-09-28 eess.SY cs.LGcs.NAcs.SYmath.NAmath.OC

Equation-Free Coarse Control of Distributed Parameter Systems via Local Neural Operators

Gianluca Fabiani , Constantinos Siettos , Ioannis G. Kevrekidis This is my paper
classification eess.SY cs.LGcs.NAcs.SYmath.NAmath.OC MSC 93C2068T0765H10
keywords equation-free methodsneural operatorslocal-in-time surrogatescoarse controldistributed parameter systemsKrylov subspace methodsdiscrete-time LQRLiouville–Bratu equation
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 tries to show that you can design a stabilizing feedback controller for a high-dimensional, spatially distributed system without ever writing down its coarse-grained equations or running a microscopic simulator on demand. Instead, a neural operator trained on short-horizon spatiotemporal data becomes the timestepper, and matrix-free Krylov methods extract the steady state, the slow subspace, and a reduced linear model from it. Standard discrete-time control design (dLQR or pole placement) then produces a gain that is applied to the full nonlinear surrogate. The demonstration on the Liouville–Bratu PDE shows the surrogate-based controller driving the system to an unstable steady state with behavior consistent with a controller designed from the known PDE. If this holds generally, it would let controllers be built from data alone for systems where first-principles simulators are unavailable or too expensive.

Core claim

The paper's central claim is that a local neural operator, trained on short-horizon spatiotemporal snapshots of an autonomous distributed parameter system, can replace the microscopic/first-principles timestepper in the equation-free methodology. Around this surrogate, the paper runs Newton–Krylov GMRES to locate an unstable coarse steady state, Arnoldi iteration to obtain its dominant Ritz spectrum and slow subspace, and actuator derivative estimates to build a reduced discrete-time model y_{n+1}=F y_n + D z_n. On this reduced model, discrete-time LQR and pole-placement gains are designed and then lifted back to the full nonlinear neural timestepper. For the Liouville–Bratu PDE at λ=2, both

What carries the argument

The load-bearing object is the learned short-time solution operator S_Δt (a local neural operator, implemented with a random-projection architecture), used as a matrix-free oracle for directional derivatives. Around that operator, Newton–Krylov GMRES locates the unstable steady state and Arnoldi iteration exposes the dominant slow modes; the reduced open-loop model y_{n+1}=F y_n + D z_n then lets standard discrete-time control tools (dLQR, pole placement) produce a gain that is lifted to the full nonlinear timestepper.

Load-bearing premise

The load-bearing premise is that the controlled timestepper is well approximated by u_{n+1} ≈ S_Δt(u_n) + B z_n; if the true actuator dynamics differ, the designed gain may fail on the physical system.

What would settle it

Take the gain matrix K produced by the neural-timestepper reduced model and apply it to a controlled PDE integrator that evaluates actuator influence through the exact Duhamel integral (or a sufficiently fine inner-step integration), rather than the additive B z_n update. If the closed-loop L2 error to the target steady state does not decay, the end-to-end claim is refuted.

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

If this is right

  • Controllers for high-dimensional distributed parameter systems can be designed without explicit equations, using only short-horizon trajectory data.
  • The same learned timestepper serves all three stages of the pipeline—fixed-point finding, stability analysis, and control design—so no on-demand simulation calls are needed.
  • Both dLQR and pole-placement gains computed from the reduced slow model stabilize the full nonlinear surrogate in the Liouville–Bratu example.
  • Closed-loop performance is limited by surrogate accuracy: the neural-timestepper controller reaches an L2 error around 1e-7, versus near machine precision for the exact-model controller.

Where Pith is reading between the lines

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

  • The paper leaves untested how the gain behaves on a true controlled integrator where actuation enters through the semigroup (Duhamel form) rather than the additive B z_n term; a natural next step is to apply the designed gain to such a simulator and measure the remaining stabilization.
  • Because the reduced model's dimension equals the number of retained Ritz modes, the approach is most plausible when the actuators' influence is well captured by the dominant slow subspace; systems with strong coupling between actuators and fast, stable modes may require extending the subspace or an output-feedback variant.
  • The pipeline suggests a general template: any matrix-free coarse-level task that traditionally wraps a fine-scale simulator—bifurcation continuation, sensitivity analysis, observer design—can in principle be rerun around a neural timestepper learned from data, extending the paper's control result to a broader equation-free toolbox.

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

3 major / 5 minor

Summary. The paper proposes a data-driven, equation-free control pipeline for distributed parameter systems. A local RandONet surrogate, trained on short-horizon autonomous simulation data, replaces the fine-scale timestepper. Newton-Krylov GMRES is used to compute unstable steady states, and Arnoldi iterations provide a low-dimensional slow subspace and a reduced linearized model (F,D). Discrete-time LQR and pole-placement controllers are designed on this reduced model and lifted to the full nonlinear timestepper. The method is demonstrated on the Liouville-Bratu PDE with three Gaussian actuators, comparing RandONet-based controllers with finite-difference-based controllers. The core limitation, acknowledged in Section IV-A, is that the RandONet controllers are tested only on the approximate additive-actuation surrogate u_{n+1}=S_Δt(u_n)+Bz_n, not on the actual controlled PDE; the abstract's claim of 'consistent performance between the learned surrogate and the true system' is therefore not supported by the reported experiments.

Significance. If fully validated, the approach would be a useful contribution: it extends local neural-operator surrogates from trajectory generation and bifurcation analysis to coarse feedback control, and it uses only short-horizon autonomous data plus a prescribed actuation model. The matrix-free Newton-Krylov and Arnoldi components are standard and appear correctly applied. The paper is honest in stating that actuator effects are not learned from data and that robustness to plant-model mismatch is future work. However, the end-to-end claim is currently stronger than the evidence: the closed-loop test for the RandONet controller exercises the same additive surrogate used to synthesize the gain, so the stabilization result is partly circular. The significance of the paper would increase substantially if the NO-designed gain were applied to the true controlled PDE (e.g., the FD timestepper with inner integration) or if actuator effects were learned from controlled data.

major comments (3)
  1. [Section IV-A, Eq. (25), Fig. 2(d)] The central validation is missing. The RandONet closed-loop trajectories are generated using u_{n+1}=S_Δt(u_n)+Bz_n, with B prescribed from the PDE. The only controller applied to the actual controlled PDE is the FD-based one. Consequently, Fig. 2(d)'s comparison between 'RandONet' and 'FD' does not establish that the NO-designed controller stabilizes the true physical system. The manuscript itself states that the RandONet controllers 'were designed and tested under the assumption that the control action is applied in the approximated actuator effect form, and not to the actual physical model.' This directly contradicts the abstract's claim of 'consistent performance between the learned surrogate and the true system.' Please add a closed-loop simulation in which the RandONet-designed gain is applied to the exact controlled timestepper (e.g., the FD model with inner Δt_FD=0.0001), or remo
  2. [Introduction, Section IV-A, Conclusion] The claim of an 'end-to-end equation-free control pipeline where the only requirement is a dataset of short-horizon simulations' is overstated. The actuator structure B in Eq. (25) is taken from the known PDE, and the paper explicitly says 'we do not learn the exact actuator effect from data.' Thus the pipeline still requires knowledge of the actuation mechanism. To support the data-only claim, the actuator effect should be learned or inferred from data, or the claim should be restricted to replacement of the autonomous timestepper. This is a load-bearing issue for the stated novelty.
  3. [Abstract and Section IV-A] The abstract promises 'quantified degradation under plant-model mismatch,' but no plant-model mismatch experiment is reported. The only related observation is that the RandONet controller saturates at an error of about 10^-7 instead of machine precision, which is a surrogate-accuracy limitation, not a quantification of closed-loop robustness to mismatch. The paper also states that robustness 'remains to be investigated in future work.' Please either add mismatch experiments (e.g., perturbing B or the surrogate) or remove this claim from the abstract.
minor comments (5)
  1. [Eq. (19)] The definition of D is under-specified. V_F is said to contain eigenvectors of F, but Eq. (17) uses the same F for y_{n+1}=F y_n. Please clarify whether y_n are coordinates in the Arnoldi basis or in the eigenbasis of F, and define H explicitly (derivatives of S_T with respect to the actuators at the steady state). Currently the dimensions and the role of V_F are ambiguous.
  2. [Section II-B and Section III-B] The symbol F is overloaded: it denotes the general operator being approximated in Eq. (5) and later the reduced Jacobian matrix in Eq. (17). This is a readability issue; consider renaming one of them.
  3. [Abstract and Introduction] The phrase 'coarse stable and unstable steady states' and 'steady and unsteady-states' should be unified and corrected; 'unsteady' usually means time-dependent, not unstable.
  4. [Fig. 2 caption] Panel (e) is described as 'dLQR RandONet convergence' in the caption text but the caption itself says 'Spatiotemporal evolution.' Please make the caption and panel labels consistent.
  5. [Conclusion] Minor typo: 'spatially structureed' should be 'spatially structured.' Also, the sentence beginning 'An important modeling challenge arises...' appears in the Conclusion but reads like a Discussion section; consider moving it before the closing paragraph.

Circularity Check

0 steps flagged

No circular derivation chain; the closed-loop validation gap is a correctness issue, not circularity.

full rationale

The paper's derivation is self-contained: a RandONet surrogate (from prior work [26]) is used in a matrix-free Newton-Krylov/Arnoldi loop to obtain a reduced linear model (F,D), from which dLQR or pole-placement gains are computed, and the closed-loop test is performed on the full nonlinear surrogate. This is not circular: the design model is a low-dimensional linearization, while the test exercises the full nonlinear timestepper, and stabilization of the full surrogate is not guaranteed by the reduced-order design. The only potentially circular-looking element is that the RandONet closed-loop test uses the same approximate controlled surrogate u_{n+1} ≈ S_Δt(u_n) + B z_n (Eq. 25) that was used to compute D and design K, so the closed-loop test is in-sample rather than an independent test on the true controlled PDE. However, the paper explicitly acknowledges this: 'We remark that the RandONet-based controllers were designed and tested under the assumption that the control action is applied in the approximated actuator effect form, and not to the actual physical model. The framework's robustness to plant-model mismatch remains to be investigated in future work' (Section IV-A). The abstract's phrase 'with quantified degradation under plant-model mismatch' is therefore unsupported and inconsistent with the in-text limitation, but that is a validation/overclaim problem, not a circular derivation. The comparison against the finite-difference-based controller provides an external benchmark for the pipeline's components (fixed point, eigenvalues, reduced-model design), and the surrogate's open-loop accuracy is checked against FD results. Self-citations, including [26] for the trained NO, are load-bearing but independently checked in this paper against a PDE solver, so they do not constitute circularity. Overall, no step in the derivation reduces by construction to its own inputs.

Axiom & Free-Parameter Ledger

4 free parameters · 4 axioms · 0 invented entities

The paper's central demonstration rests on the surrogate from [26] and on an additive actuation approximation; the latter is assumed rather than derived or learned, and the validation is performed on the same surrogate, so the evidence is partially self-referential.

free parameters (4)
  • RandONet hyperparameters (from [26]) = Not reported in this paper
    The neural timestepper SΔt used in all computations was trained in [26]; its architecture size, random feature scales, regularization, and training data are not given here, so the central results are not self-contained.
  • Slow-subspace dimension M (number of Arnoldi basis vectors) = M=5 (inferred from the 5 desired pole-placement eigenvalues)
    The reduced model dimension is a hand-chosen truncation; the paper does not justify why M=5 captures the unstable dynamics.
  • Control sampling time Δt = 0.001
    Chosen small to justify the additive first-order actuation approximation in Eq. (25); the validity of the controller depends on this choice.
  • Actuator placement/structure (centers and σ) = centers {0.25, 0.5, 0.75}, σ=0.05
    The Gaussian actuator bumps are chosen and assumed known in the validation; the framework does not learn them from data.
axioms (4)
  • domain assumption The learned RandONet SΔt accurately approximates the true solution operator of the Liouville-Bratu PDE
    All fixed-point, eigenvalue, and controller computations are performed on this surrogate; the paper cites [26] for validation rather than reproducing it.
  • ad hoc to paper The controlled timestepper can be written as SΔt(u) + Bz (Eq. 25), neglecting higher-order Duhamel terms
    This approximation is explicitly adopted in Section IV and used to build D and to run the closed-loop simulations; its error against the true controlled PDE is not quantified.
  • domain assumption The slow subspace spanned by Arnoldi vectors V_M contains the unstable modes and is sufficient for controller design
    Standard EF assumption; the paper shows the open-loop eigenvalues but does not check the reduced model's prediction error against the full surrogate.
  • standard math Standard linear algebra and control theory results (Newton-Krylov, Arnoldi, Riccati) hold as implemented
    Background from [12],[13],[32] and MATLAB implementations.

pith-pipeline@v1.3.0-alltime-deepseek · 12594 in / 13055 out tokens · 368611 ms · 2026-08-04T14:37:24.815159+00:00 · methodology

0 comments
read the original abstract

The control of high-dimensional distributed parameter systems (DPS) remains a challenge when explicit coarse-grained equations are unavailable. Classical equation-free (EF) approaches rely on fine-scale simulators treated as black-box timesteppers. However, repeated simulations for steady-state computation, linearization, and control design are often computationally prohibitive, or the microscopic timestepper may not even be available, leaving us with data as the only resource. We propose a data-driven alternative that uses local neural operators, trained on spatiotemporal microscopic/mesoscopic data, to obtain efficient short-time solution operators. These surrogates are employed within Krylov subspace methods to compute coarse stable and unstable steady states, while also providing Jacobian information in a matrix-free manner. Krylov-Arnoldi iterations then approximate the dominant eigenspectrum, yielding reduced models that capture the open-loop slow dynamics without explicit Jacobian assembly. Both discrete-time Linear Quadratic Regulator (dLQR) and pole-placement (PP) controllers are based on this reduced system and lifted back to the full nonlinear dynamics, thereby closing the feedback loop. The framework is validated by stabilizing an unstable steady-state of the Liouville-Bratu PDE, demonstrating consistent performance between the learned surrogate and the true system, with quantified degradation under plant-model mismatch.

Figures

Figures reproduced from arXiv: 2509.23975 by Constantinos Siettos, Gianluca Fabiani, Ioannis G. Kevrekidis.

Figure 1
Figure 1. Figure 1: Schematic of the equation-free control pipeline via Neural Operators (NOs). An NO, trained on spatiotemporal data, replaces the unavailable [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Equation-free coarse stabilization of an unstable steady-state of the Liouville-Bratu PDE, in Eq.(23), with [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

discussion (0)

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Reference graph

Works this paper leans on

46 extracted references · 3 linked inside Pith

  1. [1]

    Equation-free, coarse-grained multiscale computation: Enabling microscopic simulators to perform system-level analysis,

    I. G. Kevrekidis, C. W. Gear, J. M. Hyman, P. G. Kevrekidis, O. Runborg, and C. Theodoropoulos, “Equation-free, coarse-grained multiscale computation: Enabling microscopic simulators to perform system-level analysis,”Communications in Mathematical Sciences, vol. 1, no. 4, pp. 715–762, 2003

    2003

  2. [2]

    Physics-informed machine learning,

    G. E. Karniadakis, I. G. Kevrekidis, L. Lu, P. Perdikaris, S. Wang, and L. Yang, “Physics-informed machine learning,”Nature Reviews Physics, vol. 3, no. 6, pp. 422–440, 2021

    2021

  3. [3]

    Task-oriented machine learning assisted surrogates for tipping points of agent-based models,

    G. Fabiani, N. Evangelou, T. Cui, J. M. Bello-Rivas, C. P. Martin- Linares, C. Siettos, and I. G. Kevrekidis, “Task-oriented machine learning assisted surrogates for tipping points of agent-based models,” Nature Communications volume, vol. 15, no. 4117, pp. 1–13, 2024

    2024

  4. [4]

    Catastrophic regime shifts in ecosystems: linking theory to observation,

    M. Scheffer and S. R. Carpenter, “Catastrophic regime shifts in ecosystems: linking theory to observation,”Trends in ecology & evolution, vol. 18, no. 12, pp. 648–656, 2003

    2003

  5. [5]

    Time-steppers and ‘coarse’control of distributed microscopic processes,

    A. Armaou, C. I. Siettos, and I. G. Kevrekidis, “Time-steppers and ‘coarse’control of distributed microscopic processes,”International Journal of Robust and Nonlinear Control: IFAC-Affiliated Journal, vol. 14, no. 2, pp. 89–111, 2004

    2004

  6. [6]

    Data-driven control of agent-based models: An equation/variable-free machine learning approach,

    D. G. Patsatzis, L. Russo, I. G. Kevrekidis, and C. Siettos, “Data-driven control of agent-based models: An equation/variable-free machine learning approach,”Journal of Computational Physics, vol. 478, p. 111953, 2023

    2023

  7. [7]

    Discrete-time nonlinear feedback linearization via physics-informed machine learning,

    H. Vargas Alvarez, G. Fabiani, N. Kazantzis, C. Siettos, and I. G. Kevrekidis, “Discrete-time nonlinear feedback linearization via physics-informed machine learning,”Journal of Computational Physics, vol. 492, p. 112408, 2023

    2023

  8. [8]

    Nonlinear and robust control of pde systems: Methods and applications to transport-reaction processes,

    P. D. Christofides and J. Chow, “Nonlinear and robust control of pde systems: Methods and applications to transport-reaction processes,” Appl. Mech. Rev., vol. 55, no. 2, pp. B29–B30, 2002

    2002

  9. [9]

    Low-dimensional approximation and control of periodic solutions in spatially extended systems,

    S. Y . Shvartsman and I. Kevrekidis, “Low-dimensional approximation and control of periodic solutions in spatially extended systems,” Physical Review E, vol. 58, no. 1, p. 361, 1998

    1998

  10. [10]

    Equation-free: The computer-aided analysis of complex multiscale systems,

    I. G. Kevrekidis, C. W. Gear, and G. Hummer, “Equation-free: The computer-aided analysis of complex multiscale systems,”AIChE Journal, vol. 50, no. 7, pp. 1346–1355, 2004

    2004

  11. [11]

    Variable-free exploration of stochastic models: a gene regulatory network example,

    R. Erban, T. A. Frewen, X. Wang, T. C. Elston, R. Coifman, B. Nadler, and I. G. Kevrekidis, “Variable-free exploration of stochastic models: a gene regulatory network example,”The Journal of chemical physics, vol. 126, no. 15, p. 04B618, 2007

    2007

  12. [12]

    C. T. Kelley,Iterative methods for linear and nonlinear equations. SIAM, 1995

    1995

  13. [13]

    Saad,Numerical methods for large eigenvalue problems: revised edition

    Y . Saad,Numerical methods for large eigenvalue problems: revised edition. SIAM, 2011

    2011

  14. [14]

    From interacting agents to density-based modeling with stochastic pdes,

    L. Helfmann, N. Djurdjevac Conrad, A. Djurdjevac, S. Winkelmann, and C. Sch ¨utte, “From interacting agents to density-based modeling with stochastic pdes,”Communications in Applied Mathematics and Computational Science, vol. 16, no. 1, pp. 1–32, 2021

    2021

  15. [15]

    Deep hidden physics models: Deep learning of nonlinear partial differential equations,

    M. Raissi, “Deep hidden physics models: Deep learning of nonlinear partial differential equations,”The Journal of Machine Learning Research, vol. 19, no. 1, pp. 932–955, 2018

    2018

  16. [16]

    Coarse-scale pdes from fine-scale observations via ma- chine learning,

    S. Lee, M. Kooshkbaghi, K. Spiliotis, C. I. Siettos, and I. G. Kevrekidis, “Coarse-scale pdes from fine-scale observations via ma- chine learning,”Chaos: An Interdisciplinary Journal of Nonlinear Science, vol. 30, no. 1, p. 013141, 2020

    2020

  17. [17]

    Solving and learning nonlinear pdes with gaussian processes,

    Y . Chen, B. Hosseini, H. Owhadi, and A. M. Stuart, “Solving and learning nonlinear pdes with gaussian processes,”Journal of Compu- tational Physics, vol. 447, p. 110668, 2021

    2021

  18. [18]

    Learning nonlinear operators via deeponet based on the universal approximation theorem of operators,

    L. Lu, P. Jin, G. Pang, Z. Zhang, and G. E. Karniadakis, “Learning nonlinear operators via deeponet based on the universal approximation theorem of operators,”Nature Machine Intelligence, vol. 3, no. 3, pp. 218–229, 2021

    2021

  19. [19]

    Multiscale simulations of complex systems by learning their effective dynamics,

    P. R. Vlachas, G. Arampatzis, C. Uhler, and P. Koumoutsakos, “Multiscale simulations of complex systems by learning their effective dynamics,”Nature Machine Intelligence, vol. 4, no. 4, pp. 359–366, 2022

    2022

  20. [20]

    Numerical bifurcation analysis of pdes from lattice boltzmann model simulations: a parsimonious machine learning approach,

    E. Galaris, G. Fabiani, I. Gallos, I. Kevrekidis, and C. Siettos, “Numerical bifurcation analysis of pdes from lattice boltzmann model simulations: a parsimonious machine learning approach,”Journal of Scientific Computing, vol. 92, no. 2, pp. 1–30, 2022

    2022

  21. [21]

    Hilbert’s sixth problem: derivation of fluid equations via boltzmann’s kinetic theory,

    Y . Deng, Z. Hani, and X. Ma, “Hilbert’s sixth problem: derivation of fluid equations via boltzmann’s kinetic theory,”arXiv preprint arXiv:2503.01800, 2025

    Pith/arXiv arXiv 2025

  22. [22]

    Fourier neural operator for parametric partial differential equations,

    Z. Li, N. Kovachki, K. Azizzadenesheli, B. Liu, K. Bhattacharya, A. Stuart, and A. Anandkumar, “Fourier neural operator for parametric partial differential equations,”arXiv preprint arXiv:2010.08895, 2020

    Pith/arXiv arXiv 2010

  23. [23]

    Multipole graph neural operator for parametric partial differential equations,

    Z. Li, N. Kovachki, K. Azizzadenesheli, B. Liu, A. Stuart, K. Bhat- tacharya, and A. Anandkumar, “Multipole graph neural operator for parametric partial differential equations,”Advances in Neural Infor- mation Processing Systems, vol. 33, pp. 6755–6766, 2020

    2020

  24. [24]

    Spectral neural operators,

    V . Fanaskov and I. V . Oseledets, “Spectral neural operators,” in Doklady Mathematics, vol. 108, Suppl 2. Springer, 2023, pp. S226– S232

    2023

  25. [25]

    Randonets: Shallow networks with random projections for learning linear and nonlinear operators,

    G. Fabiani, I. G. Kevrekidis, C. Siettos, and A. N. Yannacopoulos, “Randonets: Shallow networks with random projections for learning linear and nonlinear operators,”Journal of Computational Physics, vol. 520, p. 113433, 2025

    2025

  26. [26]

    Enabling local neural operators to perform equation-free system-level analysis,

    G. Fabiani, H. Vandecasteele, S. Goswami, C. Siettos, and I. G. Kevrekidis, “Enabling local neural operators to perform equation-free system-level analysis,”arXiv preprint arXiv:2505.02308, 2025

    arXiv 2025

  27. [27]

    Learning dissipative dynamics in chaotic systems,

    Z. Li, M. Liu-Schiaffini, N. Kovachki, B. Liu, K. Azizzadenesheli, K. Bhattacharya, A. Stuart, and A. Anandkumar, “Learning dissipative dynamics in chaotic systems,” inProceedings of the 36th International Conference on Neural Information Processing Systems, 2022, pp. 16 768–16 781

    2022

  28. [28]

    Long-time integration of parametric evolution equations with physics-informed deeponets,

    S. Wang and P. Perdikaris, “Long-time integration of parametric evolution equations with physics-informed deeponets,”Journal of Computational Physics, vol. 475, p. 111855, 2023

    2023

  29. [29]

    Coarse brownian dynamics for nematic liquid crystals: Bifurcation, projective integra- tion, and control via stochastic simulation,

    C. I. Siettos, M. D. Graham, and I. G. Kevrekidis, “Coarse brownian dynamics for nematic liquid crystals: Bifurcation, projective integra- tion, and control via stochastic simulation,”The Journal of chemical physics, vol. 118, no. 22, pp. 10 149–10 156, 2003

    2003

  30. [30]

    An equation-free approach to nonlinear control: Coarse feedback linearization with pole- placement,

    C. I. Siettos, I. G. Kevrekidis, and N. Kazantzis, “An equation-free approach to nonlinear control: Coarse feedback linearization with pole- placement,”International Journal of Bifurcation and Chaos, vol. 16, no. 07, pp. 2029–2041, 2006

    2029

  31. [31]

    An equation-free approach to agent-based computation: Bifurcation analysis and control of stationary states,

    C. Siettos, C. W. Gear, and I. G. Kevrekidis, “An equation-free approach to agent-based computation: Bifurcation analysis and control of stationary states,”EPL (Europhysics Letters), vol. 99, no. 4, p. 48007, 2012

    2012

  32. [32]

    Ogata,Discrete-time control systems

    K. Ogata,Discrete-time control systems. Prentice-Hall, Inc., 1995

    1995

  33. [33]

    An analytical and numerical study of the two-dimensional bratu equation,

    J. P. Boyd, “An analytical and numerical study of the two-dimensional bratu equation,”Journal of Scientific Computing, vol. 1, pp. 183–206, 1986

    1986

  34. [34]

    Universal approximation to nonlinear operators by neural networks with arbitrary activation functions and its applica- tion to dynamical systems,

    T. Chen and H. Chen, “Universal approximation to nonlinear operators by neural networks with arbitrary activation functions and its applica- tion to dynamical systems,”IEEE Transactions on Neural Networks, vol. 6, no. 4, pp. 911–917, 1995

    1995

  35. [35]

    Approximation by superpositions of a sigmoidal func- tion,

    G. Cybenko, “Approximation by superpositions of a sigmoidal func- tion,”Mathematics of control, signals and systems, vol. 2, no. 4, pp. 303–314, 1989

    1989

  36. [36]

    Training neural networks is np-hard in fixed dimension,

    V . Froese and C. Hertrich, “Training neural networks is np-hard in fixed dimension,”Advances in Neural Information Processing Systems, vol. 36, pp. 44 039–44 049, 2023

    2023

  37. [37]

    Negative results for approximation using single layer and multilayer feedforward neural networks,

    J. Almira, P. Lopez-de Teruel, D. Romero-Lopez, and F. V oigtlaender, “Negative results for approximation using single layer and multilayer feedforward neural networks,”Journal of mathematical analysis and applications, vol. 494, no. 1, p. 124584, 2021

    2021

  38. [38]

    Random projection neural networks of best approxima- tion: Convergence theory and practical applications,

    G. Fabiani, “Random projection neural networks of best approxima- tion: Convergence theory and practical applications,”SIAM Journal on Mathematics of Data Science, vol. 7, no. 2, pp. 385–409, 2025

    2025

  39. [39]

    Efficient training of deep neural operator networks via randomized sampling,

    S. Karumuri, L. Graham-Brady, and S. Goswami, “Efficient training of deep neural operator networks via randomized sampling,”arXiv preprint arXiv:2409.13280, 2024

    Pith/arXiv arXiv 2024

  40. [40]

    Extensions of lipschitz mappings into a hilbert space,

    W. B. Johnson, “Extensions of lipschitz mappings into a hilbert space,” Contemp. Math., vol. 26, pp. 189–206, 1984

    1984

  41. [41]

    Stochastic choice of basis functions in adaptive function approximation and the functional-link net,

    B. Igelnik and Y .-H. Pao, “Stochastic choice of basis functions in adaptive function approximation and the functional-link net,”IEEE Transactions on Neural Networks, vol. 6, no. 6, pp. 1320–1329, 1995

    1995

  42. [42]

    Random features for large-scale kernel ma- chines,

    A. Rahimi and B. Recht, “Random features for large-scale kernel ma- chines,”Advances in neural information processing systems, vol. 20, 2007

    2007

  43. [43]

    Uniform approximation of functions with random bases,

    ——, “Uniform approximation of functions with random bases,” in 2008 46th annual allerton conference on communication, control, and computing. IEEE, 2008, pp. 555–561

    2008

  44. [44]

    Ap- proximation with random bases: Pro et contra,

    A. N. Gorban, I. Y . Tyukin, D. V . Prokhorov, and K. I. Sofeikov, “Ap- proximation with random bases: Pro et contra,”Information Sciences, vol. 364, pp. 129–145, 2016

    2016

  45. [45]

    Numerical solution and bifurcation analysis of nonlinear partial differential equations with extreme learning machines,

    G. Fabiani, F. Calabr `o, L. Russo, and C. Siettos, “Numerical solution and bifurcation analysis of nonlinear partial differential equations with extreme learning machines,”Journal of Scientific Computing, vol. 89, no. 2, pp. 1–35, 2021

    2021

  46. [46]

    Parsimonious physics-informed random projection neural networks for initial value problems of odes and index-1 daes,

    G. Fabiani, E. Galaris, L. Russo, and C. Siettos, “Parsimonious physics-informed random projection neural networks for initial value problems of odes and index-1 daes,”Chaos: An Interdisciplinary Journal of Nonlinear Science, vol. 33, no. 4, 2023

    2023