Pith. sign in

REVIEW 3 major objections 7 minor 48 references

Stability Analysis of Reservoir Computers Dynamics via Lyapunov Functions

T0 review · 3 major / 7 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A Lyapunov comparison identifies a ball of guaranteed stability around a reservoir computer's fixed point, and training error is lowest inside that ball.

desk verdict Solid continuous-time stability criterion for reservoir computers, but the discrete-time proof has a load-bearing norm/spectral-radius gap that needs fixing before the c-region claims for nonnormal adjacency matrices can stand. read the letter →

arxiv 1908.04411 v2 pith:6AFXRIIC submitted 2019-08-09 eess.SY cs.SYnlin.AO

Afroza Shirin , Isaac S. Klickstein , Francesco Sorrentino This is my paper · ORCID
classification eess.SYcs.SYnlin.AO MSC 34D2093D0568T07
keywords reservoircomputingLyapunovstabilityc-regiontrainingerrorpolynomialnonlinearitysigmoidbasinofattractionechostatenetworks
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper tries to establish a design rule: for a reservoir computer, a scalar Lyapunov comparison gives a radius $c$ around a fixed operating point inside which the unforced dynamics is guaranteed stable, and the parameters predicted to be globally stable are the parameters where the reservoir actually performs best on training tasks. In continuous time, stability of the ball follows when the tightest quadratic bound $K^*(c,\theta)$ for the node nonlinearity is no larger than $-\alpha_{\max}$ of the symmetric part of the adjacency matrix; in discrete time, similar bounds constrain the ratio $f(r_i)/r_i$ by the eigenvalues nearest the unit circle. The authors verify this by training a 100-node reservoir on Lorenz and Duffing input-output tasks and showing low training error below the global-stability boundary. They also find that polynomial node dynamics need both an odd and an even power, for example linear plus quadratic, to work well. If the rule holds, it gives a practical parameter pre-screen: stay inside the predicted stable region and then search for performance.

What carries the argument

The named object is the $c(\theta)$-region: the ball of radius $c$ around the fixed point inside which the Lyapunov function certifies stability. What carries the argument is the scalar bound $K^*(c,\theta)$, obtained by solving $\min K$ subject to $r_i f(r_i;\theta) - K r_i^2 \le 0$ on $[-c,c]$; the optimum is the largest of $f(c)/c$, $f(-c)/(-c)$, $f'(0)$, and $f(r^*)/r^*$ at interior stationary points of the ratio. Stability in continuous time reduces to $K^*(c,\theta) \le -\alpha_{\max}(A_s)$, decoupling node dynamics from network topology. In discrete time, the same ratio is squeezed from both sides, $K_-^* \le f(r_i;\theta)/r_i \le K_+^*$, and the admissible corridor is set by the two eigenvalues of $A$ whose real parts are closest to the unit circle; the stability radius is $c_{\max} = \min\{c^+_{\max}, c^-_{\max}\}$.

What would settle it

Construct a 2-node discrete reservoir with $f(r)=0$ and adjacency matrix $A = [[0.9, 1], [0, 0.9]]$. All eigenvalues equal 0.9, so with $K=0$ the paper's condition holds, but the matrix norm exceeds 1; starting inside the unit ball on the top singular-vector direction, the state norm immediately grows past 1, so the predicted invariant ball is not invariant. That observation settles whether the discrete-time $c$-region theorem holds as stated.

Watch

Extended reading notes

Core claim

The paper's central claim is that, for a given unforced reservoir computer with a fixed point shifted to the origin, a scalar comparison function can certify nonlinear stability on an explicit ball. In continuous time, let $K^*(c,\theta)$ be the smallest $K$ such that $r_i f(r_i;\theta) \le K r_i^2$ for all $r_i \in [-c,c]$; then every trajectory starting in the ball $\|r\| \le c$ converges to the origin when $K^*(c,\theta) \le -\alpha_{\max}(A_s)$, where $A_s$ is the symmetric part of the adjacency matrix. In discrete time the same idea uses $V(r)=\|r\|$ and a corridor of two bounds $K_-^* \le f(r_i;\theta)/r_i \le K_+^*$, admissible when they fall between the shifts toward the unit circle set by the nearest eigenvalues. The paper then reports that in simulations the training error of a 100-node reservoir is markedly lower exactly in the parameter region where these global-stability conditions hold, and that for polynomial node dynamics the reservoir works when the polynomial contains both an odd and an even power, with the quadratic coefficient playing a distinctive role.

Load-bearing premise

The discrete-time argument assumes that squeezing every ratio $f(r_i)/r_i$ between the same two constants is enough to bound the norm of the whole map, and that checking eigenvalues alone controls that norm; this requires the adjacency matrix's eigenvectors to be mutually perpendicular, a condition the paper never states.

Editorial extensions

If this is right

  • In continuous-time polynomial reservoirs, the curve $K^*(\infty,\theta)=-\alpha_{\max}(A_s)$ marks the boundary below which training error is low; parameters above it risk divergence or convergence to a different attractor.
  • The $c(\theta)$-region rule separates nodal dynamics from network topology, so a designer can enlarge the stable region by adjusting $K^*$, the node nonlinearity, or by adjusting $\alpha_{\max}$, the coupling structure.
  • For discrete-time sigmoid reservoirs, global stability holds between the curves $K_+^*=\rho_c^+$ and $K_-^*=\rho_c^-$, and the training error is small only inside that corridor.
  • Polynomial node functions with only odd powers, such as linear, cubic, and fifth-order terms, perform poorly; a nonzero even-power coefficient is required, with a zero quadratic coefficient producing training error near one.
  • Because the analysis is input independent, a reservoir stabilized at the origin will remain near the operating point for any bounded input signal, not just for the training signal.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Extension: the same $K^*/\alpha_{\max}$ split suggests a direct design optimization: maximize the radius $c_{\max}$ by tuning node parameters and adjacency weights separately, since the two contributions decouple.
  • Extension: the discrete-time criterion should be tightened to use a genuine norm bound, for example the induced 2-norm, rather than eigenvalues alone for non-normal adjacency matrices; this would shrink the predicted $c_{\max}$ and can be checked against direct simulation.
  • Extension: the odd/even-power requirement may generalize to a necessary condition for polynomial activations in reservoirs: an even power breaks sign symmetry, and this could be tested on higher-degree polynomials and different reservoir topologies.
  • Extension: the basin radius could be converted into an input-amplitude bound for the forced system: normalize inputs so the driven trajectory stays inside $D(c_{\max})$, giving a tighter but input-specific performance guarantee.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 7 minor

Summary. The paper proposes a Lyapunov-based method to certify nonlinear stability of reservoir computers around a fixed point, for both continuous-time and discrete-time dynamics. For continuous time, the authors derive a sufficient condition K*(c,theta) <= -alpha_max(A_s), where K* is the maximum of f(r)/r on [-c,c] and alpha_max is the largest eigenvalue of the symmetric part of the adjacency matrix. For discrete time, they propose a condition involving eigenvalues of the adjacency matrix and bounds K- and K+ on f(r)/r, and they use these conditions to draw parameter-space stability boundaries. Numerically, the paper reports training errors for continuous-time polynomial reservoirs and discrete-time sigmoid reservoirs, finding that training error tends to be lower in the analytically predicted globally stable parameter region.

Significance. If the discrete-time arguments were correct, the paper would offer a simple, input-independent design rule for choosing reservoir parameters that guarantee stability, along with an interesting empirical link between the predicted global stability region and low training error. The continuous-time sufficient condition is derived cleanly, and the decoupling between nodal dynamics and network topology is a useful and clearly presented idea. The paper's numerical validation with Lorenz and Duffing signals is a genuine strength, and the analytical boundary curves in the figures are not fits to the error data. However, the discrete-time stability result currently rests on an invalid norm inequality and on an unjustified replacement of an operator norm by a spectral radius, so the discrete-time portion of the central claim is not supported as written.

major comments (3)
  1. [Section II.B.1, Eq. (19)] The inequality ||f(r)+Ar|| <= ||K(c,theta) r + Ar|| is asserted to follow from the componentwise bound f_i(r_i) <= K(c,theta) r_i, but this implication is false. For example, take K=1, r=(1,-1), and f(r)=(1,-100); each component satisfies f_i <= K r_i (1<=1 and -100<=-1), yet ||f(r)|| = sqrt(10001) > ||K r|| = sqrt(2). Since this norm inequality is the step that reduces the Lyapunov decrement to a check on the linear operator K I + A, the discrete-time stability condition is not established by the given argument.
  2. [Section II.B.1, Eqs. (20a)-(20b)] The passage from |K + gamma_i| <= 1 for every eigenvalue gamma_i of A to the operator norm bound ||K I + A||_2 <= 1 is valid only for normal matrices, and the random adjacency matrices constructed in Section III are generally non-normal. A concrete counterexample is A = [[0,2],[0,0]] with K=0: the spectral radius is 0, yet ||A||_2 = 2, so the Lyapunov function V(r)=||r|| can increase even though the eigenvalue condition holds. Consequently, the discrete-time c-region and the boundary curves K+*(c,theta)=rho_c^+ and K-*(c,theta)=rho_c^- in Figs. 8-9 are not justified by the arguments provided.
  3. [Section II.B.2, Eqs. (37)-(41)] In the non-homogeneous case, the reduction to scalar bounds K-* = min_i K_i^-* and K+* = max_i K_i^+* is not sufficient for the norm inequality needed with V(r)=||r||. The actual map involves a state-dependent diagonal matrix D(r) = diag(f_i(r_i)/r_i), and bounding each entry of D(r) by the global scalars K-* and K+* does not control ||D(r)+A||_2 when A is non-normal. Since the sigmoid example in Section III.C has node-dependent fixed points q_i* and hence node-dependent K_i^+* and K_i^-*, the discrete-time numerical boundaries in Fig. 8 also rely on this unjustified scalarization.
minor comments (7)
  1. [Section II.A.2] The text states that the origin is linearly stable if the largest real part of the eigenvalues of (A - p1 I) is negative; the linearization of Eq. (13) at the origin is A + p1 I, so this appears to be a sign error that should be corrected.
  2. [Eq. (20a)] The notation |K I + A| is undefined; if it denotes a determinant, it is not the induced norm needed for the Lyapunov argument, and if it denotes the matrix of absolute values, it is not what is used in Eq. (20b).
  3. [Eq. (14)] The formula for the interior critical point r* = -p2/(2p3) is singular when p3 = 0, a case that occurs in the parameter scans of Figs. 3-4; the handling of the case p3 = 0 should be stated explicitly.
  4. [Introduction] The claim that global asymptotic stability of the unforced system implies stable behavior under any bounded input is not true for general nonlinear systems; the authors should either prove an input-to-state stability property or soften this statement to avoid overclaiming.
  5. [Section III.A, Eqs. (51)-(52)] The training error formula Delta_RC = <Omega k - g> / <g> is confusing because <X> is later defined as the RMS value; please define the normalized RMS error explicitly.
  6. [Fig. 2 caption] The caption contains an apparent typo ('closest to the positive side of the unit circle Wis'), and the definitions of rho_i^+ and rho_i^- in the caption are hard to follow without repeating Eq. (22).
  7. [Section III.C, Eqs. (44)-(45)] The indices in the sums involving A_{ij} q*_i and A_{ij} q*_j appear inconsistent; the fixed-point shift should be checked to ensure the transformed dynamics is written correctly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the stability region is derived analytically from the reservoir dynamics, and the training error is an independent benchmark.

full rationale

The paper's central stability condition is derived from the reservoir dynamics and a quadratic-norm Lyapunov function: Eq. (6b) follows from bounding r_i f(r_i) by K(c, θ) r_i^2 and using the largest eigenvalue of the symmetric part of A, while the optimal K*(c, θ) is obtained from the optimization in Eqs. (7)-(10) rather than from training-error data. The reported boundary curves (Figs. 3, 4, and 8) are evaluations of K*(c→∞, θ) = −α_max(A_s) or K±*(c→∞, θ) = ρ±, which are analytical expressions in the nodal parameters and the eigenvalues of A. The training error Δ_RC is computed by an independent least-squares fit (Eqs. (48)-(52)) and then compared with the analytically predicted stable region; the paper never fits K* or c_max to the error map. There are no load-bearing self-citations: the cited items are standard Lyapunov theory and external reservoir-computing papers, and no uniqueness theorem or prior-work ansatz is invoked to force the chosen form. The discrete-time derivation's replacement of a matrix norm by its spectral radius in Eq. (20) is a mathematical gap for non-normal A, but it is not circular—it does not assume the conclusion and does not make the prediction equivalent to its inputs. Hence no circular step meets the quoted-evidence standard.

Assumptions & free parameters 6 free parameters · 6 assumptions · 0 invented entities

The derivation rests on the Lyapunov theorems and the fixed point at the origin, plus two unproven premises: the discrete-time norm bound and the claim that unforced global stability guarantees boundedness under bounded inputs. The polynomial coefficients p1-p5 and the spectral normalization 0.5 are chosen parameters that shape the reported stability boundaries.

free parameters (6)
  • p1 (linear coefficient) = -3 in continuous-time simulations
    Linear coefficient fixed, along with construction of A, to make the origin a stable fixed point; not fitted to the training error.
  • p2 (quadratic coefficient) = scanned, e.g., -10 to 10 in Figs. 3-4; p2=0 is the failure case
    Quadratic coefficient of the polynomial node dynamics; stability boundary and training error are plotted against it.
  • p3 (cubic coefficient) = scanned, e.g., -10 to 10 in Fig. 3; p3=-4 in Fig. 7
    Cubic coefficient of the polynomial node dynamics; stability boundary and training error are plotted against it.
  • p4 (quartic coefficient) = scanned in Fig. 5; zero elsewhere
    Quartic coefficient used in the additional polynomial f = p1 r + p3 r^3 + p4 r^4 simulation.
  • p5 (quintic coefficient) = scanned in Fig. 6; zero elsewhere
    Quintic coefficient used in the additional polynomial f = p1 r + p3 r^3 + p5 r^5 simulation.
  • adjacency spectral normalization = 0.5
    A is normalized so the largest real part of its eigenvalues has absolute value 0.5; this fixes alpha_max and all stability thresholds.
assumptions (6)
  • standard math Lyapunov stability theorems: existence of V with V>0 and V-dot<0 on D(c) implies asymptotic stability
    Used in Secs II.A.1 and II.B.1 as the core criterion.
  • domain assumption The origin is a fixed point and is linearly stable for the unforced dynamics
    Stated after Eq. (2): 'We assume that r*_i(t)=0 is a (linearly) stable fixed point'. The entire basin analysis is local to the origin.
  • domain assumption The reservoir nodes are homogeneous (same function f and same parameter theta) for the main derivations
    Assumed in Secs II.A and II.B; only Sec II.B.2 considers non-homogeneous dynamics.
  • domain assumption For non-homogeneous dynamics, any nonzero fixed point can be moved to the origin by coordinate transformation
    Assumed in Sec II.B.2 and used in the sigmoid example; not proven in detail.
  • ad hoc to paper Global asymptotic stability of the unforced system implies stable behavior under any bounded input
    Stated in Sec I: 'If the desired operating point is found to be globally asymptotically stable, then stability is independent of the particular stimulus, as long as it is bounded.' This requires input-to-state stability and is not generally true for nonlinear systems.
  • ad hoc to paper The discrete-time norm bound ||f(r)+Ar|| <= ||K r+Ar|| follows from f_i(r_i) <= K r_i, and ||K I + A|| is controlled by the eigenvalues of A
    Used in Eqs. (18)-(20). This step is not valid for arbitrary sign-varying f_i and non-normal A.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Stability Analysis of Reservoir Computers Dynamics via Lyapunov Functions." pith.science (2026). https://pith.science/paper/6AFXRIIC

@misc{pith2026190804411,
  author       = {Pith},
  title        = {Pith review of: Stability Analysis of Reservoir Computers Dynamics via Lyapunov Functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6AFXRIIC}},
  note         = {Machine review of arXiv:1908.04411}
}
read the original abstract

A Lyapunov design method is used to analyze the nonlinear stability of a generic reservoir computer for both the cases of continuous-time and discrete-time dynamics. Using this method, for a given nonlinear reservoir computer, a radial region of stability around a fixed point is analytically determined. We see that the training error of the reservoir computer is lower in the region where the analysis predicts global stability but is also affected by the particular choice of the individual dynamics for the reservoir systems. For the case that the dynamics is polynomial, it appears to be important for the polynomial to have nonzero coefficients corresponding to at least one odd power (e.g., linear term) and one even power (e.g., quadratic term).

Figures

Figures reproduced from arXiv: 1908.04411 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p016_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p022_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p023_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p024_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p025_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p026_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p027_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p028_9.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

48 extracted references · 46 canonical work pages

  1. [1]

    echo state

    Herbert Jaeger. The “echo state” approach to analysing and training recurrent neural networks-with an erratum note. Bonn, Germany: German National Research Center for Information Technology GMD Technical Report , 148(34):13, 2001

    work page 2001

  2. [2]

    An overview of reservoir computing: theory, applications and implementations

    Benjamin Schrauwen, David Verstraeten, and Jan Van Campenhout. An overview of reservoir computing: theory, applications and implementations. In Proceedings of the 15th european symposium on artificial neural networks. p. 471-482 2007 , pages 471--482, 2007

    work page 2007

  3. [3]

    a ger, Wolfgang Maass, and Henry Markram. The

    Thomas Natschl \"a ger, Wolfgang Maass, and Henry Markram. The" liquid computer": A novel strategy for real-time computing on time series. Special issue on Foundations of Information Processing of TELEMATIK , 8(ARTICLE):39--43, 2002

    work page 2002

  4. [4]

    Real-time computing without stable states: A new framework for neural computation based on perturbations

    Wolfgang Maass, Thomas Natschl \"a ger, and Henry Markram. Real-time computing without stable states: A new framework for neural computation based on perturbations. Neural computation , 14(11):2531--2560, 2002

    2002

  5. [5]

    Photonic nonlinear transient computing with multiple-delay wavelength dynamics

    Romain Martinenghi, Sergei Rybalko, Maxime Jacquot, Yanne K Chembo, and Laurent Larger. Photonic nonlinear transient computing with multiple-delay wavelength dynamics. Physical review letters , 108(24):244101, 2012

    work page 2012

  6. [6]

    Parallel photonic information processing at gigabyte per second data rates using transient states

    Daniel Brunner, Miguel C Soriano, Claudio R Mirasso, and Ingo Fischer. Parallel photonic information processing at gigabyte per second data rates using transient states. Nature communications , 4:1364, 2013

    work page 2013

  7. [7]

    Information processing via physical soft body

    Kohei Nakajima, Helmut Hauser, Tao Li, and Rolf Pfeifer. Information processing via physical soft body. Scientific reports , 5:10487, 2015

    work page 2015

  8. [8]

    Photonic delay systems as machine learning implementations

    Michiel Hermans, Miguel C Soriano, Joni Dambre, Peter Bienstman, and Ingo Fischer. Photonic delay systems as machine learning implementations. Journal of Machine Learning Research , 2015

    work page 2015

Show all 48 references
  1. [9]

    High-performance photonic reservoir computer based on a coherently driven passive cavity

    Quentin Vinckier, Fran c ois Duport, Anteo Smerieri, Kristof Vandoorne, Peter Bienstman, Marc Haelterman, and Serge Massar. High-performance photonic reservoir computer based on a coherently driven passive cavity. Optica , 2(5):438--446, 2015

    2015

  2. [10]

    Fully analogue photonic reservoir computer

    Fran c ois Duport, Anteo Smerieri, Akram Akrout, Marc Haelterman, and Serge Massar. Fully analogue photonic reservoir computer. Scientific reports , 6:22381, 2016

    2016

  3. [11]

    High-speed photonic reservoir computing using a time-delay-based architecture: Million words per second classification

    Laurent Larger, Antonio Bayl \'o n-Fuentes, Romain Martinenghi, Vladimir S Udaltsov, Yanne K Chembo, and Maxime Jacquot. High-speed photonic reservoir computing using a time-delay-based architecture: Million words per second classification. Physical Review X , 7(1):011015, 2017

    2017

  4. [12]

    Artificial neural networks for modelling and control of non-linear systems

    Johan AK Suykens, Joos PL Vandewalle, and Bart L de Moor. Artificial neural networks for modelling and control of non-linear systems . Springer Science & Business Media, 2012

    2012

  5. [13]

    Introduction to focus issue: intrinsic and designed computation: information processing in dynamical systems—beyond the digital hegemony, 2010

    James P Crutchfield, William L Ditto, and Sudeshna Sinha. Introduction to focus issue: intrinsic and designed computation: information processing in dynamical systems—beyond the digital hegemony, 2010

    2010

  6. [14]

    Simple recurrent networks learn context-free and context-sensitive languages by counting

    Paul Rodriguez. Simple recurrent networks learn context-free and context-sensitive languages by counting. Neural computation , 13(9):2093--2118, 2001

    2001

  7. [15]

    Lstm recurrent networks learn simple context-free and context-sensitive languages

    Felix A Gers and E Schmidhuber. Lstm recurrent networks learn simple context-free and context-sensitive languages. IEEE Transactions on Neural Networks , 12(6):1333--1340, 2001

    2001

  8. [16]

    Attractor reconstruction by machine learning

    Zhixin Lu, Brian R Hunt, and Edward Ott. Attractor reconstruction by machine learning. Chaos: An Interdisciplinary Journal of Nonlinear Science , 28(6):061104, 2018

    2018

  9. [17]

    Observing spatio-temporal dynamics of excitable media using reservoir computing

    Roland S Zimmermann and Ulrich Parlitz. Observing spatio-temporal dynamics of excitable media using reservoir computing. Chaos: An Interdisciplinary Journal of Nonlinear Science , 28(4):043118, 2018

    2018

  10. [18]

    Using a reservoir computer to learn chaotic attractors, with applications to chaos synchronization and cryptography

    Piotr Antonik, Marvyn Gulina, Ja \"e l Pauwels, and Serge Massar. Using a reservoir computer to learn chaotic attractors, with applications to chaos synchronization and cryptography. Physical Review E , 98(1):012215, 2018

    2018

  11. [19]

    Harnessing nonlinearity: Predicting chaotic systems and saving energy in wireless communication

    Herbert Jaeger and Harald Haas. Harnessing nonlinearity: Predicting chaotic systems and saving energy in wireless communication. science , 304(5667):78--80, 2004

    2004

  12. [20]

    Using machine learning to replicate chaotic attractors and calculate lyapunov exponents from data

    Jaideep Pathak, Zhixin Lu, Brian R Hunt, Michelle Girvan, and Edward Ott. Using machine learning to replicate chaotic attractors and calculate lyapunov exponents from data. Chaos: An Interdisciplinary Journal of Nonlinear Science , 27(12):121102, 2017

    2017

  13. [21]

    Model-free prediction of large spatiotemporally chaotic systems from data: A reservoir computing approach

    Jaideep Pathak, Brian Hunt, Michelle Girvan, Zhixin Lu, and Edward Ott. Model-free prediction of large spatiotemporally chaotic systems from data: A reservoir computing approach. Physical review letters , 120(2):024102, 2018

    2018

  14. [22]

    On the application of reservoir computing networks for noisy image recognition

    Azarakhsh Jalalvand, Kris Demuynck, Wesley De Neve, and Jean-Pierre Martens. On the application of reservoir computing networks for noisy image recognition. Neurocomputing , 277:237--248, 2018

    2018

  15. [23]

    Biologically plausible speech recognition with lstm neural nets

    Alex Graves, Douglas Eck, Nicole Beringer, and Juergen Schmidhuber. Biologically plausible speech recognition with lstm neural nets. In International Workshop on Biologically Inspired Approaches to Advanced Information Technology , pages 127--136. Springer, 2004

    2004

  16. [24]

    An application of recurrent nets to phone probability estimation

    Tony Robinson. An application of recurrent nets to phone probability estimation. IEEE transactions on Neural Networks , 5(2), 1994

    1994

  17. [25]

    Reservoir computing trends

    Mantas Luko s evi c ius, Herbert Jaeger, and Benjamin Schrauwen. Reservoir computing trends. KI-K \"u nstliche Intelligenz , 26(4):365--371, 2012

    2012

  18. [26]

    Information processing capacity of dynamical systems

    Joni Dambre, David Verstraeten, Benjamin Schrauwen, and Serge Massar. Information processing capacity of dynamical systems. Scientific reports , 2:514, 2012

    2012

  19. [27]

    Short term memory in echo state networks

    Herbert Jaeger. Short term memory in echo state networks. gmd-report 152. In GMD-German National Research Institute for Computer Science (2002), http://www. faculty. jacobs-university. de/hjaeger/pubs/STMEchoStatesTechRep. pdf . Citeseer, 2002

    2002

  20. [28]

    Memory traces in dynamical systems

    Surya Ganguli, Dongsung Huh, and Haim Sompolinsky. Memory traces in dynamical systems. Proceedings of the National Academy of Sciences , 105(48):18970--18975, 2008

    2008

  21. [29]

    State-dependent computations: spatiotemporal processing in cortical networks

    Dean V Buonomano and Wolfgang Maass. State-dependent computations: spatiotemporal processing in cortical networks. Nature Reviews Neuroscience , 10(2):113, 2009

    2009

  22. [30]

    Short-term memory in orthogonal neural networks

    Olivia L White, Daniel D Lee, and Haim Sompolinsky. Short-term memory in orthogonal neural networks. Physical review letters , 92(14):148102, 2004

    2004

  23. [31]

    An experimental unification of reservoir computing methods

    David Verstraeten, Benjamin Schrauwen, Michiel d’Haene, and Dirk Stroobandt. An experimental unification of reservoir computing methods. Neural networks , 20(3):391--403, 2007

    2007

  24. [32]

    Reservoir computing beyond memory-nonlinearity trade-off

    Masanobu Inubushi and Kazuyuki Yoshimura. Reservoir computing beyond memory-nonlinearity trade-off. Scientific reports , 7(1):10199, 2017

    2017

  25. [33]

    Difference between memory and prediction in linear recurrent networks

    Sarah Marzen. Difference between memory and prediction in linear recurrent networks. Physical Review E , 96(3):032308, 2017

    2017

  26. [34]

    On the quantification of dynamics in reservoir computing

    David Verstraeten and Benjamin Schrauwen. On the quantification of dynamics in reservoir computing. In International Conference on Artificial Neural Networks , pages 985--994. Springer, 2009

    2009

  27. [35]

    Mathematical control theory: deterministic finite dimensional systems , volume 6

    Eduardo D Sontag. Mathematical control theory: deterministic finite dimensional systems , volume 6. Springer Science & Business Media, 2013

    2013

  28. [36]

    Stability of analog neural networks with delay

    CM Marcus and RM Westervelt. Stability of analog neural networks with delay. Physical Review A , 39(1):347, 1989

    1989

  29. [37]

    Stability and stabilizability of fuzzy-neural-linear control systems

    Kazuo Tanaka. Stability and stabilizability of fuzzy-neural-linear control systems. IEEE Transactions on Fuzzy Systems , 3(4):438--447, 1995

    1995

  30. [38]

    Frustration, stability, and delay-induced oscillations in a neural network model

    Jacques B \'e lair, Sue Ann Campbell, and Pauline van den Driessche. Frustration, stability, and delay-induced oscillations in a neural network model. SIAM Journal on Applied Mathematics , 56(1):245--255, 1996

    1996

  31. [39]

    Stability analysis of hopfield neural networks with uncertainty

    Xinzhi Liu and Robert Dickson. Stability analysis of hopfield neural networks with uncertainty. Mathematical and Computer Modelling , 34(3-4):353--363, 2001

    2001

  32. [40]

    Attractor dynamics in feedforward neural networks

    Lawrence K Saul and Michael I Jordan. Attractor dynamics in feedforward neural networks. Neural Computation , 12(6):1313--1335, 2000

    2000

  33. [41]

    Exponential input-to-state stability of recurrent neural networks with multiple time-varying delays

    Zhichun Yang, Weisong Zhou, and Tingwen Huang. Exponential input-to-state stability of recurrent neural networks with multiple time-varying delays. Cognitive neurodynamics , 8(1):47--54, 2014

    2014

  34. [42]

    Memory versus non-linearity in reservoirs

    David Verstraeten, Joni Dambre, Xavier Dutoit, and Benjamin Schrauwen. Memory versus non-linearity in reservoirs. In The 2010 international joint conference on neural networks (IJCNN) , pages 1--8. IEEE, 2010

    2010

  35. [43]

    Nonlinear dynamical systems and control: a Lyapunov-based approach

    Wassim M Haddad and VijaySekhar Chellaboina. Nonlinear dynamical systems and control: a Lyapunov-based approach . Princeton University Press, 2011

    2011

  36. [44]

    Lyapunov design for safe reinforcement learning

    Theodore J Perkins and Andrew G Barto. Lyapunov design for safe reinforcement learning. Journal of Machine Learning Research , 3(Dec):803--832, 2002

    2002

  37. [45]

    Network structure effects in reservoir computers

    Thomas L Carroll and Louis M Pecora. Network structure effects in reservoir computers. arXiv preprint arXiv:1903.12487 , 2019

    1903 arXiv

  38. [46]

    Mutual information and the edge of chaos in reservoir computers

    Thomas L Carroll. Mutual information and the edge of chaos in reservoir computers. arXiv preprint arXiv:1906.03186 , 2019

    1906 arXiv

  39. [47]

    Reservoir observers: Model-free inference of unmeasured variables in chaotic systems

    Zhixin Lu, Jaideep Pathak, Brian Hunt, Michelle Girvan, Roger Brockett, and Edward Ott. Reservoir observers: Model-free inference of unmeasured variables in chaotic systems. Chaos: An Interdisciplinary Journal of Nonlinear Science , 27(4):041102, 2017

    2017

  40. [48]

    Chaotic motion in forced duffing system subject to linear and nonlinear damping

    Tai-Ping Chang. Chaotic motion in forced duffing system subject to linear and nonlinear damping. Mathematical Problems in Engineering , 2017, 2017

    2017

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.