REVIEW 3 major objections 5 minor 27 references
Stability of Jordan Recurrent Neural Network Estimator
T0 review · 3 major / 5 minor · reviewed 2026-08-08 · deepseek-v4-flash
Pith's one-line read A Jordan recurrent neural network state estimator is proved to have stable error dynamics—input-to-state stable with the plant state as input—so that when the plant is stable the estimation error converges to zero.
desk verdict The JRN error-dynamics stability analysis is a real but incomplete contribution: the cascade theorem is fine, and the linear case is clean, but the nonlinear verification only checks a bounded box and leaves the excluded-origin linearization argument unstated. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The Jordan recurrent network defined by $a(t+1)=\sigma(W_{ay}y(t+1)+W_{ax}\hat{x}(t))$ and $\hat{x}(t+1)=W_{xa}a(t+1)$, whose recurrent connection goes from the previous output to the next hidden layer, is the central object. Substituting the plant and measurement equations turns the estimation error into the discrete-time system $e(t+1)=g(e(t),x(t))$ with $x(t)$ as an input. The paper's main instrument is the ISS-Lyapunov function $V$, satisfying $\alpha_1(\|e\|)\le V(e)\le \alpha_2(\|e\|)$ and $V(g(e,x))-V(e)\le -\alpha_3(\|e\|)+\gamma(\|x\|)$; together with the cascade theorem, this converts a certificate on the error system plus stability of the noise-free plant into a global asymptotic stability certificate for the estimator error.
What would settle it
Sample the decrease condition $V(g(e,x))-V(e)+\alpha_3(\|e\|)-\gamma(\|x\|)\le 0$ on the down-pendulum example at points outside $[-2,2]^2$ for $e$ and $x$, and run the trained JRN from an initial error larger than 2; a single persistent violation, or a trajectory whose error fails to converge, would refute the claim.
Extended reading notes
Core claim
The main contribution is an input-to-state stability analysis of the error dynamics of JRN-based estimators. The paper derives the error system $e(t+1)=g(e(t),x(t))$ from the Jordan network's forward propagation and treats the plant state $x(t)$ as an input. Theorem 3.6 shows that if this error system is ISS and the origin of the nominal plant (3) is globally asymptotically stable, then the origin of the combined cascade is globally asymptotically stable. For linear plants the ISS property is certified by a quadratic Lyapunov function obtained from a discrete Lyapunov equation; for nonlinear plants the authors synthesize an ISS-Lyapunov function with a neural network and verify its decrease and bounds constraints with SMT solvers, using linearization near the origin. The conclusion is that the trained JRN estimator, on the tested stable systems, has error dynamics that are asymptotically stable, not merely optimized.
Load-bearing premise
The nonlinear examples' stability certificate is verified only on a finite box ([-2,2] for the error states) with linearization near the origin; if a violation of the stability condition hides in the unverified region or the linearization argument is incomplete, the claimed asymptotic stability of the JRN error is not established.
Editorial extensions
If this is right
- For any stable plant, a trained bias-free JRN estimator provably drives the estimation error to zero in the noiseless case, rather than merely giving a low mean-square error.
- Small process or measurement disturbances produce a bounded error whose size scales with the disturbance level, because the error system is input-to-state stable.
- On the three test systems, the JRN achieves lower or comparable root-mean-square error than EKF and UKF while having a much lower online testing cost.
- The linear and nonlinear Lyapunov verification procedure supplies a formal stability certificate that can be produced as part of estimator design, not just checked after training.
- The same theorem applies to any estimator whose error dynamics are ISS with the plant state as input, so the argument is not limited to the exact network architecture.
Reading between the lines
- The cascade argument would transfer to any recurrent estimator architecture that can be written with state-as-input error dynamics; an LSTM variant would inherit the same stability guarantee if its error system can be shown ISS, which may be harder than for JRNs because of hidden-state recurrence.
- The nonlinear examples' numerical verification is carried out on the box $[-2,2]^2$ for the error states and linearized near the origin; a natural stress test is to verify the decrease condition on larger boxes or with a global method, since the paper does not provide a global numerical certificate for the nonlinear cases.
- The comparison uses only three benchmark systems, so the observed advantage over EKF and UKF may be particular to low-dimensional, smooth dynamics; testing on higher-order or stiff systems would clarify whether the architecture's structural advantage persists.
- The trained ISS-Lyapunov function itself could be reused as a monitoring tool: evaluating $V(e(t))$ online gives a quantitative confidence bound on the estimation error.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proposes the use of Jordan recurrent neural networks (JRNs) for state estimation of nonlinear discrete-time systems and studies the stability of the resulting estimation error. The error dynamics are written as a cascade in which the nominal state trajectory x(t) acts as an input to the error system (5). Theorem 3.6 states that if the error system is input-to-state stable (ISS) and the origin of the nominal system (3) is globally asymptotically stable, then the origin of the cascade is globally asymptotically stable. For linear systems, a quadratic ISS-Lyapunov function is obtained from a discrete Lyapunov equation. For nonlinear systems, the authors train a neural-network Lyapunov function and verify the required inequalities with the SMT solver dReal, reporting the down pendulum and reversed Van der Pol oscillator as examples. The paper also compares the JRN estimator with KF, EKF, and UKF in terms of RMSE.
Significance. If the main claim were fully established, the paper would provide a useful stability guarantee for a class of learned recurrent-neural-network estimators. The cascade theorem itself is correct and elementary, and the linear mass-spring example is essentially complete. The authors also deserve credit for providing a reproducible code link and for using machine-checked SMT verification on the reported box. The practical significance of the paper, however, rests on the nonlinear verification, and that verification is not currently global: the paper verifies only a bounded box for the error states and explicitly defers the region near the origin to an unstated linearization argument. The reversed Van der Pol example has no verification details at all. Thus the central claimed contribution, the stability of the JRN error dynamics for nonlinear systems, is not yet established as written.
major comments (3)
- [Section V, down pendulum example] The only concrete nonlinear certificate reported in the paper is the statement that a learned ISS Lyapunov function was verified by dReal on a [-2,2] x [-2,2] region "for the error states." No verification box for the nominal state x is given, and no invariant-set argument shows that the trajectories of the nominal system remain within any verified region. Since Definition 3.5 requires the Lyapunov inequalities for all e in E and all x in X, and Theorem 3.6 concludes global asymptotic stability of the cascade, a check on a bounded error box does not by itself establish the claimed asymptotic stability of the JRN error dynamics. This is the load-bearing gap between the SMT verification and the theorem's conclusion.
- [Section III, Nonlinear systems paragraph] The text acknowledges that SMT solvers require excluding a small region around the origin, that "the linearization of the nonlinear system dominates" there, and that the linear method is applied to provide stability guarantees. However, no linearized error system, no local Lyapunov matrix, no quantitative radius for the excluded region, and no proof of domination is given for either nonlinear example. Local linear stability near the origin does not imply the ISS Lyapunov inequality on the remaining unbounded set. The transition from a local linear certificate to the global stability claim is an omitted, load-bearing step.
- [Section V, reversed Van der Pol example] The text says that "for both nonlinear systems" an ISS Lyapunov function was learned and verified by an SMT solver, but only the down pendulum is described. For the reversed Van der Pol oscillator, no Lyapunov function, comparison functions, verification box, or even an explicit statement that the origin of the nominal system is globally asymptotically stable is presented. Since Theorem 3.6 requires the nominal system to be globally asymptotically stable and the error system to be ISS, the stability claim for the reversed Van der Pol example is currently unsupported.
minor comments (5)
- [Section III, Linear systems] Equation (17) appears to contain a typo: the discrete Lyapunov equation should presumably be A^T P A - P + Q = 0, not "A^T P A - A + Q = 0" as printed.
- [Section III, Linear systems] The symbol A is used both for the state matrix in (13) and later for WxaWax in the error dynamics, which makes the Lyapunov equation (17) confusing. Please use distinct symbols for these two matrices.
- [Definitions 3.4 and 3.5] The definitions use |e| and |x| without formally defining the norm on R^n, and the same notation is reused for different quantities. It would be clearer to use \|\cdot\| throughout.
- [Section III, Nonlinear systems, Eq. (19)] The displayed falsification formula (19) should be checked: as printed, the conjunction and disjunction structure does not appear to express a standard falsification query for the Lyapunov conditions on the specified valid region. Please provide the exact SMT formula that was used.
- [Section V] In Table III, the header "E(KF)" should likely read "EKF", and the text occasionally refers to figures in an inconsistent order (e.g., "Figures 2d and 2a"). These should be corrected for clarity.
Circularity Check
No significant circularity: the stability analysis is a standard ISS-cascade argument with post-hoc Lyapunov verification, not a result forced by construction or by self-citation.
full rationale
The paper's derivation chain is not circular. Theorem 3.6 is a standard composition result: if the error system (5) is ISS with x(t) as input and the nominal system (3) is globally asymptotically stable, then the cascade is globally asymptotically stable. The proof in the paper explicitly combines the KL-bound for x(t) with the ISS bound for e(t) to obtain a KL-bound for the cascade; this is a mathematical implication, not a restatement of an assumption. The ISS-Lyapunov functions for the nonlinear examples are not fitted to the stability conclusion itself: the candidates are trained using the loss (20), which penalizes violations of the Lyapunov inequalities, and are then checked by an external SMT solver (dReal) against the actual error dynamics. This is a verification step that can fail, and the paper reports the verified region and comparison functions. The self-citations to [14], [21], and [23] are not load-bearing in a circular sense: [14] is motivational, [23] describes the Lyapunov-learning framework from prior external work, and [21] supplies a proof for the linearized local argument around the origin, but the local method itself is described in the present paper. The main weaknesses—the bounded verification box and the unresolved local linearization for the nonlinear examples—are correctness and completeness risks, not circularity. No equation or theorem in the paper reduces to its own input by construction.
Assumptions & free parameters
free parameters (5)
- Trained JRN weights (Wxa, Way, Wax) =
Not reported; trained via Adam on simulated data
- Lyapunov comparison functions alpha1, alpha2, alpha3, gamma =
alpha1 = alpha3 = 0.01|s|, alpha2 = gamma = 100|s| for nonlinear examples
- Positive definite matrix Q in linear Lyapunov equation =
Q = I
- Noise covariances in simulated data =
0.01I for process, measurement, and initial condition
- Training hyperparameters =
Learning rates 0.01/0.001, 50 hidden units, batch 40, patience 10/25
assumptions (5)
- domain assumption f and h are sufficiently smooth
- domain assumption Origin of the nominal system (3) is globally asymptotically stable
- domain assumption dReal SMT solver soundly verifies the Lyapunov inequalities on the chosen box
- ad hoc to paper Linearized error dynamics dominate near the origin
- domain assumption The JRN is bias-free with identity or tanh activation
Cite this review
Pith. "Pith review of Stability of Jordan Recurrent Neural Network Estimator." pith.science (2026). https://pith.science/paper/4QYWLXVJ
@misc{pith2026250204551,
author = {Pith},
title = {Pith review of: Stability of Jordan Recurrent Neural Network Estimator},
year = {2026},
howpublished = {\url{https://pith.science/paper/4QYWLXVJ}},
note = {Machine review of arXiv:2502.04551}
}
read the original abstract
State estimation refers to determining the states of a dynamical system that starts from a noisy initial condition and evolves under process noise, based on noisy measurements and a known system model. For linear dynamical systems with white Gaussian noises of known mean and variance, Kalman filtering is a well-known method that leads to stable error dynamics for detectable systems. There are some non-optimal extensions to nonlinear systems. Recent work has used neural networks to develop estimators for nonlinear systems that optimize a criterion. Stability of the error dynamics is even more important than optimality. Jordan recurrent neural networks (JRNs) have a structure that mimics that of a dynamical system and are thus appealing for estimator design. We show that a JRN performs better than an extended Kalman filter(EKF) and unscented Kalman filter(UKF) for several examples. The main contribution of this paper is an input-to-state stability analysis of the error dynamics of JRNs. The stability of the error dynamics of several examples is shown.
Figures
Reference graph
Works this paper leans on
-
[1]
Simon, Optimal state estimation: Kalman, H ∞, and nonlinear approaches
D. Simon, Optimal state estimation: Kalman, H ∞, and nonlinear approaches. John Wiley & Sons, 2006
work page 2006
-
[2]
Long short-term memory-based deep recurrent neural networks for target tracking,
C. Gao, J. Yan, S. Zhou, P. K. Varshney, and H. Liu, “Long short-term memory-based deep recurrent neural networks for target tracking,” Information Sciences , vol. 502, pp. 279–296, 2019
work page 2019
-
[3]
Neural network based nonlinear ob- servers,
T. Breiten and K. Kunisch, “Neural network based nonlinear ob- servers,” Systems & Control Letters , vol. 148, p. 104829, 2021
work page 2021
-
[4]
Deep learning-based Luenberger observer design for discrete-time nonlinear systems,
J. Peralez and M. Nadri, “Deep learning-based Luenberger observer design for discrete-time nonlinear systems,” 60th IEEE Conference on Design and Control , pp. 4370–4375, 2021
work page 2021
-
[5]
Nonlinear modelling and prediction with feed- forward and recurrent networks,
R. Gencay and T. Liu, “Nonlinear modelling and prediction with feed- forward and recurrent networks,” Physica D: Nonlinear Phenomena , vol. 108, no. 1-2, pp. 119–134, 1997
work page 1997
-
[6]
Deep filtering with dnn, cnn and rnn,
B. Xie and Q. Zhang, “Deep filtering with dnn, cnn and rnn,” in 2024 43rd Chinese Control Conference(CCC) , pp. 8363–8368, IEEE, 2024
work page 2024
-
[7]
The new trend of state estimation: from model-driven to hybrid-driven methods,
X.-B. Jin, R. J. Robert Jeremiah, T.-L. Su, Y .-T. Bai, and J.-L. Kong, “The new trend of state estimation: from model-driven to hybrid-driven methods,” Sensors, vol. 21, no. 6, p. 2085, 2021
work page 2021
-
[8]
State estimation for recurrent neural networks with unknown delays: a robust analysis approach,
Z. Wang, J. Wang, and Y . Wu, “State estimation for recurrent neural networks with unknown delays: a robust analysis approach,” Neuro- computing, vol. 227, pp. 29–36, 2017
work page 2017
Show all 27 references
-
[9]
State estimation and tracking problems: A comparison between kalman filter and recurrent neu- ral networks,
S. K. Chenna, Y . K. Jain, H. Kapoor, R. S. Bapi, N. Yadaiah, A. Negi, V . S. Rao, and B. L. Deekshatulu, “State estimation and tracking problems: A comparison between kalman filter and recurrent neu- ral networks,” in Neural Information Processing: 11th International Conferen...
2004
-
[10]
Analysis of recurrent neural network and predictions,
J. Park, D. Yi, and S. Ji, “Analysis of recurrent neural network and predictions,” Symmetry, vol. 12, no. 4, p. 615, 2020
2020
-
[11]
Mobile phone customers churn prediction using elman and jordan recurrent neural network,
Z. Kasiran, Z. Ibrahim, and M. S. M. Ribuan, “Mobile phone customers churn prediction using elman and jordan recurrent neural network,” in 7th International Conference on Computing and Conver- gence Technology(ICCCT), pp. 673–678, IEEE, 2012
2012
-
[12]
Time series analysis of human brucellosis in mainland China by using Elman and Jordan recurrent neural networks,
W. Wu, S.-Y . An, P. Guan, D.-S. Huang, and B.-S. Zhou, “Time series analysis of human brucellosis in mainland China by using Elman and Jordan recurrent neural networks,” BMC infectious diseases , vol. 19, pp. 1–11, 2019
2019
-
[13]
A convergence result for learning in recurrent neural networks,
C.-M. Kuan, K. Hornik, and H. White, “A convergence result for learning in recurrent neural networks,” Neural Computation , vol. 6, no. 3, pp. 420–440, 1994
1994
-
[14]
State estimator design using Jordan based long short-term memory networks,
A. Kaur and K. A. Morris, “State estimator design using Jordan based long short-term memory networks,” arXiv preprint arxiv:2502.04518 , 2025
2025 arXiv
-
[15]
Goodfellow, Y
I. Goodfellow, Y . Bengio, and A. Courville, Deep learning. MIT Press, 2016
2016
-
[16]
On the input-to-state stability property,
E. D. Sontag, “On the input-to-state stability property,” European .L Control, pp. 24–36, 1995
1995
-
[17]
Output-to-state stability and detectabilty of nonlinear systems,
E. D. Sontag and Y . . Wang, “Output-to-state stability and detectabilty of nonlinear systems,” Systems & Control Letters , vol. 29, pp. 279– 290, 1997
1997
-
[18]
Extended Kalman filter-based observer design for semilinear infinite-dimensional systems,
S. Afshar, F. Germ, and K. Morris, “Extended Kalman filter-based observer design for semilinear infinite-dimensional systems,” IEEE Transactions on Automatic Control , vol. 69, no. 6, pp. 3631–3646, 2023
2023
-
[19]
Gauthier and I
J.-P. Gauthier and I. Kupka, Deterministic Observation Theory and Applications. Cambridge University Press, 2001
2001
-
[20]
Performance evaluation of ukf-based nonlinear filtering,
K. Xiong, H. Y . Zhang, and C. W. Chan, “Performance evaluation of ukf-based nonlinear filtering,” Automatica, 2006
2006
-
[21]
Physics-informed neural network Lyapunov functions: PDE characterization, learning, and verification,
J. Liu, Y . Meng, M. Fitzsimmons, and R. Zhou, “Physics-informed neural network Lyapunov functions: PDE characterization, learning, and verification,” Automatica, vol. 175, p. 112193, 2025
2025
-
[22]
J. N. Knight, Stability analysis of recurrent neural networks with applications. Colorado State University, 2008
2008
-
[23]
Neural Lyapunov control of unknown nonlinear systems with stability guarantees,
R. Zhou, T. Quartz, H. De Sterck, and J. Liu, “Neural Lyapunov control of unknown nonlinear systems with stability guarantees,” Ad- vances in Neural Information Processing Systems , vol. 35, pp. 29113– 29125, 2022
2022
-
[24]
Neural Lyapunov control,
Y .-C. Chang, N. Roohi, and S. Gao, “Neural Lyapunov control,” Advances in Neural Information Processing Systems , vol. 32, 2019
2019
-
[25]
Input-to-state stability for discrete-time nonlinear systems,
Z.-P. Jiang and Y . Wang, “Input-to-state stability for discrete-time nonlinear systems,” Automatica, vol. 37, no. 6, pp. 857–869, 2001
2001
-
[26]
FOSSIL: a software tool for the formal synthesis of Lyapunov func- tions and barrier certificates using neural networks,
A. Abate, D. Ahmed, A. Edwards, M. Giacobbe, and A. Peruffo, “FOSSIL: a software tool for the formal synthesis of Lyapunov func- tions and barrier certificates using neural networks,” in Proceedings of the 24th International Conference on Hybrid Systems: Computation and Contro...
2021
-
[27]
dreal: An SMT solver for nonlinear theories over the reals,
S. Gao, S. Kong, and E. M. Clarke, “dreal: An SMT solver for nonlinear theories over the reals,” in International Conference on Automated Deduction , pp. 208–214, Springer, 2013
2013
Reviewed August 8, 2026 · model on record in the stance chip above.
Discussion (0). Continue with ORCID to comment.