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State estimator design using Jordan based long short-term memory networks

T0 review · 3 major / 5 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read A Jordan-style LSTM filters states as accurately as an Elman LSTM but trains far faster, and both beat the extended Kalman filter on nonlinear systems.

desk verdict A modest but real LSTM variant whose claimed speed advantage is confounded by unequal learning rates; worth refereeing, not citing yet. read the letter →

arxiv 2502.04518 v1 pith:OXGQG2IO submitted 2025-02-06 math.OC

Avneet Kaur , Kirsten Morris This is my paper · ORCID
classification math.OC MSC 93E1168T0793B07
keywords stateestimationlongshort-termmemorynetworksJordanrecurrentneuralElmanextendedKalmanfilternonlinearfilteringtrainingtimeuniversalapproximation
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

State estimation for nonlinear systems is hard because optimal filters require solving a difficult Hamilton-Jacobi-Bellman equation, so this paper asks whether recurrent neural networks can serve as practical estimators. It introduces a Jordan-style long short-term memory network (JLSTM) in which the previous state estimate, rather than the previous hidden state, is fed back into the LSTM gates, a structure that mirrors the dynamics of the system being estimated. The paper reports that on a ten-spring linear system and two nonlinear systems (down pendulum and reversed Van der Pol oscillator), the JLSTM achieves normalized mean-square error comparable to an Elman LSTM while taking substantially less training time, and both networks beat the extended Kalman filter on the nonlinear examples. The theoretical part shows that Jordan recurrent networks are universal approximators for a broad class of state estimators, and the paper positions the work as a step toward optimal estimation of high-order nonlinear systems.

What carries the argument

The central object is the JLSTM cell, defined by forward-propagation equations in which the four gate and cell-update operations use $\hat{x}(t-1)$, the previous state estimate, in place of the previous hidden vector $a(t-1)$ used by an ELSTM. This recurrent connection type is inherited from Jordan recurrent neural networks, whose estimator form is $\hat{x}(t+1)=W_{xa}\sigma(W_{ay}y(t)+W_{ax}\hat{x}(t)-b)$; the paper proves a universal approximation theorem showing that such recurrences can approximate any continuous state estimator of the form $\bar{x}(t+1)=F(\bar{x}(t),y(t))$. The architecture matters because it makes the network's recurrence conform to the structure of the discrete-time dynamical system, and that structural match is what the paper credits for faster loss minimization during training.

What would settle it

Train both networks on the down-pendulum or van der Pol example using the same learning rate, the same early-stopping patience, and the same number of hidden units, then record epochs and NMSE; if JLSTM does not reach comparable error in fewer epochs, the paper's central claim about faster training is unsupported.

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Extended reading notes

Core claim

The central discovery is empirical: changing the feedback path in an LSTM state estimator from hidden-state feedback (Elman) to previous-estimate feedback (Jordan) preserves estimation accuracy while reducing training time. For the three systems tested, the normalized mean-square errors of JLSTM and ELSTM are close, and the training time for JLSTM is smaller in every example (15983 versus 16298 seconds for the springs, 95644 versus 99473 for the pendulum, and 619 versus 3389 for the van der Pol oscillator). Both LSTM filters have lower NMSE than the extended Kalman filter for the nonlinear systems and NMSE comparable to the Kalman filter for the linear system. The authors conclude that using a JLSTM instead of an ELSTM appears preferable because the JLSTM reaches the same error with considerably less training, and they attribute the advantage to the Jordan structure's direct use of the previous state estimate, which resembles the discrete-time dynamics being estimated.

Load-bearing premise

The training-time comparison assumes that the different learning rates chosen for ELSTM and JLSTM, selected so the final errors would match, are fair choices, and that the faster convergence comes from the Jordan feedback rather than from the more favorable learning rate.

Editorial extensions

If this is right

  • On the two nonlinear test systems, both JLSTM and ELSTM yield lower normalized mean-square error than the extended Kalman filter, suggesting that the nonlinear activation in LSTM cells reduces the linearization error that limits the EKF.
  • When initial conditions lie outside the training range, both LSTM estimators still beat the EKF on the nonlinear examples, so the learned filters appear to generalize beyond the training distribution in at least these cases.
  • For the linear spring system, JLSTM and ELSTM errors are close to the Kalman filter's error, while the KF remains the best estimator when testing outside the training range.
  • JLSTM requires substantially less training time than ELSTM across all three examples while reaching similar error, making it the preferable structure for these problems.
  • Testing time for both LSTM filters is much smaller than for the KF and EKF on the tested systems, with the LSTM filters taking 0.2 to 2.7 seconds versus 0.8 to 38.2 seconds for ten test sequences.

Reading between the lines

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

  • If the faster training is caused by the Jordan feedback rather than by unequal learning-rate settings, the same advantage should appear when both networks are trained with identical learning-rate schedules; this is a direct test of the architectural claim.
  • The structural match between the JLSTM and the state-space model suggests the architecture may extend to higher-order nonlinear systems, the paper's stated future goal, but this remains untested because the nonlinear examples are only second-order.
  • Because the JLSTM feeds back the state estimate itself, its recurrence resembles an observer, so existing observer theory could potentially yield convergence and stability guarantees rather than purely empirical accuracy.
  • The observation that both LSTM filters outperform the EKF outside the training range on nonlinear examples hints that learned estimators may be less sensitive to initial-condition mismatch than local linearization, which would matter for applications with uncertain starting states.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper introduces a Jordan-based long short-term memory (JLSTM) estimator for discrete-time state estimation, in which the previous state estimate, rather than the previous hidden state, is fed back into the gate and cell-input units. It states a universal approximation theorem for Jordan recurrent networks (Theorem 2.11), describes the ELSTM and JLSTM implementations, and compares them with the Kalman filter on a 20-state linear spring system and with the extended Kalman filter on a pendulum and a reversed van der Pol oscillator. The reported results claim that both LSTM filters achieve lower NMSE than the EKF on the nonlinear examples, that JLSTM matches ELSTM estimation error, and that JLSTM trains considerably faster than ELSTM.

Significance. If the claims were established, the paper would make a modest but useful contribution: a simple architectural variant that feeds the state estimate back into LSTM gates, with evidence that recurrent neural filters can outperform the EKF on low-order nonlinear systems and that the Jordan feedback structure can reduce training time. The paper's strengths are the use of standard benchmark examples, the inclusion of out-of-distribution initial-condition tests, and the transparent reporting of the network hyperparameters and training/test times. However, the central comparative claim is not yet supported: the training-time comparison is confounded by unequal learning rates chosen to equalize final errors, and the linear-system result appears inconsistent with Kalman optimality. The universal approximation theorem, although presented as supporting motivation, also has a proof gap. These issues are local and can be addressed with additional experiments and a revised proof, so the manuscript merits revision rather than rejection.

major comments (3)
  1. [Section 2, Theorem 2.11] The proof of Theorem 2.11 establishes only a one-step uniform approximation of the map F on a compact set K, as in Eq. (7), and then asserts that continuity of F yields the recurrent bound ||\bar{x}(t) - \hat{x}(t)|| < epsilon. This does not follow: even if the network approximates F when evaluated at the same argument, the recurrent error can grow through the recursion unless a Lipschitz or contraction bound is imposed and an induction over t is carried out. In addition, K is assumed to contain both trajectories, but the network-generated trajectory \hat{x}(t) cannot be assumed a priori to remain in K. As written, the theorem is unproved.
  2. [Section 5 and Table 2] The paper's main conclusion, stated in Section 6 as 'Using a JLSTM instead of an ELSTM appears preferable because the JLSTM network has considerably smaller training time to achieve the same error,' rests on Table 2. For the down pendulum and reversed van der Pol examples, Sections 5.2 and 5.3 explicitly set different Adam learning rates for ELSTM and JLSTM (1e-4 versus 1e-3, and 1e-3 versus 1e-2) in order to make the final NMSE values comparable. A higher learning rate can reduce the number of epochs needed to reach a given loss, so the reported 5.5x training-time advantage on the van der Pol example cannot be attributed to the Jordan feedback structure. The other two examples show only 2-4% differences. No repeated runs, seeds, epoch counts, or error bars are reported, and training time is wall-clock time, which mixes epoch count with per-epoch cost. The authors should compare both architectures at matched learning rates, or report full training-time-versus-error curves over a range of learning rates with several random seeds.
  3. [Table 1, connected springs] The connected-springs model is linear with Gaussian noises, for which the Kalman filter is optimal. Yet Table 1 reports NMSE 0.0153 for ELSTM and 0.0162 for JLSTM, both below the Kalman filter's 0.0174. Unless the KF was deliberately initialized with a mismatched covariance, or the finite test set makes the difference statistically insignificant, this contradicts KF optimality. Please clarify the KF initialization, report per-sequence variance or confidence intervals, and reconcile this result with the statement in Section 5.1 that 'theory shows the KF to be the best estimator for a linear system.'
minor comments (5)
  1. [Section 6] The first paragraph of Section 6 refers to 'a 100 dimensional linear system,' but the connected-springs example has a 20-dimensional state vector; please correct this.
  2. [Equation (12)] The quantity in Eq. (12) is called NMSE, but no normalization by the variance of the true signal is displayed; please define the normalization explicitly or rename the metric.
  3. [Section 3] The statement 'simple recurrent networks are a special case of long short term memory networks where the gates are not involved' is asserted without showing how the gate equations (8) and (9) reduce to the JRN/ERN recursions (2) and (3); a one-sentence construction would make the transfer of Theorem 2.11 to LSTMs transparent.
  4. [Figure 2] The caption of Figure 2 mentions a bias 'by' that does not appear in Eq. (3); please clarify whether this is a typo or whether a bias term is missing from the forward-propagation equations.
  5. [Table 3] The testing-time comparison in Table 3 may be unfair if the KF/EKF is implemented in Python on a CPU while the LSTM models run on a T4 GPU; please state the hardware and software used for all estimators.

Circularity Check

1 steps flagged · score 2.0 of 10

No central circularity: empirical LSTM comparisons are measurements; one auxiliary universal-approximation proof has a self-referential compact-set step.

  1. other [Theorem 2.11 proof, after Eq. (7)]
    "Let ϵ > 0 and K ⊆ Rn × Rm be a compact set which contains (¯x(t), y(t)) and (ˆx(t), y(t)), ∀t = 1, 2, ..., T. From Theorem 2.8 and Corollary 2.9, we know that ... a function N N(¯x(t), y(t)) = Wxaσ(Wayy(t) + Wax ¯x(t) − b) ... exists, such that ... sup ∥F (¯x(t), y(t)) − N N(¯x(t), y(t))∥∞ < δ."

    The compact set K on which Hornik's theorem is invoked is required to contain the trajectory \xhat(t) of the very recurrent network NN that the theorem is supposed to construct. Since NN is chosen using K, and K is defined using \xhat (which is generated by NN), the proof has a self-referential dependency: it presupposes the approximating network's trajectory in order to define the domain of approximation. A non-circular proof would fix K from the target estimator F and then prove, by continuity and small approximation error, that the constructed NN's trajectory stays in a slightly enlarged compact set. This is a technical proof gap in an auxiliary theoretical motivation, not in the empirical claims.

full rationale

The paper's central claims are empirical: NMSE values (Table 1), training and testing times (Tables 2 and 3), and out-of-distribution NMSE (Table 4) are measurements, not quantities forced by construction. The JLSTM/ELSTM training-time comparison is weakened by unequal learning rates chosen to equalize final error (Section 5: 1e-4 vs 1e-3 for the pendulum, 1e-3 vs 1e-2 for van der Pol), but that is an experimental-control concern, not a circular derivation; the learning rates are disclosed and the claim is falsifiable by matched-learning-rate runs. The self-citation to Kaur et al. (2024) is motivational and not load-bearing for the empirical results. The one genuine circularity signal is in the proof of the auxiliary universal approximation theorem for JRNs (Theorem 2.11), where the compact set K is defined in terms of the approximating network's trajectory before that network is constructed. Because this theorem is used only as motivation and the empirical comparison is self-contained against external benchmarks (KF/EKF, ELSTM), the overall circularity score is low.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The only new object is the JLSTM architecture itself, which is a computational structure rather than a postulated physical entity; no new particles, forces, dimensions, or conserved quantities are introduced. The empirical claim depends on hand-chosen network hyperparameters and on domain assumptions about the model and noise used for data generation.

free parameters (3)
  • Learning rates = 1e-3/1e-3 (springs), 1e-4/1e-3 (pendulum), 1e-3/1e-2 (van der Pol) for ELSTM/JLSTM
    Chosen by hand to make final NMSE values comparable; the training-time comparison hinges on these choices.
  • Number of hidden units = 50 for both networks
    Set by hand for all examples; affects capacity and training time but is not swept.
  • Early stopping patience and max epochs = patience 50/50/15, max epochs 8000/3000/3000
    Hand-chosen per example; early stopping directly determines the measured training time.
assumptions (4)
  • standard math Hornik-Stinchcombe-White universal approximation theorem for feedforward networks
    Invoked in the proof of Theorem 2.11 as the external approximation benchmark.
  • domain assumption F and h are measurable/continuous and relevant trajectories lie in a compact set
    Used in Theorem 2.11 to apply uniform approximation and continuity arguments; not proved for the specific estimators or systems.
  • ad hoc to paper Simple recurrent networks are a special case of LSTMs when gates are not involved
    Section 3 uses this assertion to transfer the JRN universal approximation result to LSTM architectures, but the LSTM gating equations do not literally reduce to the JRN update without specifying gate values; the transfer is asserted, not shown.
  • domain assumption Initial condition, process noise, and measurement noise are Gaussian with known covariance 0.01I
    Section 4 generates training data from the known model under this noise model; if deployment noise differs, the performance claims need not hold.

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Pith. "Pith review of State estimator design using Jordan based long short-term memory networks." pith.science (2026). https://pith.science/paper/OXGQG2IO

@misc{pith2026250204518,
  author       = {Pith},
  title        = {Pith review of: State estimator design using Jordan based long short-term memory networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OXGQG2IO}},
  note         = {Machine review of arXiv:2502.04518}
}
read the original abstract

State estimation of a dynamical system refers to estimating the state of a system given an imperfect model, noisy measurements and some or no information about the initial state. While Kalman filtering is optimal for estimation of linear systems with Gaussian noises, calculation of optimal estimators for nonlinear systems is challenging. We focus on establishing a pathway to optimal estimation of high-order systems by using recurrent connections motivated by Jordan recurrent neural networks(JRNs). The results are compared to the corresponding Elman structure based long short-term memory network(ELSTM) and the KF for linear and EKF for nonlinear systems. The results suggest that for nonlinear systems, the use of long short-term memory networks can improve estimation error and also computation time. Also, the Jordan based long short-term memory networks(JLSTMs) require less training to achieve performance similar to ELSTMs.

Figures

Figures reproduced from arXiv: 2502.04518 by the authors.

Figure 1
Figure 1. The structure of an Elman recurrent neural network(ERN) for state estimation. It uses hidden to hidden recurrent connections. The symbols y (t) , a(t) , x(t) and ˆx (t) represent input measurement vector, hidden layer vector, true state vector and estimated state vector respectively at time t. The cost function J is considered to be mean squared error(MSE). The weights and biases are represented by Way, Waa, Wxa, ba… view at source ↗
Figure 2
Figure 2. The structure of the Jordan recurrent network(JRN) proposed for state estimation. It uses output￾to-hidden recurrent connections similar to the filter’s dynamical system. The symbols y (t) , a(t) , x(t) and ˆx (t) represent input measurement vector, hidden layer vector, true state vector and estimated state vector respec￾tively at time t. Cost function J (t) is the mean squared error at time t. The weights and biase… view at source ↗
Figure 3
Figure 3. Structure of an Elman long short term memory network(ELSTM) for state estimation. It uses hidden to hidden recurrent connections for state estimation of a nonlinear dynamical system. The symbols y (t) , a(t) and ˆx (t) represent input measurement vector, hidden layer vector and estimated state vector respec￾tively.The forget, input and output gates are represented by f (t) , i(t) and o (t) . The cell state vector an… view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: Structure of a Jordan long short term memory network(JLSTM) for state estimation. It uses output to hidden recurrent connections. The symbols y (t) , a(t) and ˆx (t) represent input measurement vector, hidden layer vector and estimated state vector respectively. The fo…
Figure 5
Figure 5. Figure 5: This figure compares the performance of KF, JLSTM and ELSTM using the average errors over 10 test sequences for 50 seconds of 10 connected springs with a noisy Gaussian initial condition. at time-step t is given by y (t) =       x (t) 1 x (t) 2 . . . x (t) 10  …
Figure 6
Figure 6. Figure 6: This figure compares the performance of KF, JLSTM and ELSTM using the average errors over 10 test sequences for 50 seconds of 10 connected springs with a noisy Gaussian initial condition outside the training region. 5.2. Nonlinear system: down pendulum We considered an…
Figure 7
Figure 7. Figure 7: This figure compares the performance of EKF, JLSTM and ELSTM using the normalised mean squared errors over 10 test sequences for 40 seconds of down pendulum with a noisy Gaussian initial condition [PITH_FULL_IMAGE:figures/full_fig_p015_7.png]
Figure 8
Figure 8. Figure 8: This figure compares the performance of EKF, JLSTM and ELSTM using the normalised mean squared errors over 10 test sequences for 40 seconds of down pendulum with a noisy Gaussian initial condition outside the training region [PITH_FULL_IMAGE:figures/full_fig_p015_8.png]
Figure 9
Figure 9. Figure 9: This figure compares the performance of EKF, JLSTM and ELSTM using the average errors over 10 test sequences for 30 seconds of reversed Van der Pol oscillator with a noisy Gaussian initial condition. 5.3. Nonlinear system: reversed Van der Pol oscillator We considered …
Figure 10
Figure 10. Figure 10: This figure compares the performance of EKF, JLSTM and ELSTM using the average errors over 10 test sequences for the first 30 seconds of reversed Van der Pol oscillator with a noisy Gaussian initial condition outside the training region [PITH_FULL_IMAGE:figures/full_…

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    2018

Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.