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REVIEW 4 major objections 6 minor 37 references

Systems of ODEs Parameters Estimation by Using Stochastic Newton-Raphson and Gradient Descent Methods

T0 review · 4 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read This paper claims that ODE parameters can be estimated directly by fitting finite-difference derivatives to the system's right-hand side with Newton-Raphson or gradient descent updates, and that these direct methods beat nonlinear least…

desk verdict Derivative matching with a non-uniform grid twist, but the 'outperforms NLS' claim rests on a constrained baseline that is clearly stuck at bounds. read the letter →

arxiv 2501.12856 v1 pith:K7DNDZOD submitted 2025-01-22 math.NA cs.NA

S. Syafiie , Aries Subiantoro , Vivi Andasari , Fernando Tadeo This is my paper · ORCID
classification math.NAcs.NA MSC 65L0965K1034A55
keywords parameterestimationsystemsofODEsNewton-RaphsongradientdescentstochasticdiscretederivativeTaylorexpansionnonlinearleastsquares
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 simpler route to estimating unknown parameters in systems of ordinary differential equations: instead of repeatedly solving the ODE and matching predicted states to measurements, it replaces the derivative $x'(t)$ by a finite difference computed from the data and solves the residual equation $f(t,x,a)-x'(t)=0$ with Newton-Raphson or gradient descent. This direct formulation avoids numerical ODE integration during fitting, and the paper argues it yields parameters that are closer to the true values than those from the standard nonlinear least squares approach. The authors test the idea on a predator-prey system, the chaotic Lorenz system, and an activator-inhibitor model, reporting that Newton-Raphson converges in very few iterations while gradient descent handles chaotic dynamics robustly, and that stochastic subsampled versions scale to large data. If the claim holds, it offers a fast, simple alternative for fitting ODE models to noisy observations.

What carries the argument

The object that carries the argument is the residual vector $E(t,x,\bar a)=f(t,x,\bar a)-x'(t)$, with the derivative $x'(t)$ obtained from a finite-difference formula (first-order from two consecutive time points, or second-order from three points). Linearizing $f$ by Taylor expansion converts the residual into a linear system in $\Delta a$, enabling the Newton-Raphson update $a_{i+1}=a_i-\gamma\,\nabla_a F\,E$ and the gradient descent update $a_{i+1}=a_i-\eta\,\nabla_a G\,G$. The machinery includes the Jacobian $\nabla_a F$ (static for systems linear in parameters, updated each iteration otherwise), the explicit step-size rule for GD, and the random subsampling that defines the stochastic variants SNR and SGD.

What would settle it

Generate Lorenz-system data with known parameters and additive noise, compute $x'(t)$ using the paper's equation (12), and check whether the NR/GD estimates converge to the true parameters as the number of data points $N$ grows. If the estimates remain biased even for large $N$, or if the residual objective's global minimum is not at the true parameter vector, the central claim fails.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that parameter estimation for systems of ODEs can be posed as a direct residual-minimization problem in parameter space using discrete derivatives and Taylor expansion, solvable by Newton-Raphson (NR) and gradient descent (GD). The residual is $E(t,x,\bar a)=f(t,x,\bar a)-x'(t)$, where $x'(t)$ is approximated by the first- or second-order finite differences in equations (6) and (12); expanding $f$ in a Taylor series in $a$ gives the linear system $E=-\nabla_a F\,\Delta a$, which NR solves with the pseudo-inverse step $\gamma=(\nabla_a F^\top\nabla_a F)^{-1}$. GD instead uses the explicit step size $\eta=\delta^\top G/\delta^\top\delta$ with $\delta=\nabla_a G\,\nabla_a G^\top G$. Stochastic versions randomly select $r$ of the $n$ ODE equations each iteration to cut cost. Across three numerical examples, the estimated parameters from both methods are closer to the generating parameters than those from constrained NLS, with error metrics closer to zero and $R^2$ closer to one.

Load-bearing premise

The finite-difference derivative computed from noisy data is close enough to the true derivative that the residual $f(t,x,a)-x'(t)$ has its minimum at the true parameters, an assumption that is especially fragile for chaotic systems where state errors amplify.

Editorial extensions

If this is right

  • Newton-Raphson on the direct derivative residual can converge in one or two iterations for systems linear in parameters, giving very fast estimates from clean or mildly noisy data.
  • Gradient descent with the explicit step size provides a robust fallback for chaotic systems, where Newton-Raphson may jump to a poor local solution.
  • Stochastic subsampling of the ODE equations makes the method scalable to large data sets by reducing the per-iteration Jacobian and residual size.
  • If the accuracy comparison to NLS holds, the direct method is a practical choice for real data fitting when the ODE lacks an explicit solution and numerical integration is expensive.
  • The finite-difference formulas accommodate irregular sampling intervals, so the method applies directly to experimental data with varying time steps.

Reading between the lines

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

  • Beyond the paper: the direct derivative-matching objective is a variant of gradient matching, and its bias under measurement noise is well known; a natural extension would be to derive the asymptotic bias of the NR/GD estimates and to add smoothing or regularization to the derivative estimate.
  • Beyond the paper: the authors compare against NLS with the same initial guesses, but do not report the basin of attraction for GD on the Lorenz system; measuring how often random restarts converge to the true parameters would sharpen the robustness claim.
  • Beyond the paper: the stochastic variants randomly subsample equations, not data points; an alternative extension is to subsample time points or use mini-batches of the residual, which could further reduce cost on very long time series.
  • Beyond the paper: the finite-difference derivative amplifies noise in chaotic systems, so a testable extension is to repeat the Lorenz experiment at several noise levels and check whether the parameter estimates remain closer to truth than NLS when the derivative is noisier.
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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

4 major / 6 minor

Summary. The paper proposes a direct derivative-matching approach for estimating parameters in systems of ODEs. The ODE residual is formed by comparing a finite-difference approximation of the state derivative (Eqs. (6) and (12)) with the model right-hand side f(t,x,a), and the parameters are updated by Newton-Raphson (Eq. (16)) or gradient descent (Eq. (30)) iteratively, with stochastic variants defined in Algorithms 2 and 4. The method is demonstrated on population dynamics, Lorenz, and activator-inhibitor examples, and the abstract claims that the proposed methods outperform nonlinear least squares (NLS). Convergence theorems are stated in Section 2.1 (Theorem 1) and Section 2.2 (Theorem 2).

Significance. If the central claim were established, the derivative-matching formulation would offer a computationally cheap alternative to NLS for ODE parameter estimation. However, the manuscript has load-bearing deficiencies: the convergence proof contains an impossible inequality, the numerical comparison to NLS is not apples-to-apples because the NLS baseline is constrained while the proposed methods are unconstrained, the stochastic variants are never numerically evaluated, and the treatment of noisy finite-difference derivatives is not analyzed. The paper does not currently provide a credible demonstration of its headline claim of improved accuracy over NLS.

major comments (4)
  1. [Section 2.2, Eq. (39)] Equation (39) states "0 ≥ δ⊤δ = G(t, x, ai)⊤ Π² G(t, x, ai) + O(∥μi∥³)". Since δ⊤δ is a sum of squares, it is always nonnegative, so the inequality cannot be correct as written; the only consistent statement would be 0 ≤ δ⊤δ. This sign error invalidates the derivation of the linear convergence bound in Eq. (41), which relies on the incorrect direction of the inequality. This is a load-bearing error in the GD convergence analysis.
  2. [Section 3.1.4, 3.2.4, and 3.3.4] The claim that the proposed methods outperform NLS is not supported by the reported experiments because the comparison is not fair: the NLS baseline is solved as a constrained optimization with bounds (Eqs. (52), (57), (62)), while the proposed methods are unconstrained. The reported NLS solutions sit at or near the active bounds (a3=2 in Eq. (58) and a3≈0.052 in Eq. (63)), suggesting that the NLS solver is finding constrained local optima rather than the true unconstrained optimum. No multi-start, solver tolerance, or convergence diagnostics are reported for NLS. Therefore Tables 3, 7, and 10 do not establish the paper's central claim of outperforming NLS.
  3. [Section 3 (numerical implementation)] The stochastic versions SNR and SGD are defined in Algorithms 2 and 4, but the numerical examples only apply the deterministic NR and GD methods (Algorithms 1 and 3). No results are reported for the stochastic variants, despite the title, abstract, and conclusion emphasizing their contribution for large-scale systems. The statement in the introduction to Section 3 that the application of Algorithms 2 and 4 is presented in Python is not followed by any stochastic numerical experiment. Consequently, the claims about scalability and reduced computational cost for big data are unsupported.
  4. [Section 2, Eqs. (6) and (12)] The objective E in Eq. (14) is built from finite-difference derivatives of noisy data. The paper does not analyze how noise propagates through these derivative approximations or how that affects the estimated parameters. For the chaotic Lorenz system in Section 3.2, small state errors amplify exponentially, and the reported results are based on single realizations, so the accuracy claims are not demonstrable. Since the entire method rests on the reliability of these derivative approximations, this missing analysis is a load-bearing gap.
minor comments (6)
  1. [Title and throughout] The title contains a typo: "ODE S" should read "ODEs".
  2. [Section 3.1, Eq. (45)] The second equation writes "dx2/dx" instead of "dx2/dt".
  3. [Section 2.1, Eq. (16)] The step length γ is defined as the inverse of the Gram matrix of the Jacobian, which is appropriate only when the Jacobian has full column rank; the paper does not state this assumption.
  4. [Section 2.2, Eq. (31)] The citation "Barzilai and Bowein, 1988" should be "Barzilai and Borwein, 1988".
  5. [Section 3.2.3, text near Table 4] The sentence "The optimal estimated parameters ¯a int Table 4" contains a typo: "int" should be "in".
  6. [Section 3.2.3, Table 4] The layout of the NR row in Table 4 is confusing because the convergent parameter vector is placed in the last cell, making it difficult to associate it with the corresponding initial guess.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the derivative-residual parameter estimation is self-contained; reported estimates are simulation fits, not predictions forced by construction.

full rationale

The paper's derivation chain builds an objective E(t,x,a) = f(t,x,a) - discrete derivative (Eq. (14), with discrete derivative from Eq. (6)/(12)) and then solves for parameters via Newton-Raphson (Eq. (16)) or gradient descent (Eq. (30)). This is a standard gradient-matching formulation, and the parameter updates are derived from Taylor expansions of the residual, not from the target parameters themselves. The numerical experiments generate data from known parameters, add noise, and fit parameters, reporting the fitted values as estimates; no held-out prediction is claimed, and no fitted parameter is renamed as a prediction. The comparison to NLS is a benchmark comparison, not part of the derivation, and any weakness in that comparison (constrained NLS, possible local minima) is a correctness concern, not circularity. The paper contains no self-citations that carry a load-bearing role, and no uniqueness theorem or prior-work ansatz is imported to force the chosen formulation. The central method's behavior therefore rests on its own stated equations and data, so the derivation is not equivalent to its inputs by construction.

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

The method introduces no new entities. The central claim rests on the accuracy of finite-difference derivatives from noisy data, an unstated tolerance, and a specific ad hoc step size for gradient descent. The stochastic variants add the unstated mini-batch size r.

free parameters (4)
  • Taylor expansion order n = not specified
    The paper derives first-order (Eq. 6) and second-order (Eq. 12) derivative formulas, but never states which is used in each experiment. This choice affects the residual E and therefore the fitted parameters.
  • GD step size eta (Barzilai-Borwein style delta) = computed by Eq. (31)
    The step size is set by an explicit formula introduced ad hoc to guarantee the descent property (32). The convergence proof relies on this choice.
  • Stopping tolerance epsilon = not specified
    Algorithms 1-4 terminate when |a_i - a_{i-1}| > epsilon, but epsilon is never given. All reported iteration counts depend on this unstated value.
  • Stochastic mini-batch size r for SNR/SGD = not specified
    The SNR and SGD methods require selecting r ODE equations per iteration, but r is never defined and the stochastic methods are not tested.
assumptions (4)
  • standard math Taylor's theorem applies and the remainder is negligible on the observed grid
    Section 2, Eq. (4)-(12): the discrete derivative formulas follow from truncated Taylor expansions. The paper does not bound the remainder.
  • domain assumption The finite-difference derivative of noisy data approximates the true derivative well
    Section 2, Eq. (14): the residual E uses x'(t_i) from data. Noise amplification in finite differences is not analyzed, and no regularization is applied.
  • ad hoc to paper The ODE model f is twice continuously differentiable with non-vanishing Jacobian at the optimum
    Theorem 1 assumes this to state quadratic convergence; it is not established for the example systems, especially the non-smooth cases.
  • domain assumption Noise is additive, Gaussian, and not filtered
    Section 3: 'data are added noise with normal distribution' and 'data are not filtered'. The results are for this specific noise model only.

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Cite this review

Pith. "Pith review of Systems of ODEs Parameters Estimation by Using Stochastic Newton-Raphson and Gradient Descent Methods." pith.science (2026). https://pith.science/paper/K7DNDZOD

@misc{pith2026250112856,
  author       = {Pith},
  title        = {Pith review of: Systems of ODEs Parameters Estimation by Using Stochastic Newton-Raphson and Gradient Descent Methods},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/K7DNDZOD}},
  note         = {Machine review of arXiv:2501.12856}
}
read the original abstract

Ordinary differential equations (ODEs) are widely used to describe the time evolution of natural phenomena across various scientific fields. Estimating the parameters of these systems from data is a challenging task, particularly when dealing with nonlinear and high-dimensional models. In this paper, we propose novel methodologies for parameter estimation in systems of ODEs by using the Newton-Raphson (NR) method and Gradient Descent (GD) method. By leveraging the discrete derivative and Taylor expansion, the problem is formulated in a way that enables the application of both methods, allowing for flexible, efficient solutions. Additionally, we extend these approaches to stochastic versions - Stochastic Newton-Raphson (SNR) and Stochastic Gradient Descent (SGD) - to handle large-scale systems with reduced computational cost. The proposed methods are evaluated by using numerical examples, including both linear and nonlinear parameter models, and compare the results to the well-known Nonlinear Least Squares (NLS) method. While NR converges rapidly to the optimal solution, GD demonstrates robustness in handling chaotic systems, though it may occasionally lead to suboptimal results. Overall, the proposed methods provide improved accuracy in parameter estimation for ODE systems, outperforming NLS in terms of error metrics such as bias, mean absolute error (MAE), mean absolute percentage error (MAPE), root mean square error (RMSE), and coefficient of determination R2. These methods offer a valuable tool for fitting ODE models, particularly in scenarios involving big data and complex dynamics.

Figures

Figures reproduced from arXiv: 2501.12856 by the authors.

Figure 1
Figure 1. population dynamic Example: scattered plots are original data, solid lines are estimated data [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. first example NLS fitting: dotted is data, continuous line is approximated data [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Lorenz’s system example: marked plots are original data, solid lines are estimated data [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Lorenz’s system example NLS method: marked plots are original data, solid lines are estimated data by using [PITH_FULL_IMAGE:figures/full_fig_p015_4.png]
Figure 5
Figure 5. Figure 5: Activator-inhibitor example implementation: scatter plots are original data, solid lines are estimated data [PITH_FULL_IMAGE:figures/full_fig_p017_5.png]
Figure 6
Figure 6. Figure 6: Activator-inhibitor example: marked plots are original data, solid lines are estimated data by using NLS [PITH_FULL_IMAGE:figures/full_fig_p017_6.png]

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