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REVIEW 5 major objections 5 minor 26 references

Discovering Interpretable Ordinary Differential Equations from Noisy Data

T0 review · 5 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper claims that a homogeneous linear constant-coefficient ODE can be recovered from noisy data by fitting its general solution, smoothing with a B-spline, and taking the null space of the resulting gradient matrix.

desk verdict New smoothing-plus-SVD pipeline for linear ODE discovery that works when the order is known but lacks a valid null-space identification when the order is over-specified, and the empirical case studies lean on manual thresholding and synthetic data. read the letter →

arxiv 2507.21841 v1 pith:X3ADYSMX submitted 2025-07-29 cs.LG physics.comp-ph

Rahul Golder , M. M. Faruque Hasan This is my paper
classification cs.LGphysics.comp-ph MSC 34A3065D0765F20
keywords ODEdiscoverysparseidentificationnoisydataB-splineapproximationsingularvaluedecompositiongeneticalgorithmmodelinversemodeling
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 proposes a three-step pipeline that recovers a governing ordinary differential equation from noisy measurements without pre-specifying a library of candidate basis functions. It assumes the true equation is a homogeneous linear constant-coefficient ODE, and works by fitting the analytical general solution of that ODE family to the data, smoothing the fitted curve with an adaptive B-spline, and reading the ODE coefficients out of the null space of a gradient matrix built from spline derivatives. On spring-mass benchmarks and on experimental photolysis data for estrogen-disrupting chemicals, the recovered equations match the expected second-order or first-order forms and yield physical parameters such as mass, damping, and rate constants. The paper reports that sparsity in the recovered coefficients emerges without any regularization, that the method tolerates Gaussian noise better than derivative-based sparse regression baselines, and that higher-order or variable-coefficient systems remain outside its current scope.

What carries the argument

The load-bearing object is the gradient matrix $\mathbf G$ whose entries are spline derivatives $G_{p,k}=d^p y(x_k)/dx^p$; because the spline is a polynomial of order $P$, its derivatives $x^0, x^1, \dots, x^P$ are linearly independent, so when the data obey a unique governing equation the reduced matrix $\tilde{\mathbf G}$ is rank-deficient with a one-dimensional null space. Singular value decomposition $\tilde{\mathbf G}=\mathbf U\boldsymbol\Sigma\mathbf V^\top$ yields the coefficient vector $\mathbf C$ as the last column of $\mathbf V$. The approximate general solution step and the adaptive-knot B-spline fit exist to make this null-space extraction possible: the former removes noise by fitting the general solution form of the assumed ODE family, and the latter supplies a polynomial whose derivatives are well-conditioned and linearly independent.

What would settle it

Generate clean data from $y''' - 6y'' + 11y' - 6y = 0$, run the pipeline with a higher order such as $P=7$, and check whether the null-space vector converges to $(-6, 11, -6, 1)$ up to scale as noise tends to zero; failure to converge within the reported tolerances would refute the central claim that the method recovers homogeneous linear constant-coefficient ODEs from noisy data.

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Extended reading notes

Core claim

The central claim is that for data generated by a homogeneous linear ODE of order $P$ with constant coefficients, the coefficients $C_p$ can be estimated accurately from noisy measurements by a three-stage procedure: first, optimize the coefficient set $\hat C$ of the general solution form $$y=\sum_q D_q\left(\sum_{j=0}^{\alpha_q}x^j\right)$e^{{\beta_q x}}$(\cos(\gamma_q x)+\sin(\gamma_q x))$$ using a genetic algorithm and a pseudoinverse fit; second, approximate that smooth function by a B-spline with adaptive knot selection so that the derivatives entering the gradient matrix are linearly independent; third, assemble the gradient matrix $\mathbf G$ from spline derivatives, compute its singular value decomposition, and take the ODE coefficients as the right-singular vector spanning the null space. The paper argues that this recovers the governing equation without a basis library or regularization, produces sparse coefficient vectors by construction, and remains accurate when Gaussian noise is added to the data.

Load-bearing premise

The load-bearing premise is Eq. (1), that the governing equation is a homogeneous linear ODE of order $P$ with constant coefficients; the authors' own conclusion notes that higher-order and variable-coefficient systems remain limitations.

Editorial extensions

If this is right

  • Equations of the form $\sum_{p=0}^P C_p y^{(p)}=0$ can be identified from noisy time-series without choosing a candidate basis beforehand.
  • Physical parameters can be read off the recovered coefficients: spring-mass tests recover mass and damping, with the best noisy-data agreement in underdamped responses, and photolysis tests recover first-order rate constants close to the literature values.
  • The recovered coefficients are sparse by construction: higher-order terms fall below threshold in both case studies with no regularization term added.
  • The same pipeline extends naturally to second-order harmonic-oscillator-type systems and to first-order degradation kinetics, while the authors state that higher-order and variable-coefficient systems remain limitations.
  • Because the intermediate genetic-algorithm coefficients are nonphysical, the method's output is the final null-space vector, not the intermediate smooth fit.

Reading between the lines

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

  • The method's smoothing step effectively pre-filters the data with a model from the target ODE family, so for genuinely nonlinear systems a 'discovered' linear ODE should be read as a local linearization rather than the true law.
  • The genetic-algorithm fit of the general solution is only a means to a smooth curve; a cheaper deterministic smoothing approach could replace it and make the pipeline faster and more reproducible.
  • The EDC case study adds 1000 synthetic points by linear interpolation of a small number of experimental points before discovery; a natural test is to vary the original sample count and noise level to see when this interpolation distorts the recovered rate constants.
  • The same fit-smooth-nullspace design could be extended to nonhomogeneous or weakly nonlinear equations by fitting a particular solution along with the homogeneous general solution, but the paper does not do so.
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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

5 major / 5 minor

Summary. The paper proposes a three-step method for discovering homogeneous linear constant-coefficient ordinary differential equations from noisy data. First, a genetic algorithm fits an approximate general solution of the assumed ODE family (exponential/sinusoidal eigenfunctions) to the data. Second, an adaptive B-spline approximates this fitted function to provide a polynomial form whose derivatives are used to build a gradient matrix. Third, singular value decomposition of a reduced gradient matrix yields a null vector that is taken as the ODE coefficient vector. The method is demonstrated on synthetic spring-mass data (overdamped, critically damped, underdamped) and on experimental photolytic degradation data for estrogenic chemicals. The paper claims high accuracy, noise robustness, and sparsity without regularization.

Significance. If the method worked as described, it would offer a useful alternative to SINDy-style approaches for a restricted but common class of ODEs (homogeneous, linear, constant-coefficient), with the advantage of not requiring a pre-specified basis library and of smoothing noisy data before differentiation. The numerical results on the spring-mass system and the EDC case show plausible coefficient recovery in some regimes. However, the significance is limited by the narrowness of the assumed ODE class, the failure to provide code or data, and several theoretical gaps in the derivation, which I detail below. The paper is honest about some limitations (e.g., variable-coefficient and higher-order systems), but the central mechanism for coefficient extraction is not rigorously justified as written.

major comments (5)
  1. [2.3, Eqs. (22)-(26)] The rank-deficiency justification for the gradient matrix is internally inconsistent. The text states that the spline S is a polynomial of order P with P+1 linearly independent basis functions (x^0,...,x^P). If that is the case, the gradient matrix G (with rows representing derivatives of orders 0 through P) has full row rank P+1, so the equation G^T C = 0 admits only the trivial solution C=0, and the paper's asserted nullspace does not exist. If instead the B-spline order ω=P gives piecewise polynomials of degree P-1, then the P-th derivative row is identically zero, making C_P unidentifiable and the nullspace multidimensional. The assumption Γ=1 is therefore unsupported; when the user chooses P above the true order, as the paper explicitly does, the last right-singular vector is an arbitrary vector in a multidimensional nullspace and need not recover the true ODE coefficients. The reduced matrix G̃ is never defined concretely, and its column selection is arbitrary, which further undermines the reproducibility of the claimed extraction.
  2. [2.1, Eq. (6)] The eigenfunction basis in Eq. (6) is not the most general solution of a homogeneous linear constant-coefficient ODE when eigenvalues have multiplicity or come in complex-conjugate pairs. For a repeated complex root, one requires independent coefficients for x^j e^{βx} cos(γx) and x^j e^{βx} sin(γx) for each power j; Eq. (6) couples the cosine and sine terms with a single coefficient D_q. This restriction narrows the function class used in the approximate general solution and may bias the subsequent ODE discovery, especially for underdamped oscillatory systems with repeated roots.
  3. [3.2, Tables 6 and 7] The EDC case study is not a clean demonstration on real noisy data. The authors state that the experimental data contain too few points for the genetic algorithm, so they logarithmically transform the data, generate 1000 new samples by linear approximation, add noise of scale 0.001 to those synthetic samples, and then run the full pipeline. The recovered rate constants in Table 7 are computed from data that were themselves derived from the same experimental observations used to define the 'original' first-order rate constants; the comparison is therefore partly tautological. The claim of noise robustness in real experimental settings is accordingly overstated for this case.
  4. [Abstract and Section 3.1, Figures 7 and 11] The claim that the method 'promotes sparsity in the solution without using any regularization techniques' is contradicted by the manual thresholding applied throughout the results. In Figure 7, coefficients below 10^-4 are set to zero; in Table 6, terms below 10^-5 are neglected; in Figure 11, the threshold epsilon=10^-4 is applied. This is a hard-thresholding post-processing step, which is a form of sparsity enforcement. The SVD step alone does not produce sparse coefficient vectors, as the null space vector is generally dense.
  5. [Table 3] The reported coefficient recovery for the noiseless overdamped case is not accurate: the predicted ODE (2.03 d²x/dt² + 1.03 dx/dt + 1.00 x = 0) differs from the original (2 d²x/dt² + 4 dx/dt + 1.00 x = 0) by a factor of about four in the damping coefficient, yet Section 3.1 reports R²=1.0 for the regression of mass and damping factor (Figure 5). The discrepancy suggests a possible identifiability issue in the overdamped regime that the paper does not explain, and it weakens the general claim of high-accuracy ODE recovery.
minor comments (5)
  1. [2.1] The index range 'q = 0, 2, · · ·, Q' appears to be a typo; it should likely read 'q = 0, 1, · · ·, Q' or 'q = 1, 2, · · ·, Q'.
  2. [2.2] The word 'suuitable' should be corrected to 'suitable'.
  3. [2.3, Eq. (24)] The SVD dimensions are incorrect: for G̃ ∈ R^{(P+1)×L}, the correct factorization has U ∈ R^{(P+1)×(P+1)}, Σ ∈ R^{(P+1)×L}, and V ∈ R^{L×L}. As written, the dimensions of U and Σ are swapped.
  4. [2.3] The statement that the last column of V corresponds to the 'least squares solution with minimum Euclidean norm' is not accurate. The minimum-norm solution to G̃^T C = 0 is the zero vector, while the last right-singular vector is the direction of the smallest singular value; the citation to Trefethen and Bau does not support the usage as stated.
  5. [3.1, Table 4] The error metric used in the SINDy comparison is not defined; it is unclear whether the values correspond to mean squared error of the coefficients, of the trajectory, or of some other quantity. This makes the comparison difficult to interpret.

Circularity Check

2 steps flagged · score 6.0 of 10

The method's 'discovered' ODE is the characteristic equation of the fitted general-solution ansatz, and the EDC rate constants are fitted, not predicted, from data constructed by linear interpolation of log concentrations.

  1. self definitional [Section 2.1, Eqs. (1)-(7); Section 2.3, Eqs. (23)-(26)]
    "We assume this governing equation to be a homogeneous linear ODE of P th order with coefficients Cp ... PX p=0 Cp dpy dxp = 0. (1) ... For a given coefficient set Cˆ, it is known that Eq. 1 takes a solution of the form: y = A exp(λx). ... The approximate general solution is postulated using the same functional form as the analytical solution of a general homogeneous, linear, constant-coefficient ODE."

    The target output C is the coefficient vector of Eq. (1), and the fitted ansatz in Eq. (7) is, by construction, the general solution of exactly that equation family. The GA fits the parameters of Eq. (7) to the data, and the later SVD step reads an ODE coefficient vector off the derivatives of that same fitted function. Any function of the Eq. (7) form satisfies a homogeneous linear constant-coefficient ODE whose coefficients are determined by the fitted exponents, so the 'discovered' ODE is a repackaging of the fitted general-solution parameters, not an independent result. The method cannot leave the pre-assumed ODE class, and the claimed discovery reduces to fitting the ansatz.

  2. fitted input called prediction [Section 3.2, data augmentation paragraph and Table 7]
    "For this reason, we first considered a logarithmic transformation of the normalized concentrations with respect to the initial concentration. Following this transformation, we added 1000 new data samples based on a linear approximation of the experimental data points. ... We use the entire dataset since the purpose is to discover a mathematical description of the underlying system. ... Table 7: Predicted Rate Constant."

    The 'predicted rate constants' in Table 7 are not held-out predictions. The training set was built by linearly interpolating log-concentration data, i.e., by assuming exponential/first-order decay, and then the entire dataset was used to fit the exponential ansatz. The rate constant in the discovered ODE is the decay rate of that fitted exponential, so comparing it to Puma et al.'s first-order rate constants is a fit-to-fit comparison. The predicted value is statistically forced by the input construction and by the assumed solution form.

full rationale

The central reduction is that the paper postulates the approximate general solution using the analytical solution form of the target equation family, then reports the ODE coefficients read from that fitted function. Because Eq. (7) is the general solution of Eq. (1), the output coefficients are the characteristic-polynomial coefficients of the fitted exponents; the spline/SVD machinery is an elaborate way to extract the same information already encoded in the fitted ansatz. This is a partial circularity: the data do determine the numerical coefficients, but the 'discovery' is restricted to a pre-imposed class and within that class the ODE is equivalent to the fitted solution. The EDC case strengthens the concern: the data are augmented by linear interpolation of log concentrations, which presupposes first-order kinetics, and the resulting rate constants are then called 'predicted.' There is no load-bearing self-citation here; the comparison against SINDy (Table 4) is an independent external benchmark and does not contribute to the circularity score. Overall, the strong claim of 'discovering ODEs' reduces in these cases to fitting the general solution of the assumed ODE family, so a score of 6 is appropriate.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The method's central claim rests on the assumption that the data come from a homogeneous linear constant-coefficient ODE (Eq. 1), on the completeness of the eigenfunction basis of Eq. 7, and on an incorrect claim about the rank of the spline gradient matrix. The GA-fitted characteristic coefficients and least-squares eigenfunction weights are free parameters that are fitted to data and later discarded; the final coefficients C and the sparsity thresholds are also chosen or fitted by hand. No new physical entities are introduced.

free parameters (5)
  • ODE order P = P=7 for EDC case; unspecified for spring (GA vector has 6 entries, implying P=5)
    Chosen by hand as a flexible high order; the method requires P to define the model class, the spline order, and the null-space dimension.
  • Characteristic polynomial coefficients c_hat = e.g., (0.0654, -3.4774, 7.5927, -0.7251, 7.9382, 9.9778) for overdamped spring
    Fitted by a genetic algorithm (bounds [-10,10]) to minimize the MSE between the approximate general solution and the data; these coefficients are later discarded.
  • Eigenfunction weights D = not reported
    Computed by Moore-Penrose pseudoinverse (Eq. 9) as the least-squares weights of the eigenfunction basis; part of the smoothing step.
  • Sparsity thresholds = 1e-4 in Fig. 7; 1e-5 in Section 3.2
    Coefficients below these thresholds are set to zero manually; this produces the claimed sparsity.
  • Knot refinement threshold tau = 1e-6
    Chosen by hand to decide when to insert new spline knots.
assumptions (5)
  • domain assumption The true system is governed by a homogeneous, linear, constant-coefficient ODE of finite order (Eq. 1)
    Section 1 states 'We assume this governing equation to be a homogeneous linear ODE of Pth order'. This defines the model class for the entire pipeline.
  • standard math Eq. 7's eigenfunction basis spans all solutions of Eq. 1
    This is a standard result for constant-coefficient linear ODEs, though the specific (cos+sin) form requires treating complex-conjugate eigenvalues as separate unique eigenvalues.
  • ad hoc to paper The spline is a global polynomial of order P with P+1 independent basis functions, so the gradient matrix is rank-deficient
    Section 2.3 uses this to justify the null-space calculation, but a B-spline with interior knots is piecewise polynomial and its gradient matrix is generically full rank; the rank deficiency actually comes from the underlying function satisfying a low-order ODE.
  • domain assumption The data are generated from a unique governing equation, so a single null-space direction suffices (Gamma=1)
    Section 2.3: 'We assume that the data are generated from a unique governing equation Gamma = 1'. This justifies taking only the last column of V.
  • domain assumption The genetic algorithm converges to a sufficiently accurate smooth approximation
    The GA is stochastic and no convergence guarantee is given; the paper relies on it producing a low-MSE functional fit that then drives the recovered ODE.

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Cite this review

Pith. "Pith review of Discovering Interpretable Ordinary Differential Equations from Noisy Data." pith.science (2026). https://pith.science/paper/X3ADYSMX

@misc{pith2026250721841,
  author       = {Pith},
  title        = {Pith review of: Discovering Interpretable Ordinary Differential Equations from Noisy Data},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/X3ADYSMX}},
  note         = {Machine review of arXiv:2507.21841}
}
read the original abstract

The data-driven discovery of interpretable models approximating the underlying dynamics of a physical system has gained attraction in the past decade. Current approaches employ pre-specified functional forms or basis functions and often result in models that lack physical meaning and interpretability, let alone represent the true physics of the system. We propose an unsupervised parameter estimation methodology that first finds an approximate general solution, followed by a spline transformation to linearly estimate the coefficients of the governing ordinary differential equation (ODE). The approximate general solution is postulated using the same functional form as the analytical solution of a general homogeneous, linear, constant-coefficient ODE. An added advantage is its ability to produce a high-fidelity, smooth functional form even in the presence of noisy data. The spline approximation obtains gradient information from the functional form which are linearly independent and creates the basis of the gradient matrix. This gradient matrix is used in a linear system to find the coefficients of the ODEs. From the case studies, we observed that our modeling approach discovers ODEs with high accuracy and also promotes sparsity in the solution without using any regularization techniques. The methodology is also robust to noisy data and thus allows the integration of data-driven techniques into real experimental setting for data-driven learning of physical phenomena.

Figures

Figures reproduced from arXiv: 2507.21841 by the authors.

Figure 1
Figure 1. Proposed Methodology. While the approximate general solution step allows to obtain smooth derivative information directly from the approxi￾mated function, for many functions, we may not be able to obtain linearly independent derivative information. For linear model such as Eq. 1, we need linearly independent basis in the gradient matrix. Derivatives of nonlinear functions, such as trigonometric and exponential, are … view at source ↗
Figure 2
Figure 2. Comparison of spline fitting with and without noise. [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Spring Mass system. The dataset used for analysis is synthetically generated using an analytical solution of a spring mass system under different scenarios (see Appendix A). We performed analysis on the spring mass system for both noisy data and data without noise. The configuration for generating the datasets for all three is provided in [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: Comparison of model predictions for overdamped, critically damped, and underdamped spring-mass systems. [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: Regression plots for damping factor and mass without noise. [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: Regression plots (log scale) comparing original and predicted damping (top row) and mass (bottom row) for [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]
Figure 7
Figure 7. Figure 7: Sparsity in model prediction [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]
Figure 8
Figure 8. Figure 8: Concentration profiles for different EDC components under photolysis. [PITH_FULL_IMAGE:figures/full_fig_p014_8.png]
Figure 9
Figure 9. Figure 9: Comparison of original and predicted degradation profiles for EDC components under UVA photolysis. [PITH_FULL_IMAGE:figures/full_fig_p015_9.png]
Figure 10
Figure 10. Figure 10: Comparison of original and predicted degradation profiles for EDC components under UVC photolysis. [PITH_FULL_IMAGE:figures/full_fig_p016_10.png]
Figure 11
Figure 11. Figure 11: Sparsity in model prediction. coefficients for a P th (here P = 7) order system and observe if the model is still able to discover the order of the underlying system. It is known to us that the system follows an apparent first order kinetic expression. For visualizing…

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