Equation discovery framework EPDE: Towards a better equation discovery

Alexander Hvatov; Mikhail Maslyaev

REVIEW 4 major objections 6 minor 22 references

Equation discovery framework EPDE: Towards a better equation discovery

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

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Pith's one-line read Evolutionary search beats SINDy on noisy equation discovery.

desk verdict A useful evolutionary equation-discovery framework, but its central noise-robustness claim is contradicted by the paper's own Table 7. read the letter →

arxiv 2501.14768 v1 pith:HZBAVBUA submitted 2024-12-28 cs.NE cs.AIcs.LG

Mikhail Maslyaev , Alexander Hvatov This is my paper · ORCID

classification cs.NEcs.AIcs.LG

keywords equationdiscoveryevolutionaryoptimizationmulti-objectiveSINDypartialdifferentialequationsnoiserobustnesssymbolicregressionordinary

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

This paper tries to establish that a stochastic, evolutionary approach to differential equation discovery can outperform the widely used sparse-regression method SINDy, especially when the data are noisy. The key idea is that instead of fitting a fixed library of candidate terms, the algorithm evolves equations from basic building blocks like derivatives and elementary functions, letting the term set adapt during the search. The authors enhance the EPDE framework with multi-objective optimization, improved handling of equation systems, and neural-network-based derivative estimation, then test it on Burgers, Korteweg-de Vries, Van der Pol, and Lotka-Volterra equations. If correct, this would make equation discovery less reliant on prior knowledge about the equation's form and more robust to experimental noise, at the price of much longer computation time.

What carries the argument

The engine of the method is an equation encoding as a tree whose leaves are tokens: derivatives and elementary functions such as trigonometric or polynomial factors, grouped into families. Terms are formed by multiplying tokens, and the equation is a linear combination of terms; a sparsity constant and complexity metric control parsimony. The search uses a multi-objective evolutionary algorithm based on dominance and decomposition, where each population member is associated with a weight vector in the objective space and a penalty-based intersection (PBI) value balances convergence and diversity. Fitness is computed either from the L2 discrepancy of the differential operator on the data grid or from the error of a neural-network-solved candidate equation, with weighted regression for coefficient estimation.

What would settle it

A direct test would be to take a noisy dataset for a known equation, enumerate all candidate equations of the same length and token families that the evolutionary search could produce, and compute their discrepancy fitness; if any wrong equation scores better than the true one, then the algorithm's occasional successes depend on lucky search rather than on the fitness criterion, and the claimed noise robustness would fail in that regime.

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Extended reading notes

Core claim

The central claim is that dynamic term generation coupled with multi-objective evolutionary search distinguishes significant equation terms from noise more effectively than deterministic sparse regression on a predefined library. In the experiments, EPDE recovered the correct governing equation at noise levels up to 5% for the Lotka-Volterra system and with a higher frequency than SINDy across most tested cases, while SINDy reliably succeeded only on noiseless data. The paper also reports that when the correct structure is found, EPDE's coefficient estimates are comparable or slightly better in relative error, but that this robustness comes with a dramatic increase in runtime, from milliseconds to tens of seconds or more.

Load-bearing premise

That a few dozen generations of evolution over a small population can reliably locate the true equation in the combinatorially large term space, and that the chosen fitness function ranks that true equation as optimal even on noisy data.

Editorial extensions

If this is right

The term library adapts to the problem, so equations whose forms are not anticipated by a fixed candidate set can, in principle, be discovered.
At moderate noise levels, the evolutionary approach can recover correct structures more often than SINDy, including systems of ODEs.
When the correct structure is found, coefficient errors are comparable to or better than those of sparse regression.
The multi-objective formulation yields a Pareto set of candidate equations, letting a researcher inspect the trade-off between equation complexity and data fit.
The main practical drawback is computational cost, so SINDy may remain preferable when a reliable library is known and the equation is first-order in time.

Reading between the lines

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

If the central claim holds, equation discovery becomes feasible for experimental datasets where a functional form is unknown and noise is non-negligible, such as ecology or climate data.
A natural test of the mechanism would be to compare EPDE against a weak-form or ensemble variant of SINDy, which were not included in the paper's comparison.
The reported 'wrong equation is optimal' failure at high noise suggests the fitness landscape, rather than the search algorithm alone, is the limiting factor; improving the discrepancy metric could extend the noise ceiling.
The evolutionary search could be hybridized with sparse regression as a local refinement step to reduce the runtime that currently limits practical use.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Two things to know. First, this is a real engineering contribution: the EPDE framework now combines multi-objective evolutionary search, dynamic term generation, and neural/Chebyshev derivative preprocessing, with code on GitHub and a systematic comparison against pySINDy on four benchmark equations. Second, the paper's central claim—that EPDE notably surpasses SINDy in noise robustness—is not supported by its own numbers. Read Table 7 and you'll see SINDy actually reports higher positive-discovery rates at 1% and 2.5% noise (25% vs 10% and 2.5%). The 5% 'success' for EPDE is a single run out of forty, inflated by the 'one positive in experiment' column. That column is transparent but it rewards a lucky run, not systematic behavior.

What's deserved: the integration is genuinely new compared to the authors' earlier EPDE and multi-objective papers. The comparison details, the run statistics, and the appendix with hyperparameters are useful. The authors are also honest about limitations: the method needs manual parameter setting, is slow, and they concede in Section 4 that high noise leads the wrong equation to be optimal. That concession is in tension with the abstract's 'robust equation extraction' claim, but it's there.

Soft spots: (1) The headline overstates the evidence. For Burgers at 1% noise, they say the algorithm 'reliably converges,' but the key nonlinear term uu'_x is found in only 10% of runs. (2) The comparison is asymmetric: SINDy is deterministic and gets one shot; EPDE gets ten stochastic runs and the authors count the best. A fair comparison would report median or worst-case performance, or at least the full distribution. (3) The evolutionary search hyperparameters (populations of 8–12, 55–85 epochs) are small; there is no sensitivity analysis showing the results are robust to those choices. None of these flaws makes the framework worthless; they make the noise-robustness claim unproven.

Who should read it: anyone working on equation discovery, especially on the practical side of evolutionary search and derivative preprocessing. It deserves peer review—the code and the benchmark comparison merit scrutiny—but the revision should re-analyze Table 7, either dropping the 'notably surpass' claim or providing a comparison that supports it. If I were the editor, I'd send it to review with a strong request for major revision.

Referee Report

4 major / 6 minor

Summary. The paper extends the EPDE evolutionary equation-discovery framework by adding multi-objective optimization, improved handling of differential equation systems, a neural-network-based differentiation/preprocessing route, and automated hyperparameter management. The authors validate the method on Burgers, Korteweg-de Vries, Van der Pol, and Lotka-Volterra systems with noise levels from 0% to 5%, comparing term-recovery rates, coefficient errors, and full-equation recovery against pySINDy. The central claim is that stochastic, dynamically generated term libraries allow EPDE to notably surpass deterministic sparse-regression methods such as SINDy in noise robustness.

Significance. If the central claim were supported, the paper would make a useful contribution by demonstrating an evolutionary alternative to library-based sparse regression, with public code and benchmark comparisons. The framework's multi-objective formulation, dynamic term generation, and treatment of missing differentials are potentially valuable directions. However, the headline comparative claim is not supported by the paper's own summary table, and the evaluation procedure is asymmetric between the deterministic and stochastic algorithms. The contribution is therefore promising but needs a substantially revised and more careful empirical assessment before the stated conclusions can be accepted.

major comments (4)

[Table 7, rows for 1%, 2.5%, and 5% noise] The reported positive-discovery rates contradict the claim that EPDE 'notably surpasses' SINDy. At 1% and 2.5% noise, SINDy recovers the correct equation in 25% of cases while EPDE's per-run rate is 10% and 2.5%, respectively; at 5% noise, EPDE's 2.5% positive rate and 25% one-positive rate correspond to a single successful run in one of the four benchmark experiments. The authors should either revise the comparative claim or provide a statistically justified reason why best-of-ten stochastic scoring is the appropriate basis for 'notably surpass'.
[Section 4, scoring procedure] The comparison is asymmetric: SINDy is deterministic and produces one model after hyperparameter selection, while EPDE is given ten stochastic runs and scored by the best run through the 'one positive in experiment' column. This metric rewards lucky runs and, with ten runs per experiment, cannot distinguish systematic superiority from chance. Please report the full distribution of outcomes over the ten runs, or use a matched criterion such as median or mean success rate, success in at least k of ten runs, or a statistical test with confidence intervals.
[Section 3.1.1, Table 1] The statement that 'the algorithm can reliably converge to the correct equation ... only with noise levels equal to or lower than 1%' is not supported by the table. At 1% noise, EPDE recovers u'_t in 90% of runs but uu'_x in only 10%, and the reported coefficient for u'_t is far from the ground-truth value. The text should distinguish term-level inclusion from full-equation recovery and define 'reliably' operationally, since the table shows a rapid loss of full-structure recovery even at 1% noise.
[Section 4, last paragraph] The concession that 'high noise contamination leads to the wrong equation being optimal from an optimization criteria standpoint' directly undermines the noise-robustness claim in the regime where EPDE is presented as superior. If the true equation is not optimal under the fitness criterion, the paper should explain how a practitioner can identify the rare successful run without already knowing the true equation, or present a criterion under which the true equation is optimal on noisy data.

minor comments (6)

[Section 2, organization] The introduction of Section 2 refers to 'Subsection 2.2' twice; the second reference should be to Subsection 2.3.
[Equation (6)] The definition of F_i repeats the same grid-node expression twice; it should list a range over grid nodes rather than duplicating the first entry.
[Table 3, 0.5% noise row] The confidence intervals for u^2 u'_t and u'_t are reported as 33.6e5 and 11.9e5, which are implausibly large relative to the coefficient estimates; please explain the source of this dispersion.
[Appendix A] The notation 'Deriv. ord=(2,3)' is ambiguous; for the KdV experiments, the maximal derivative order should be stated explicitly as 3.
[Section 2.3] Several phrases appear to be typographical errors, including 'P of f spr' (likely 'offspring') and 'autonomous tokes' (likely 'tokens').
[Table 7] The coefficient-error column is labeled 'best' for EPDE but the rule for selecting this best run is not specified; please state how the best run is chosen and report the variability across runs.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: EPDE's comparison with SINDy rests on standard external benchmarks; self-citations are supporting infrastructure, not load-bearing reductions.

full rationale

The paper's derivation chain is self-contained with respect to its central comparative claim. The fitness objective (Eqs. 4-8) is a standard weighted linear regression discrepancy; the evolutionary search is an implementation of a published MOEA framework with external references [20,21]; and the validation compares EPDE against pySINDy on standard benchmark equations (Burgers, KdV, Van der Pol, Lotka-Volterra) using public data and code. The authors' self-citations ([12] original EPDE, [14] single- vs multi-objective comparison, [19] PINN solver) are supporting infrastructure: [14] motivates the multi-objective design but is a separate published, externally falsifiable comparison, and the paper's headline result (EPDE vs SINDy noise robustness) is not derived from any of these citations. No equation in the paper is defined in terms of the quantity it is claimed to predict, and no fitted parameter is renamed as a prediction. The asymmetry in Table 7 (ten stochastic EPDE runs scored by 'one positive' vs deterministic SINDy) is a statistical-comparison concern, not a circularity, because the EPDE successes are measured against the same ground-truth equations used for SINDy rather than against EPDE's own outputs.

Assumptions & free parameters 7 free parameters · 5 assumptions · 0 invented entities

The method's success depends on hand-set token families, derivative-order bounds, complexity bounds, sparsity ranges, and fitness/preprocessing choices that are tuned per experiment. The representational assumption of Eq. (1) and the differentiability assumption are standard for the field, but the optimality of the fitness under noise is explicitly conceded to fail at high noise (Section 4).

free parameters (7)

Token families and maximal derivative order = Set per experiment (e.g., Deriv. ord=(2,3), Coord., Sine/Cosine for Burgers)
Hand-chosen search-space limits directly constrain which equations can be discovered, as detailed in Appendix A.
Fitness criterion selection = Solution-based for Burgers, discrepancy-based for others
The paper states the choice should depend on expected noise, but the selection rule is informal (Section 2.2, Section 3.1.1).
Preprocessing method = Chebyshev for Burgers/KdV, ANN for VdP/Lotka-Volterra
Derivative estimation method is case-specific and affects noise robustness (Section 3, Appendix A).
Population size and epochs = 8-12 population, 55-85 epochs
Computation budget hyperparameters that bound the search; no sensitivity analysis is provided (Appendix A).
Equation length and factor count = 5-6 terms, 2-4 factors
Upper bounds on model complexity chosen per experiment; changing them changes the search outcome (Appendix A).
Sparsity interval = (1e-9, 100) or (1e-12, 1e-4)
Range from which the sparsity constant is drawn; affects term selection (Appendix A).
Weight function g(x) = Not specified
Introduced in Eq. (5) to down-weight boundary errors but never defined or used in the experiments.

assumptions (5)

domain assumption The observed field is differentiable up to the maximal derivative order used.
Section 2 states this assumption before defining the search space.
domain assumption The true equation is a linear combination of products of elementary functions and derivative tokens (Eq. 1).
This representational assumption restricts the class of discoverable equations.
domain assumption Noise is zero-mean Gaussian with variance proportional to signal magnitude.
The synthetic data generation in Section 3 defines noise as N(0, k*u(t,x)), which may not represent real experimental noise.
ad hoc to paper The chosen fitness metric ranks the true structure as optimal under noise.
Section 4 concedes that high noise makes a wrong equation optimal, so this assumption fails exactly in the regime the paper claims robustness.
domain assumption The PINN-based solution fitness (Eqs. 9-10) provides a reliable approximation of solution error.
Used only for Burgers; computational cost limits its use, and its reliability is not independently validated.

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Cite this review

Pith. "Pith review of Equation discovery framework EPDE: Towards a better equation discovery." pith.science (2026). https://pith.science/paper/HZBAVBUA

@misc{pith2026250114768,
  author       = {Pith},
  title        = {Pith review of: Equation discovery framework EPDE: Towards a better equation discovery},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HZBAVBUA}},
  note         = {Machine review of arXiv:2501.14768}
}

read the original abstract

Equation discovery methods hold promise for extracting knowledge from physics-related data. However, existing approaches often require substantial prior information that significantly reduces the amount of knowledge extracted. In this paper, we enhance the EPDE algorithm -- an evolutionary optimization-based discovery framework. In contrast to methods like SINDy, which rely on pre-defined libraries of terms and linearities, our approach generates terms using fundamental building blocks such as elementary functions and individual differentials. Within evolutionary optimization, we may improve the computation of the fitness function as is done in gradient methods and enhance the optimization algorithm itself. By incorporating multi-objective optimization, we effectively explore the search space, yielding more robust equation extraction, even when dealing with complex experimental data. We validate our algorithm's noise resilience and overall performance by comparing its results with those from the state-of-the-art equation discovery framework SINDy.

Figures

Figures reproduced from arXiv: 2501.14768 by the authors.

**Figure 2.** Scheme of the cross-over operator. The cross-over operator is based on the interactions between the structures representing the equations describing the same variable. Each pair of parents produces two offspring. Two types of operators are introduced since the encoding contains the symbolic graphs of the equations and the values of the sparsity constants. For the sparsity constant, the new values are selected in t… view at source ↗

**Figure 3.** Scheme of the mutation operator. must be affected by additional mutation operations until the uniqueness is achieved. The mutation of equation construction parameters involves the addition of increments obtained from the pre-defined distribution. The update procedure involves insertion of the sub-population of new candidate solutions P of fspr = {S of fspr 1 (u), ... , Sof fspr n of fspr(u)} and deletion of the les… view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Example of Van der Pol oscillator state predictions, based on obtained differential [PITH_FULL_IMAGE:figures/full_fig_p021_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

S. L. Brunton, J. L. Proctor, J. N. Kutz, Discovering governing equa- tions from data by sparse identification of nonlinear dynamical systems, Proceedings of the national academy of sciences 113 (15) (2016) 3932– 3937

2016

[2] [2]

S. H. Rudy, S. L. Brunton, J. L. Proctor, J. N. Kutz, Data-driven dis- covery of partial differential equations, Science advances 3 (4) (2017) e1602614

2017

[3] [3]

D. A. Messenger, D. M. Bortz, Weak sindy for partial differential equa- tions, Journal of Computational Physics 443 (2021) 110525. 26

work page 2021

[4] [4]

Fasel, J

U. Fasel, J. N. Kutz, B. W. Brunton, S. L. Brunton, Ensemble-sindy: Robust sparse model discovery in the low-data, high-noise limit, with ac- tive learning and control, Proceedings of the Royal Society A 478 (2260) (2022) 20210904

2022

[5] [5]

Z. Long, Y. Lu, B. Dong, Pde-net 2.0: Learning pdes from data with a numeric-symbolic hybrid deep network, Journal of Computational Physics 399 (2019) 108925

2019

[6] [6]

Stephany, C

R. Stephany, C. Earls, Pde-read: Human-readable partial differential equation discovery using deep learning, Neural Networks 154 (2022) 360– 382

2022

[7] [7]

Raissi, P

M. Raissi, P. Perdikaris, G. E. Karniadakis, Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, Journal of Computational physics 378 (2019) 686–707

2019

[8] [8]

L. Lu, P. Jin, G. Pang, Z. Zhang, G. E. Karniadakis, Learning nonlinear operators via deeponet based on the universal approximation theorem of operators, Nature Machine Intelligence 3 (3) (2021) 218–229

2021