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REVIEW 2 major objections 1 minor 23 references

A Data-Free Symbolic Regression Approach for Solving Equations

T0 review · 2 major / 1 minor · reviewed 2026-06-27 · grok-4.3

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Pith's one-line read Symbolic Equation Solver recovers exact analytical expressions from the governing equation and conditions alone, without any training data.

desk verdict SES turns equation solving into data-free symbolic optimization but gives almost no implementation details, so the reliability claim rests on four success cases. read the letter →

arxiv 2606.07152 v1 pith:476OBWNS submitted 2026-06-05 cs.NE cs.SC

Sergei Garmaev , Vinay Sharma , Olga Fink This is my paper · ORCID
classification cs.NEcs.SC
keywords symbolicregressiondata-freelearningequationsolvingdifferentialequationsexpressionsoptimization
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 introduces a framework that recasts equation solving as an optimization task over differentiable symbolic models. The objective is built directly from the equation's residual together with initial or boundary conditions, so no paired input-output examples are required. The method is tested on algebraic systems, transcendental equations, ordinary differential equations, and partial differential equations under varying conditions. If successful, the approach yields compact symbolic expressions that match known analytical solutions and can be inspected or manipulated further.

What carries the argument

Optimization of differentiable symbolic models whose loss is assembled from the equation residual plus initial or boundary conditions.

What would settle it

Apply SES to the wave equation with standard initial and boundary conditions and check whether it returns the exact known closed-form solution rather than an approximate or alternative expression.

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Extended reading notes

Core claim

The Symbolic Equation Solver (SES) formulates equation solving as an optimization problem over differentiable symbolic models. SES constructs its objective from the equation together with initial or boundary conditions, eliminating the need for paired input-output data. The learned model is expressed in explicit symbolic form, enabling further analysis. Across these settings, SES recovers compact symbolic expressions that match the corresponding analytical solutions.

Load-bearing premise

The optimization procedure over the space of symbolic expressions will reliably locate the global solution that satisfies the equation rather than converging to a local minimum or an incorrect but numerically small residual expression.

Editorial extensions

If this is right

  • SES produces explicit symbolic solutions for systems of algebraic equations.
  • It recovers symbolic forms for equations containing transcendental functions.
  • It yields closed-form solutions for ordinary differential equations from the equation and conditions alone.
  • It handles partial differential equations under different initial or boundary conditions while returning compact expressions.

Reading between the lines

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

  • The same optimization approach could be tried on nonlinear equations whose analytical solutions are still unknown.
  • The recovered symbolic expressions might be inserted into larger models for stability or sensitivity analysis without re-deriving them numerically.
  • Because the method needs no external data, it could serve as a verification step that checks whether a candidate symbolic form satisfies a governing equation exactly.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 1 minor

Summary. The manuscript proposes the Symbolic Equation Solver (SES), a data-free framework that casts equation solving as an optimization problem over differentiable symbolic models. The objective is built directly from the governing equation plus initial/boundary conditions, eliminating the need for paired training data. The authors evaluate the approach on a system of algebraic equations, a transcendental equation, an ODE, and PDEs with varying conditions, claiming that SES recovers compact symbolic expressions matching the known analytical solutions.

Significance. If the optimization procedure can be shown to reliably locate the exact global symbolic solution rather than a numerically small but incorrect expression, the method would offer a novel route to closed-form solutions for equations that currently require numerical treatment. This could complement existing symbolic regression techniques by removing the data requirement and enabling direct use of the governing equation. The current manuscript, however, provides insufficient detail on the symbolic library, parameterization, and optimizer to assess whether this reliability holds.

major comments (2)
  1. Abstract and method description: the central claim that SES recovers the analytical solutions rests on the optimization locating the exact symbolic expression. The space of symbolic expressions is discrete and non-convex; without explicit mechanisms (exhaustive search, provable global optimality, or multi-start statistics with failure-case reporting) the success on four representative cases does not establish that the procedure will avoid local minima whose residuals are numerically small but whose expressions are incorrect. This issue is load-bearing for the validity of the framework.
  2. Method section (implied by abstract claims): the manuscript does not specify the symbolic library, the differentiable parameterization of expressions, the optimization algorithm, convergence criteria, or how the search is initialized. These omissions make it impossible to reproduce or evaluate whether the reported recoveries are robust or merely fortunate on the chosen examples.
minor comments (1)
  1. The abstract would benefit from a brief statement of the size of the symbolic library and the form of the objective function to give readers an immediate sense of the search space.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments. We address each major point below, indicating revisions where appropriate to strengthen the manuscript.

read point-by-point responses

  1. Referee: Abstract and method description: the central claim that SES recovers the analytical solutions rests on the optimization locating the exact symbolic expression. The space of symbolic expressions is discrete and non-convex; without explicit mechanisms (exhaustive search, provable global optimality, or multi-start statistics with failure-case reporting) the success on four representative cases does not establish that the procedure will avoid local minima whose residuals are numerically small but whose expressions are incorrect. This issue is load-bearing for the validity of the framework.

    Authors: We agree that the non-convex discrete nature of the search space makes local minima a valid concern and that success on four cases alone does not fully establish robustness. The manuscript reports consistent recovery of the known analytical solutions but does not include multi-start statistics or explicit failure-case analysis. We will revise the manuscript to add a dedicated subsection on optimization reliability, including results from multiple random initializations, success rates across runs, and any observed cases where the optimizer reaches a numerically small but symbolically incorrect expression. This provides empirical support without claiming theoretical global optimality guarantees. revision: partial

  2. Referee: Method section (implied by abstract claims): the manuscript does not specify the symbolic library, the differentiable parameterization of expressions, the optimization algorithm, convergence criteria, or how the search is initialized. These omissions make it impossible to reproduce or evaluate whether the reported recoveries are robust or merely fortunate on the chosen examples.

    Authors: The referee correctly identifies that these details are required for reproducibility. The current manuscript emphasizes the high-level framework and results. We will expand the Method section with a complete specification of the symbolic library (operators and terminals), the differentiable parameterization, the optimization algorithm, convergence criteria, and initialization strategy. These additions will enable independent reproduction and assessment of robustness. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: objective constructed directly from input equation

full rationale

The paper defines SES as an optimization problem whose loss is assembled from the governing equation plus initial/boundary conditions; the symbolic expression is the decision variable, not a fitted parameter renamed as a prediction. No self-citations are invoked to justify uniqueness or to smuggle an ansatz. The central claim is an empirical demonstration on four equation classes rather than a derivation that reduces to its inputs by construction. This matches the default expectation of a self-contained method.

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

Abstract-only review provides no information on free parameters, axioms, or invented entities; ledger left empty.

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Cite this review

Pith. "Pith review of A Data-Free Symbolic Regression Approach for Solving Equations." pith.science (2026). https://pith.science/paper/476OBWNS

@misc{pith2026260607152,
  author       = {Pith},
  title        = {Pith review of: A Data-Free Symbolic Regression Approach for Solving Equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/476OBWNS}},
  note         = {Machine review of arXiv:2606.07152}
}
read the original abstract

Many equations arising in science currently cannot be solved by available analytical techniques and are therefore solved numerically, without yielding explicit symbolic expressions. Existing symbolic regression approaches can recover symbolic expressions, but require training data obtained from the underlying process, rather than the governing equation alone. We propose the Symbolic Equation Solver (SES), a framework that formulates equation solving as an optimization problem over differentiable symbolic models. SES constructs its objective from the equation together with initial or boundary conditions, eliminating the need for paired input-output data. The learned model is expressed in explicit symbolic form, enabling further analysis. We evaluate SES on representative algebraic and differential equations, including a system of algebraic equations, an equation with transcendental terms, an ordinary differential equation, and partial differential equations with different initial or boundary conditions. Across these settings, SES recovers compact symbolic expressions that match the corresponding analytical solutions.

Figures

Figures reproduced from arXiv: 2606.07152 by the authors.

Figure 1
Figure 1. Schematic overview of the Symbolic Equation Solver (SES) framework. Given a governing [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗

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