{"id":"90081014-f0c8-4bfe-8796-49d351fad846","arxiv_id":"2606.07152","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"SES recovers explicit symbolic solutions for algebraic, transcendental, ODE and PDE equations by optimizing symbolic expressions to satisfy the governing equation and boundary conditions without any paired data.","lead":"The paper introduces Symbolic Equation Solver (SES), which turns solving equations into an optimization task over differentiable symbolic models using only the equation and its conditions. This removes the need for training data from the process itself.","discovery_kind":"unclear","skeptic_critique":{"model":"grok-4.3","headline":"Optimization over symbolic expressions may converge to local minima with small residuals but incorrect expressions","rationale":"The reader's weakest assumption is precisely the load-bearing point for a method whose correctness rests on non-convex optimization over expression space; the abstract provides no counter-evidence (multiple runs, failure rates, or theoretical guarantees), so the UNVERDICTED status is appropriate.","tokens_in":1600,"tokens_out":268,"duration_ms":9935,"concrete_test":"Re-run SES on the four evaluated equations (algebraic system, transcendental, ODE, PDE) using 20 independent random seeds each; report the fraction of runs that recover an expression symbolically identical to the known analytical solution (exact match, not just small residual).","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim requires that the optimization procedure (formulated from the governing equation plus initial/boundary conditions) reliably recovers the exact compact symbolic solution rather than an alternative expression whose residual is numerically small. The space of symbolic expressions is discrete and highly non-convex; without explicit mechanisms guaranteeing escape from local minima (e.g., exhaustive enumeration, provable global optimality, or exhaustive multi-start statistics), success on the four representative cases does not establish that the method will locate the analytical solution in general.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","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.","tokens_in":1704,"tokens_out":443,"duration_ms":19143,"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":[{"comment":"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.","section":null},{"comment":"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.","section":null}],"minor_comments":[{"comment":"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.","section":null}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their constructive comments. We address each major point below, indicating revisions where appropriate to strengthen the manuscript.","responses":[{"response":"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_made":"partial","referee_comment":"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."},{"response":"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_made":"yes","referee_comment":"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."}],"tokens_in":1321,"tokens_out":431,"duration_ms":23453,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main point is that this paper removes the data requirement from symbolic regression by building the objective straight from the governing equation plus initial or boundary conditions. That formulation is the actual novelty; most prior work samples points first.\n\nOn the positive side, the abstract shows the method recovering exact known solutions for an algebraic system, a transcendental equation, an ODE, and a couple of PDEs. If the full paper supplies the library, the optimizer, and the training procedure, that could be a useful practical step for cases where data are expensive or unavailable.\n\nThe soft spot is exactly the one the stress test flags. The space of symbolic expressions is discrete and non-convex; nothing in the abstract indicates exhaustive search, provable global optimality, or even systematic multi-start statistics. Success on four hand-picked examples does not establish that the optimizer will consistently land on the compact analytical solution rather than some other expression whose residual is numerically small. Without those details the central claim is hard to evaluate.\n\nThe paper is aimed at researchers who already work on symbolic regression or equation discovery and want to explore data-free variants. It is coherent on its own terms and shows clear engagement with the problem, so it deserves a serious referee who can check the implementation and the failure modes.","headline":"SES turns equation solving into data-free symbolic optimization but gives almost no implementation details, so the reliability claim rests on four success cases.","tokens_in":2143,"tokens_out":327,"would_cite":false,"duration_ms":10184,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Symbolic Equation Solver recovers exact analytical expressions from the governing equation and conditions alone, without any training data.","keywords":["symbolic regression","data-free learning","equation solving","differential equations","symbolic expressions","optimization"],"falsifier":"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.","tokens_in":2510,"feed_emoji":"🧮","tokens_out":571,"duration_ms":17294,"temperature":0.7,"pith_summary":"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.","feed_headline":"Solver finds exact symbolic answers from equations alone","feed_subtitle":"Optimization over symbolic models uses only the governing equation and boundary conditions to match analytical solutions for algebraic and d","key_machinery":"Optimization of differentiable symbolic models whose loss is assembled from the equation residual plus initial or boundary conditions.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["Data-free symbolic solver recovers equation solutions","Optimization over symbols solves equations exactly","Symbolic regression from equation alone matches analytics","SES derives compact symbolic expressions without data"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Data-free symbolic solver recovers equation solutions","Optimization over symbols solves equations exactly","Symbolic regression from equation alone matches analytics","SES derives compact symbolic expressions without data"]},"model":"grok-4.3","cost_usd":0.003602,"raw_usage":{"total_tokens":1839,"prompt_tokens":581,"num_sources_used":0,"completion_tokens":48,"cost_in_usd_ticks":36024500,"prompt_tokens_details":{"text_tokens":581,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1210,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":581,"tokens_out":48,"duration_ms":7965,"temperature":1.0,"reasoning_tokens":1210,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-27T20:23:22.374084+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"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.","supporting_citations":[],"review_version":1}