{"id":"c23240cf-619d-42ee-9ecd-ebf06759c7c1","arxiv_id":"2602.23561","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"VaSST uses variational inference over continuously relaxed symbolic trees to recover closed-form expressions from noisy data, reporting competitive structural recovery and predictive accuracy on simulated and Feynman benchmarks.","lead":"VaSST introduces a new statistical method for symbolic regression that replaces discrete mathematical expression search with a smooth, gradient-based variational inference problem, yielding candidate formulas with quantified uncertainty. It is relevant to scientists who want to automatically discover closed-form equations from data and to understand the reliability of the recovered expression.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Exact-recovery claim in §5.2 is internally contradicted: reported VaSST min-RMSE expressions for FCE/CPE/FTC (Appendices K.1, K.4) are not the true equations, so the strongest claim fails on the paper's own evidence.","rationale":"Reader's weakest_assumption is the unquantified relaxation gap between Eq. (14) and Eq. (9). That is a legitimate theoretical concern, but the more load-bearing, checkable failure is empirical: the paper's headline recovery claim is contradicted by its own appendix. The strongest_claim as extracted is exactly the §5.2 sentence; if that sentence is false, the central contribution as advertised collapses, regardless of whether the variational objective is well founded. The reported FCE expression is not the true FCE even in the noiseless case, where the true model should achieve RMSE 0 and be the minimum in-sample RMSE sample. The extra sin(2Bv/E_f) and θ terms, and coefficient 1.068, cannot be dismissed as rounding. Similarly CPE/FTC min-RMSE expressions are approximate. This is internal inconsistency, not a disagreement with consensus, so it should be prioritized. I still credit the method: conjugate NIG marginalization is standard, soft-tree evaluation is clearly specified, and reproducibility details are good. But the exact-recovery claim needs either a precise definition and supporting frequency counts, or removal. Because this target is the paper's strongest claim and fails on the paper's own reported outputs, I would move the verdict toward reject-as-stated; a revised version that reframes claims as approximate recovery and reports exact-recovery rates might be acceptable. Hence I only partially agree with the reader: the relaxation gap is real, but the appendix contradiction is more directly decisive.","tokens_in":34339,"tokens_out":7015,"duration_ms":61737,"concrete_test":"Run VaSST on the FCE noiseless task with Appendix H settings; for each of H=2000 sampled hard symbolic ensembles, use symbolic simplification (e.g., SymPy) to test equivalence to q*E_f + q*B*v*sin(θ), and record its in-sample RMSE. If zero samples are exactly equivalent, or if the minimum-RMSE sample is not exactly equivalent, the §5.2 claim is false. Also recompute Table 3 using a stated exact-recovery criterion (symbolic equivalence after constant merging) and compare counts with Appendix K expressions.","verdict_should_be":"REJECT","load_bearing_attack":"The paper's central claim is the unconditional sentence in §5.2: 'Across all equation datasets and noise levels, the minimum RMSE model produced by VaSST recovers the correct symbolic expression.' The paper's own appendix contradicts this. For FCE (noiseless), Appendix K.7 reports VaSST's recovered expression as -0.00285 + 0.00318 sin(2Bv/E_f) + 0.0012θ + 1.068(q sinθ Bv + E_f q), and Appendix K.4.3 lists this same approximate form as the min-RMSE top-1; this is not q(E_f + vB sinθ) — coefficient 1.068 ≠ 1 plus spurious sin(2Bv/E_f) and θ terms. For CPE and FTC, the top-1 expressions in K.4.2/K.4.4 contain extra/missing terms and non-unit coefficients. Thus Table 3's ✓ marks are not supported by the reported symbolic outputs. This is a stronger and more direct problem than the relaxed-ELBO gap: even if Eq. (14) were a faithful surrogate, the empirical outputs do not match the 'exact symbolic recovery' claim. The paper never defines 'correct symbolic expression' (exact vs. approximate/tolerance), and no posterior probabilities or recovery frequencies are given for the H=2000 hard-tree samples.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes VaSST, a variational-inference framework for symbolic regression in which discrete symbolic trees are relaxed into 'soft symbolic trees' via Binary Concrete and Gumbel-Softmax reparameterizations. The authors derive an ELBO consisting of an analytically intractable marginal-likelihood term approximated by Monte Carlo and a fully analytic KL divergence, and optimize it with black-box variational inference. They claim that the method scales better than MCMC-based Bayesian SR, provides uncertainty quantification over symbolic structures, and, in Section 5.2, that 'across all equation datasets and noise levels, the minimum RMSE model produced by VaSST recovers the correct symbolic expression.' The paper evaluates VaSST on two simulated equations and four Feynman equations against BMS, BSR, QLattice, gplearn, and DEAP.","tokens_in":34780,"tokens_out":3502,"duration_ms":36468,"significance":"If the central claims were fully supported, the contribution would be significant: a scalable, fully probabilistic SR method with a principled prior over tree depth, an analytic KL divergence, and a coherent mechanism for sampling hard symbolic trees for uncertainty quantification. The authors ship a derivation of the marginal likelihood and KL terms that is standard and appears correct, and the reported predictive RMSEs are competitive while runtimes are substantially lower than the Bayesian competitors. The conceptual move of relaxing discrete symbolic structure for variational SR is interesting and worth publishing. However, the headline claim of exact symbolic recovery is contradicted by the paper's own appendix, and the variational surrogate is not shown to be faithful to the discrete posterior; these issues are load-bearing for both the recovery and the uncertainty-quantification claims.","major_comments":[{"comment":"The claim that 'across all equation datasets and noise levels, the minimum RMSE model produced by VaSST recovers the correct symbolic expression' is contradicted by the expressions reported in Appendix K. For FCE under the noiseless setting, Table K.7 gives VaSST's min-RMSE expression as −0.00285 + 0.00318 sin(2Bv/E_f) + 0.0012θ + 1.068(q sin(θ)Bv + E_f q). This is not q(E_f + vB sinθ): the coefficient on the main term is 1.068 rather than 1, and there are spurious sin(2Bv/E_f) and θ terms. The same non-exact expression is the top-1 sample in Table K.16. For CPE noiseless, the top-1 expression in Table K.15 contains a spurious G²m₁³m₂r₁⁻² term and coefficients −0.997 and 1.011; for FTC noiseless, the top-1 expression in Table K.17 has coefficients 0.999 and −0.905 plus an intercept and omits the exact coefficient 1 on κAT₂/d. Hence the ✓ entries for VaSST in Table 3 are not supported by","section":"§5.2, Table 3, Appendix K"},{"comment":"The stochastic ELBO replaces the hard design matrix T in log p(y|T) with the soft design matrix T_soft obtained from Binary Concrete/Gumbel-Softmax draws, but no bound or quantitative justification is given for treating this surrogate as faithful to the discrete ELBO in Eq. (9). The optimization is performed at temperatures annealed only to τ=0.5, so the final objective is still evaluated at continuous, mixed trees rather than hard symbolic trees. Unless the relaxation gap is shown to be small at the final temperature, the optimized variational parameters φ* need not represent a posterior over discrete symbolic structures, and the subsequent sampling of hard trees in Algorithm F.5 does not inherit a valid posterior interpretation. This affects both the structural-recovery claims and the uncertainty-quantification claims. The authors should either provide a theoretical bound or an empiric","section":"§4, Eq. (14)"},{"comment":"The reported 'minimum RMSE model' is selected post hoc as the best of H=2000 hard-tree samples ranked by in-sample RMSE. This is a selection procedure, not a posterior summary, and it can substantially inflate apparent recovery: among 2000 draws, the single best-fitting structure may match the data by chance even if the posterior mass on the true expression is negligible. The paper should report the posterior probability or frequency of the true symbolic expression among the H samples, or use Bayesian model averaging / posterior-weighted predictive summaries, before claiming that VaSST 'recovers' the correct expression. Without this, the central recovery claim is not a statement about the inferred posterior.","section":"§4, Algorithm F.5 and §5.2"}],"minor_comments":[{"comment":"The term 'unary operator pruning' in the deterministic pruning step is described only in words; a formal definition of the pruning map p would improve reproducibility, especially because the right subtree is removed when o_jζ is unary.","section":"§3.2 / Eq. (3)"},{"comment":"The cross-reference 'last two expressions in (4)' should refer to the KL derivations in Appendix G.2, not to Eq. (4); the wording is confusing.","section":"Appendix G.2"},{"comment":"The text says VaSST 'accurately recovers the true structure' for Eq. (15), but the reported coefficients in Table 1 and Appendix I.1 are not the true coefficients (e.g., 0.61 vs. 0.5 in Table 1). If the claim is structural recovery up to coefficient estimation, this should be stated explicitly and a tolerance or coefficient-error metric should be supplied.","section":"Section 5.1"},{"comment":"Several top-5 expressions contain duplicate terms (e.g., Table I.3: '+0.001381 sin(x1) + 0.001381 sin(x1)') and unusual factorizations (e.g., Table K.17, rank 2). While not incorrect, the lack of simplification makes reported expressions harder to audit; a canonical simplification step would help.","section":"Appendix I.3 / K.4"},{"comment":"There are minor typos: 'Dension et al., 1998' should be 'Denison et al., 1998'; 'constat' in Table I.3; 'BMSfailed' in Table K.10; and the abstract says 'superior performance' while the conclusion says 'strong structural recovery'. These should be corrected.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The paper has a solid variational derivation and a plausible algorithmic contribution, but the central empirical claim is overstated and directly contradicted by the appendix on the paper's own evidence. I would ask the authors to revise the claims, define 'correct symbolic recovery' precisely, report recovery frequencies or posterior probabilities, and address the relaxation-gap issue. This is within the scope of a major revision; the underlying idea is not irreparably flawed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The method is genuinely new: variational inference over soft symbolic trees for symbolic regression, with a depth-dependent parsimony prior, continuous relaxations, and an analytic marginalization over regression coefficients. The derivations (Appendix G) are standard and correct, the KL terms are worked out, the algorithms are clearly specified, and the runtime advantage over BMS/BSR is plausible. That part deserves credit.\n\nThe problem is the central empirical claim. Section 5.2 says: \"Across all equation datasets and noise levels, the minimum RMSE model produced by VaSST recovers the correct symbolic expression.\" The paper's own Appendix K contradicts this. For FCE (noiseless), the top-1 VaSST expression is -0.00285 + 0.00318 sin(2Bv/E_f) + 0.0012θ + 1.068(q sinθ Bv + E_f q), which is not q(E_f + vB sinθ). CPE and FTC show the same pattern: coefficients like 1.017, 1.068, 0.905, and spurious additive terms. So Table 3's checkmarks are not supported by the reported symbolic outputs. The paper never defines what \"correct\" means (exact? tolerance?), and no recovery frequencies or posterior probabilities over the H=2000 samples are given. The selection of the minimum in-sample RMSE is a form of post-hoc selection, not posterior-weighted prediction, which further inflates the apparent recovery.\n\nThe relaxed ELBO gap is a real concern but secondary: even if the surrogate were faithful, the empirical outputs do not match the exact-recovery claim. The paper also doesn't discuss bias between Eq. (14) and the true ELBO, so the UQ claims are not independently validated.\n\nIf the authors scale back to \"competitive structural recovery with occasional spurious terms,\" this is a useful contribution. As written, the headline claim is overstated and internally contradicted.\n\nRecommendation: send to peer review rather than desk reject — the method is novel and the flaws are fixable. The referee should ask for a precise recovery criterion, recovery rates over repeats, posterior probabilities over top structures, and either a bound on the relaxation gap or a tempered claim. I would not cite it until those are addressed.","headline":"Solid variational SR machinery, but the 'exact recovery' headline fails on the paper's own appendix; worth refereeing, not worth citing yet.","tokens_in":35174,"tokens_out":1563,"would_cite":false,"duration_ms":16525,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62F15","62J05"],"pacs":[],"model":"deepseek-v4-flash","headline":"VaSST claims that symbolic regression can be solved by variational inference over continuously relaxed expression trees, recovering true equations from noisy data with quantified uncertainty.","keywords":["symbolic regression","variational inference","soft symbolic trees","uncertainty quantification","continuous relaxation","Bayesian inference","scientific discovery","parsimony"],"falsifier":"A direct check is to enumerate all hard trees for a small problem (depth 2–3, few features), compute the exact ELBO at the optimized variational parameters, and compare it to the relaxed ELBO value and to the true posterior mode. If the relaxed optimum is far in function value or in structure from the true posterior, the UQ claim fails. Additionally, test VaSST on a low-noise dataset generated by an expression with nested divisions or exponentiation that the soft relaxation may represent poorly; failure to recover the true expression would undercut the recovery claim.","tokens_in":34255,"feed_emoji":"🧮","tokens_out":4449,"duration_ms":43310,"temperature":0.7,"pith_summary":"In symbolic regression, the goal is to recover a closed-form equation from data, but the space of expressions is combinatorial and existing probabilistic methods are either heuristic or slow. VaSST claims to solve this by treating each node of a symbolic tree as a soft probability distribution over operators and features, which turns the discrete search into a smooth optimization problem inside a variational inference framework. The method's load-bearing assertion is empirical: across two simulated laws and four Feynman equations, at noiseless and noisy levels, the minimum-RMSE model from VaSST recovers the correct symbolic expression while also producing a ranked list of plausible alternative structures. If this holds, it would give scientists a practical, uncertainty-aware tool for equation discovery that is both faster and more parsimonious than current Bayesian approaches.","feed_headline":"VaSST recovers correct equations on all Feynman benchmarks","feed_subtitle":"A variational relaxation of symbolic trees yields parsimonious physical laws plus uncertainty bounds, even under noise.","key_machinery":"Soft symbolic trees: a continuous relaxation of a symbolic expression tree in which each node carries a soft one-hot distribution over expansion indicators (Binary Concrete), operators, and input features (Gumbel-Softmax). The tree is evaluated with a soft-gating equation that mixes leaf features and unary/binary operations; this makes the entire design matrix differentiable in the variational parameters. The depth-dependent Bernoulli prior p_ζ = α(1+d_ζ)^{-δ} provides the parsimony mechanism.","core_discovery":"The paper's central claim is that symbolic regression can be recast as a scalable variational inference problem. VaSST replaces discrete expression trees with soft symbolic trees, in which each node's operator and feature choices are continuous probability vectors (via Binary Concrete and Gumbel-Softmax relaxations). The variational objective—a stochastic approximation of the ELBO that evaluates the marginal likelihood at the soft design matrix—is optimized by gradient descent under a temperature annealing schedule. After optimization, hard trees are sampled from the learned variational distribution and ranked by minimum in-sample RMSE. On two simulated laws and four equations from the Feynm","pith_inferences":["The relaxed ELBO could systematically favor smooth blends of operators; a testable consequence is that VaSST's structural recovery may degrade on equations requiring sharp discontinuities or nested compositions that the soft-gating evaluation at temperature 0.5 cannot approximate.","Ranking candidate expressions by minimum in-sample RMSE is a pragmatic selection rule but may overfit to noise; an alternative would weight structures by posterior evidence rather than predictive fit, which the paper does not explore.","The depth-dependent prior is the main parsimony lever, so a sensitivity analysis on α and δ would reveal how much of the recovery success is due to prior regularization versus the variational objective itself.","Because hard trees are sampled after optimization, the framework could be extended to full posterior predictive distributions and Bayesian model averaging over expressions, a direction the paper leaves implicit."],"forward_implications":["Symbolic regression becomes a gradient-based optimization problem rather than a discrete combinatorial search, scaling to larger datasets and higher-dimensional feature spaces.","The method yields a posterior distribution over symbolic structures, enabling uncertainty quantification in equation discovery instead of a single point estimate.","The depth-dependent split prior acts as a built-in parsimony regularizer, favoring simpler expressions and reducing overfitting.","On the tested benchmarks, VaSST recovers exact ground-truth equations across all noise levels, whereas many competing methods produce overly complex or incorrect forms.","VaSST is substantially faster than existing Bayesian symbolic regression methods (BMS and BSR), making fully probabilistic SR practical on moderately sized datasets."],"fun_headline_variants":["Soft symbolic trees turn symbolic regression into gradient descent","VaSST: variational inference for symbolic regression, no search needed","Uncertainty-aware symbolic regression via soft trees and ELBO","VaSST finds physical laws with gradient-based symbolic regression","From discrete trees to soft probabilities: VaSST for symbolic regression"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The method maximizes a relaxed ELBO that evaluates the marginal likelihood at soft symbolic trees; the paper gives no bound on how far this surrogate optimum is from the true posterior over hard trees, so the structural recovery and uncertainty claims rest on the relaxation gap being small.","fun_headline_variants_meta":{"raw":{"variants":["Soft symbolic trees turn symbolic regression into gradient descent","VaSST: variational inference for symbolic regression, no search needed","Uncertainty-aware symbolic regression via soft trees and ELBO","VaSST finds physical laws with gradient-based symbolic regression","From discrete trees to soft probabilities: VaSST for symbolic regression"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000555,"raw_usage":{"total_tokens":2445,"prompt_tokens":673,"completion_tokens":1772,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":417,"completion_tokens_details":{"reasoning_tokens":1698}},"tokens_in":417,"tokens_out":1772,"duration_ms":10933,"temperature":1.0,"reasoning_tokens":1698,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T20:17:38.689865+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct check is to enumerate all hard trees for a small problem (depth 2–3, few features), compute the exact ELBO at the optimized variational parameters, and compare it to the relaxed ELBO value and to the true posterior mode. If the relaxed optimum is far in function value or in structure from the true posterior, the UQ claim fails. Additionally, test VaSST on a low-noise dataset generated by an expression with nested divisions or exponentiation that the soft relaxation may represent poorly; failure to recover the true expression would undercut the recovery claim.","supporting_citations":[],"review_version":1}