{"id":"3199918e-019d-41b9-877c-49b2471613ef","arxiv_id":"2505.07956","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"Vision-capable LLMs can propose useful function forms from plots alone, and combining this with KANs gives a viable route to multivariate symbolic regression.","lead":"This paper reports a symbolic regression method that shows a plot of a function to a vision-capable language model, which proposes candidate formulas whose parameters are then fitted numerically and evolved in a genetic algorithm. Pairing that idea with Kolmogorov-Arnold networks extends the trick to multi-variable formulas, turning fitted network edges back into human-readable expressions.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"KAN-LEx's multivariate extension depends on trained KAN edge functions being simple; the paper's own footnote concedes no guarantee, and the three examples do not establish this.","rationale":"The reader's weakest_assumption correctly identifies the KAN edge simplicity assumption as the central risk. My stress-test agrees with this assessment. The paper's own footnote admits the lack of theoretical guarantee, and the empirical evidence (three hand-picked multivariate examples) is far too thin to establish the general claim. The univariate part of the paper is a stronger proof of concept: it demonstrates reproducible exact recoveries on a set of simple functions, with code and prompts available, and honestly notes limitations such as noise sensitivity and the cost of inference. The multivariate extension, by contrast, is the most load-bearing part because it supports the headline claim of extending the method to multivariate functions, and it is the least tested. The three examples are all 'nice' functions where the KAN edges turn out to be simple (linear, log, exp, sin); no counterexample is attempted, and no stress test of the edge-simplicity assumption is reported. A missing comparison to modern symbolic regression baselines (e.g., PySR, Operon) is a real weakness for the empirical claims, but it does not threaten the internal logic of the method as directly as the edge-simplicity assumption does. The lack of quantified LLM stochasticity is also secondary: the genetic algorithm is explicitly designed to exploit stochasticity as mutation, and the paper reports multiple runs in several places. Therefore, the single most load-bearing concern is the unresolved premise that trained KAN edge functions are simple enough for the vision LLM to identify symbolically. Since the reader has already conditioned the verdict on this assumption, and my review does not find an additional independent flaw, the verdict should remain CONDITIONAL. I recommend UNCHANGED. A concrete test (e.g., two simple but non-cherry-picked multivariate targets like sin(x*y) or x^2+y^2+sin(x*y)) would settle whether the assumption holds widely or only in favorable cases.","tokens_in":13704,"tokens_out":4110,"duration_ms":45242,"concrete_test":"Train and prune a [2,4,4,1] KAN on f(x,y)=sin(x*y) over [0,1]^2, then run the paper's LLM-LEx pipeline on each edge with gpt-4o. Check whether the final simplified expression recovers sin(x*y) to score < 1e-5. Repeat on f(x,y)=x^2+y^2+sin(x*y) as a second case. If edge ansätze do not match the actual KAN edge functions, or the composed expression fails the threshold, the edge-simplicity assumption is not generally valid.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim that 'univariate is all you need for symbolic regression' transfers to multivariate functions only if each edge of a trained, pruned KAN is a univariate function simple enough for a vision LLM to propose a correct symbolic ansatz. This is not guaranteed by the Kolmogorov-Arnold theorem. The paper itself concedes in Section 3.1, footnote 3: 'there is no theoretical guarantee the required univariate functions are simple or remotely well behaved.' In practice, a KAN edge function is a spline fitted to the target data; its complexity depends on training initialization, regularization, and pruning, not on the symbolic simplicity of the target. If the edge is a complicated spline, the LLM will propose an ansatz that does not match it, and the simplification step cannot recover from a fundamentally wrong edge form. The three multivariate examples (Section 3.2) are low-dimensional and carefully chosen—exp[sin(pi x) + y^2], xy, and xy/z—and the paper reports no failure case. The univariate results in Table 1 already show several functions that LLM-LEx fails to recover exactly, so the edge-simplicity assumption is not automatically satisfied. The whole multivariate pipeline therefore rests on an empirical hope rather than on a demonstrated property of KAN training. This is the load-bearing weakness: if it fails for a generic multivariate target, the 'univariate is all you need' claim is unsupported for multivariate symbolic regression.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces LLM-LEx, a symbolic regression method in which a vision-capable LLM is prompted with a plot of a univariate function and two parent candidate expressions and asked to propose a Python lambda ansatz; free parameters are fitted numerically and the population is evolved with a genetic algorithm inspired by FunSearch. The authors benchmark LLM-LEx against Mathematica's FindFormula on 20 generated functions, test open-source LLMs, study robustness to noise, and show that special functions can be recovered through prompt engineering. They then propose KAN-LEx, which trains and prunes a Kolmogorov-Arnold network, fits each edge function with LLM-LEx, composes the resulting univariate expressions, and simplifies the result with sympy and an LLM. Three multivariate examples are reported: exp[sin(pi x)+y^2], xy, and xy/z. The paper includes a public GitHub repository and explicitly disclaims state-of-the-art performance.","tokens_in":13979,"tokens_out":9376,"duration_ms":87759,"significance":"The central idea—using an LLM's visual pattern recognition as a symbolic-ansatz proposer inside an evolutionary loop—is novel and well-motivated, and the authors are appropriately modest about positioning the work as a proof of concept rather than a competitive benchmark. The availability of code and notebooks is a strength, as is the demonstration that open-source models can run the method locally. If the results are reproducible, the paper makes a useful contribution to the growing literature on LLM-guided symbolic regression. However, the multivariate claim that 'univariate is all you need' is currently under-supported: it rests on an empirical assumption about KAN edge simplicity that the paper itself concedes is unproven, and the three multivariate examples are all favorable cases. The univariate evaluation is also thin, with no comparison to standard genetic-programming baselines such as PySR or Operon. These issues, together with a concrete error in the scoring formula, mean that the paper's significance depends on the requested revisions.","major_comments":[{"comment":"The multivariate extension rests on the assumption that each edge function of a trained, pruned KAN is simple enough for the plot-based LLM to propose a correct symbolic ansatz. The paper itself concedes in footnote 3 that 'there is no theoretical guarantee the required univariate functions are simple or remotely well behaved,' and the three examples in §3.2—exp[sin(pi x)+y^2], xy, and xy/z—are all low-dimensional and all succeed. This is not sufficient to support the abstract's claim that 'univariate is all you need' for symbolic regression, especially since Table 3 reports only 12/20 exact recoveries for gpt-4o on univariate functions; a KAN edge from a general target could be no simpler. I ask for a systematic multivariate evaluation with reported success/failure rates on a broader set of targets, including cases where the Kolmogorov-Arnold decomposition is not visually simple, or, alternatively, a substantial rewording of the multivariate claim to state that the method applies when the trained KAN edge functions are simple.","section":"§3.1 (footnote 3), §3.2"},{"comment":"The definition of MADy as median({yi - median({yj})}) is mathematically incorrect: the median of the centered values yi - median(y) is identically zero for any finite sample. Consequently, the max in global-scale is vacuous and global-scale always reduces to alpha * mean(|yi|), contrary to the stated intention of a 'non-vanishing characteristic scale.' The presumably intended quantity is the median absolute deviation, median(|yi - median(y)|). Since the score in Eq. (3) drives selection and termination in every reported experiment, this error is load-bearing; the definition should be corrected and the authors should state whether any reported scores or thresholds change as a result.","section":"§2.2, Eq. (4)"},{"comment":"The central claim that LLM-LEx is 'viable and surprisingly successful' is supported only by a comparison to Mathematica's FindFormula on 20 functions, with one or two stochastic LLM runs per function and manual inspection of correctness. The paper cites PySR [1] and Operon [2] but does not compare against them, so it is unclear how the method fares relative to standard symbolic regression packages on the same benchmark. Because the LLM responses are probabilistic, the reported 'Total Correct' counts in Table 3 need to be accompanied by repeated-run success rates (e.g., success probability over at least several runs per function) and ideally confidence intervals. This additional evidence is necessary to substantiate the univariate viability claim.","section":"§2.3, Tables 1 and 3"}],"minor_comments":[{"comment":"Typo: 'throuhout' should be 'throughout'; in §2.3, 'at the expensive of some quality' should be 'at the expense of some quality.'","section":"§2, first paragraph"},{"comment":"The notation 'mean({|yi|}, ϵ)' is undefined; please specify how the small hyperparameter ϵ enters the mean (for example, as a floor or an additive offset).","section":"§2.2, Eq. (4)"},{"comment":"The caption states 'All runs with LLM-LEx were single shot,' which appears to conflict with Table 1's '# LLM-LEx runs' column reporting values of 1 and 2; please clarify the intended meaning of 'single shot' and reconcile the two tables.","section":"Table 3 caption"},{"comment":"The list of returned expressions is missing a comma between 'y*x' and 'x**1 * y**1', making the list syntactically confusing.","section":"§3.2, Example 2, Eq. (17)"},{"comment":"The caption says that below the dashed red line the genetic algorithm is stopped early, but the score threshold corresponding to the line is not stated; please add this value.","section":"Figure 5"}],"recommendation":"major_revision","confidential_remarks":"This is a promising proof-of-concept with public code and an honest discussion of limitations. The main risk is overclaiming the multivariate extension from three favorable examples, and the scoring formula in Eq. (4) contains a real error that affects all experiments. I would recommend major revision rather than rejection because the issues are fixable within the paper's scope: correct the MAD definition, add a PySR/Operon comparison with repeated runs, and provide a broader multivariate evaluation or carefully limit the claim."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. The univariate pipeline—a vision LLM proposing a Python lambda ansatz from a plot, a FunSearch-style genetic loop evolving a population, scipy fitting the free parameters—is new in that exact combination, and it genuinely works on a surprising fraction of a 20-function benchmark. The multivariate extension via KAN edge fitting is far less established: it rests on the hope that trained KAN edges are simple enough for the LLM to guess, and the paper itself concedes (footnote 3) there is no guarantee. The stress-test note is right that this is the load-bearing assumption, and the three clean multivariate examples do not lift it.\n\nCredit where due. The paper is honestly scoped as a proof of concept, not a state-of-the-art competitor. Code and prompts are public. The benchmark reports both exact recoveries and failures, the Mathematica comparison is fair, and the noise experiments are sensible. The authors are upfront about the probabilistic nature of the LLM and the resulting run-to-run variation.\n\nSoft spots, in proportion. The benchmark is small: 20 functions, mostly one or two runs per setting, no comparison to PySR or Operon. LLM stochasticity is not quantified with repeated seeds. Eq. 4 is missing an absolute value in the median absolute deviation, a typo but one that matters for the score. And the abstract's 'demonstrate that univariate is all you need' overreaches: three examples are a proof of concept, not a demonstration of a general method. The multivariate pipeline has no reported failure cases, so you cannot gauge its reliability.\n\nWho it helps: people thinking about LLM-guided scientific discovery, interpretable ML, and symbolic regression as a tool. The univariate result is credible enough to build on. The multivariate claim needs to be presented as a feasibility study rather than a general solution.\n\nMy recommendation: send it to peer review with room for revision. A referee should ask for repeated runs or a confidence bound on the univariate success rate, a baseline comparison with PySR on the same 20 functions, and a softening of the multivariate claim.","headline":"A genuinely new vision-LLM + FunSearch + KAN pipeline for symbolic regression; the univariate part is solid and honestly scoped, the multivariate extension is a proof of concept resting on an unproven simplicity assumption.","tokens_in":14522,"tokens_out":3070,"would_cite":false,"duration_ms":31115,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A vision-capable LLM shown only a plot can propose ansätze that evolve into exact closed forms, and a Kolmogorov–Arnold decomposition lifts the trick to multivariate functions.","keywords":["symbolic regression","multimodal large language models","Kolmogorov-Arnold networks","genetic algorithm","visual function discovery","prompt engineering","special functions","exact expression recovery"],"falsifier":"Train a KAN on dense samples of $f(x,y)=\\sqrt{|x-y|}$, prune it, and run KAN-LEx on each edge for twenty independent seeds with the default prompts. If the composed expression recovers the target at the paper's default score threshold of $10^{-5}$ rarely or not at all, the claim that edge simplicity is not an obstacle in practice fails for functions whose Kolmogorov–Arnold decomposition forces a cusped univariate edge.","tokens_in":13509,"feed_emoji":"📈","tokens_out":11756,"duration_ms":114257,"temperature":0.7,"pith_summary":"The paper tries to establish that symbolic regression can be run by showing a vision-capable large language model a plot of a function and asking it to write an ansatz, then fitting the free parameters and evolving the ansätze with a genetic algorithm. The authors argue this works because the LLM's pretraining instils a human-like simplicity bias: it proposes compact functional forms without being given any basis-function library. They then extend the same idea to multivariate targets through Kolmogorov–Arnold Networks, fitting each univariate edge function separately, composing the results, and compressing them with symbolic simplification and a second LLM pass. If the approach holds, interpretable expressions can be recovered from pure visual input, with prompt conditioning as a cheap way to steer the search.","feed_headline":"Plot-reading LLM finds exact formulas; KAN lifts to multivariate","feed_subtitle":"Show a graph, evolve the ansätze with a genetic loop, and closed forms emerge—even for multivariate functions.","key_machinery":"The machinery has two parts. First, a genetic algorithm over LLM proposals: a population of lambda functions is scored by a scale-normalised residual that emphasises shape rather than large values; pairs are sampled by softmax over scores and handed to the LLM as two example candidates plus the plot, and the reply is the next generation's ansatz. Second, a Kolmogorov–Arnold Network—a layered graph whose edges carry univariate spline functions instead of weighted sums—provides the decomposition: pruning the trained KAN reduces the architecture, and each surviving edge becomes a univariate symbolic-regression problem. The final pass compounds symbolic simplification, coefficient refitting, and an LLM-guided step that drops negligible terms and recognises Taylor-series patterns, which matters because per-edge constants are not unique and composition does not commute with fitting.","core_discovery":"The paper claims that symbolic regression can be decomposed into univariate regressions and solved by a vision-capable LLM: shown a plot of a univariate function, the model proposes a compact Python-$\\lambda$ ansatz; numerical fitting fixes its parameters; and a genetic loop, seeded with two scored candidates and prompted to give an improved ansatz, evolves a population until exact recovery. On a 20-function benchmark this loop recovers exact expressions more often than the proprietary symbolic-search baseline. The multivariate extension, KAN-LEx, is the sharper claim: after training and pruning a Kolmogorov–Arnold Network, the target function is a sum of composed univariate edge functions, and applying the same plot-driven fitting edge by edge yields, after symbolic simplification and refitting, the exact closed form in all three worked examples ($\\exp[\\sin(\\pi x)+y^2]$, $xy$, and $xy/z$).","pith_inferences":["Our inference: the boundary of the method is the model's pretraining vocabulary; fine-tuning a small vision-language model on synthetic plots of specialised functions should extend exact recovery to domains where a general model only approximates.","Our inference: plotting is an information bottleneck—features below plot resolution are invisible to the LLM, so multi-resolution crops or zoomed views of the same function would be a cheap test of whether the vision input is the limiting step.","Our inference: the same loop could be aimed at physics-law discovery by prompting for expected symmetries, units, or leading terms; the paper lists domain prompting as future work, but treating it as a tunable prior is a natural next experiment."],"forward_implications":["On a 20-function benchmark, the plot-only loop recovers the exact expression in 12 cases, while the proprietary symbolic-search baseline recovers 8; exact recovery is therefore not an occasional accident.","Because no basis-function library is specified, the search can be conditioned by prompt wording alone: one prompt steers the model to return an exact Bessel-function call rather than a trigonometric approximation.","For multivariate targets, pruning a KAN narrows the architecture before per-edge fitting, and the worked examples—an exponential of sine-plus-square, a product, and Newton's gravitational law—all reduce to the correct closed form after composition and simplification.","The method tolerates modest noise and still finds the ground-truth trend, but the user must raise the exit threshold to match the noise level, otherwise the loop overfits.","Open-weight local models can match the proprietary model's exact-recovery count but take substantially longer, so the approach does not depend on one commercial API."],"supporting_citations":[{"why":"Supplies the evolutionary loop in which LLM-generated programs are selected and recombined; LLM-LEx adapts that loop for plot-based ansätze.","marker":"[4]"},{"why":"Defines Kolmogorov–Arnold Networks and their pruning routines, which decompose multivariate targets into univariate edge functions.","marker":"[16]"},{"why":"Provides the symbolic simplification pass that composes per-edge expressions, cancels redundant constants, and prepares the final refit.","marker":"[17]"},{"why":"The proprietary symbolic search tool used as the baseline in the benchmark tables on exact-expression recovery.","marker":"[6]"},{"why":"Supplies the genetic-algorithm background for the population evolution and selection step.","marker":"[7]"}],"fun_headline_variants":["Plot-reading LLM evolves exact formulas; KAN extends to multivariate","LLM reads plot, evolves ansatz; KAN turns univariate into multivariate","Vision LLM + genetic loop + KAN: exact symbolic regression","Univariate is all you need: LLM reads plots, KAN scales to multivariate","Show a graph, LLM proposes ansatz, genetic loop finds the formula"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that a pruned KAN's edge functions are simple enough that a vision LLM can guess their closed forms from plots; the paper admits there is no guarantee of this, only that its examples suggest it.","fun_headline_variants_meta":{"raw":{"variants":["Plot-reading LLM evolves exact formulas; KAN extends to multivariate","LLM reads plot, evolves ansatz; KAN turns univariate into multivariate","Vision LLM + genetic loop + KAN: exact symbolic regression","Univariate is all you need: LLM reads plots, KAN scales to multivariate","Show a graph, LLM proposes ansatz, genetic loop finds the formula"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001158,"raw_usage":{"total_tokens":4770,"prompt_tokens":891,"completion_tokens":3879,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":507,"completion_tokens_details":{"reasoning_tokens":3779}},"tokens_in":507,"tokens_out":3879,"duration_ms":27275,"temperature":1.0,"reasoning_tokens":3779,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T22:07:33.971956+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Train a KAN on dense samples of $f(x,y)=\\sqrt{|x-y|}$, prune it, and run KAN-LEx on each edge for twenty independent seeds with the default prompts. If the composed expression recovers the target at the paper's default score threshold of $10^{-5}$ rarely or not at all, the claim that edge simplicity is not an obstacle in practice fails for functions whose Kolmogorov–Arnold decomposition forces a cusped univariate edge.","supporting_citations":[{"cited_title":"Mathematical discoveries from program search with large language models,","cited_arxiv_id":null,"evidence_quote":"Supplies the evolutionary loop in which LLM-generated programs are selected and recombined; LLM-LEx adapts that loop for plot-based ansätze."},{"cited_title":"Sympy: symbolic computing in python,","cited_arxiv_id":null,"evidence_quote":"Provides the symbolic simplification pass that composes per-edge expressions, cancels redundant constants, and prepares the final refit."},{"cited_title":"A survey of parallel genetic algorithms,","cited_arxiv_id":null,"evidence_quote":"Supplies the genetic-algorithm background for the population evolution and selection step."}],"review_version":1}