Know When the Equation Is Wrong: Structural Uncertainty for LLM-Guided Symbolic Regression under Extrapolation

Mostafa Cham; Bayu Adhi Tama; Jianwu Wang

cs.LGOct 30, 2026

Know When the Equation Is Wrong: Structural Uncertainty for LLM-Guided Symbolic Regression under Extrapolation

Authors: Mostafa Cham, Bayu Adhi Tama, Jianwu Wang

Organizations: Department of Information Systems, University of Maryland, Baltimore County, Baltimore, MD, USA.

Abstract

Large language models (LLMs) can help discover mathematical equations from data, but most LLM-guided symbolic regression methods focus on a single best equation without quantifying uncertainty over alternative functional forms. This is challenging because distinct equations can fit the observed data similarly while producing substantially different extrapolations. Our experiments on the LLM-SRBench dataset show that standard conformal intervals achieve nominal in-distribution coverage but frequently fail under extrapolation. To tackle this challenge, we introduce LLM-SR-UQ, a framework that retains competing equation structures and uses their disagreement to characterize uncertainty about the underlying functional form. We extend the standard fixed-width conformal approach to quantify structural intervals, which measure disagreement among candidate equations and identify where prediction intervals should widen. Extensive experiments show our framework substantially improves coverage attainment by incorporating structural disagreement while preserving in-distribution coverage. Our approach also shows good generalizability when it is applied to other symbolic-regression methods.

Figures & tables

Figure 1: Overview of LLM-SR-UQ. Candidate equations are proposed (A), fitted and retained in a weighted buffer (B). An optional critic guides subsequent proposals without modifying the buffer (C). The retained equations are combined for prediction, while their disagreement informs conformally calibrated structural intervals (D).

Figure 2: Problem-level OOD coverage under split-conformal and structural intervals for A0, A1, LLM SR top-24 BMA, and Ensemble-SINDy. Each point represents a held-out problem, with split-conformal coverage on the horizontal axis and structural coverage on the vertical axis. The diagonal indicates equal coverage. Dashed lines mark the attainment threshold of 0.88, corresponding to the nominal target of 0.90 with tolerance

\epsilon=0.02

. Panel headings report counts of coverage increases and decreases and transitions into or out of attainment.

		NMSE		Coverage		OOD attainment	OOD sharpness	OOD IS
Method	Interval	ID	OOD	ID	OOD	$A_{0.1}$	$S_{0.1}$	IS
Direct, Llama-8B	split	2.1e-2	2.84	0.902	0.629	0.55 (60/109)	4.45e-1	9.08e-1
Direct, GPT-4o-mini	split	2.0e-2	2.76	0.894	0.646	0.57 (62/109)	3.83e-1	7.94e-1
Ensemble-SINDy	split	3.2e-7	6.3e-2	0.899	0.365	0.26 (28/109)	2.03e-4	5.58e-2
Ensemble-SINDy	structural	3.2e-7	6.3e-2	0.900	0.762	0.62 (68/109)	9.29e-3	1.89e-2
LLM-SR, single best	split	4.2e-6	2.2e-3	0.891	0.627	0.52 (57/109)	4.82e-3	1.54e-2

Table 1: Prediction accuracy and uncertainty performance across symbolic-regression methods.

Figure 3: Empirical cumulative distribution of OOD coverage across 109 held-out problems. Each curve shows the fraction of problems with empirical coverage at or below the horizontal-axis value. A0, A1, LLM-SR top-24 BMA, and Ensemble-SINDy use structural intervals. Direct-prompting baselines and LLM-SR single-best use split-conformal intervals. The vertical dotted line marks the attainment threshold of 0.88. Legend values report coverage attainment

A_{0.1}

Appendix figures & tables14 assets

Supplementary material from the paper’s appendix.

Appendix

Split	Content	Labels	Role
$\mathcal{D}_{\mathrm{fit}}$	$80\%$ of the training rows ( $n=3{,}200$ )	yes	fit and score structures
$\mathcal{D}_{\text{cal-ID}}$	the other $20\%$ ( $m=800$ )	yes	calibrate intervals
$\mathcal{D}_{\text{cal-EXT}}$	points just beyond the training range	no	measure disagreement (critic only)
$\mathcal{D}_{\mathrm{test}}$	benchmark test set, inside the range	yes	final evaluation
$\mathcal{D}_{\mathrm{OOD}}$	benchmark test set, beyond the range	yes	final evaluation

Appendix

Table 2: Data splits. ID: in-distribution (inputs inside the training range). EXT and OOD: extrapolation and out-of-distribution (inputs beyond it).

Figure 4: Two held-out problems at A1, error against the most-shifted input.

		untempered		tempered
Configuration	$n$	median $\tilde{w}_{1}$	median $B_{\mathrm{eff}}$	median $w_{1}$	median $B_{\mathrm{eff}}$
A0	109	1.0000	1.00	0.240	8.40
A1	109	1.0000	1.00	0.227	8.40
A2 ( $8$ B critic)	108	1.0000	1.00	0.229	8.40
A3 ( $8$ B critic)	107	1.0000	1.00	0.219	8.40
A2 (GPT-4o critic)	107	1.0000	1.00	0.243	8.40

Appendix

Table 3: Concentration of the retained buffer. The tempered columns are pinned by construction (

B_{\mathrm{eff}}=8.4

) and are shown only so the gap is visible; the untempered columns are what the evidence says.

Rung	Subset	$n$	$A_{0.1}$ split	$A_{0.1}$ $S$
A0	evidence degenerate	86	0.58	0.86
A0	some spread	23	0.57	0.74
A1	evidence degenerate	62	0.57	0.88
A1	some spread	47	0.49	0.98

Appendix

Table 4: Attainment under split and structural intervals, split by whether the evidence weights are degenerate. The construction delivers its gain in both regimes.

Configuration	with ref.	FR@1	FR@3	FR@10	FR@3 $\wedge\,\neg$ FR@1
A0	105	8	8	8	0
A1	104	12	12	12	0
A2 ( $8$ B critic)	104	13	14	15	1
A3 ( $8$ B critic)	103	11	13	13	2
A2 (GPT-4o critic)	103	10	10	10	0
A3 (GPT-4o critic)	104	10	11	12	1

Appendix

Table 5: Functional recovery by rank cutoff. The column that matters is the last: problems on which the retained set contains a recovery that its top-ranked member misses.

Figure 5: Attainment against the tempering target, held-out, all six configurations. The grey band spans the split-conformal reference across configurations. The entire effect is the step from a point mass (

B_{\mathrm{eff}}=1

) to any genuine spread; above

B_{\mathrm{eff}}=2

the curves are flat to within

0.11

and every one of them beats split conformal by

20

–

40

points. Hollow rings mark targets the bisection could not reach for some problems, which is why the

B_{\mathrm{eff}}=24

column is not a clean uniform-weighting condition: it is unreachable on

9

–

46

problems per configuration, against at most

5

at every smaller target.

Rung	Feedback to proposer	Critic	Isolates
A0	frozen prompt (problem description only)	—	search + calibration
A1	+ mechanical diagnostics (fixed template)	—	plumbing
A2	+ critic guidance from point-fit diagnostics	yes	critic from fit alone
A3	+ label-free calibration diagnostics (divergence	yes	the claim
	beyond the data, interval growth)

Appendix

Table 6: Ablation ladder. Every rung reports coverage, so uncertainty is not a late addition, and every rung has the same budget of

125

language-model calls. A1 is the control: if A2

\approx

A1, mechanical feedback was the whole story.

	NMSE		OOD	$A_{0.1}$
Rung	ID	OOD	cov.	$S$	split	$S_{0.1}$	FR@1	params / plateau
A0	3.3e-6	2.4e-3	0.906	0.83 (91/109)	0.58	1.27e-2	8/105	6.7 / 0.52
A1	2.0e-8	6.0e-5	0.964	0.92 (100/109)	0.53	1.12e-3	12/104	8.2 / 0.42
critic $=$ the proposer model (Llama-3.1-8B-Instruct)
A2	9.3e-9	2.0e-5	0.924	0.87 (91/105)	0.52	1.87e-3	13/104	8.1 / 0.46
A3	4.4e-7	4.0e-4	0.947	0.88 (93/106)	0.56	7.73e-3	11/103	7.4 / 0.48

Appendix

Table 7: The ablation ladder, held-out, at a matched budget of

125

language-model calls per problem (A2/A3:

101

proposer rounds and

24

critic calls). NMSE are medians; coverage and

A_{0.1}

use structural intervals, and the split column gives

A_{0.1}

of split conformal around the same predictor. Parameters are the mean coefficient count of the reported equation. Plateau is the fraction of rounds after the last improvement of the best score.

Figure 6: Search behaviour across the ladder, held-out. (a) Progress of the best buffer score, normalized per problem so that

0

is the first round and

1

the final best, plotted against the language-model call index (critic calls included), median over problems. (b) Fraction of proposer rounds spent after the last improvement, per problem. The six configurations are nearly indistinguishable: every one reaches

90\%

of its final score by a median of

32

41

calls and then spends roughly half its budget without improving.

critic calls	24	edits proposed	72
failed calls	0	edits valid	44
prompt tokens	39,120	edits fitted	44
completion tokens	10,245	edits entering buffer	0
directives proposed	72	bans proposed	111
directives valid	52	bans valid	110
slots trimmed	19	proposals under ban	701

Appendix

Table 8: Critic activity on the problem of Section D.2 :

24

calls over

101

proposer rounds, costing about

49

k tokens in total.

round	leading structure	log-evidence	families
1	c0+c1*(Pc2)/(c3+tc4)	5,496	8
2	c0+c1Pc2/(c3+tc4)+c5P**c6	16,847	16
5	c0+c1Pc2+c3Pc4	16,855	24
6	c0+c1P*c2+c3Pc4+c5tc6	17,207	24
7	c0+c1Pc2+c3Pc4+c5softplus(c6t)**c7	18,320	24
39	c0+c1P*c2+c3Pc4+c5c6c7/(c8+t**c9)	26,398	22

Appendix

Table 9: Every improvement in the leading structure, and the final round. Buffer size is capped at

24

; the effective model count is pinned at

8.4

by the temperature controller of Section 3.5 , as designed.

rank	best expression in family	log-evidence	weight	coeffs
1	c0+c1P*c2+c3Pc4+c5c6c7/(c8+t**c9)	26,398	0.313	10
2	c0+c1P*c2+c3tc4+c5Pc6	17,564	0.059	7
3	c0+c1P**c2+c2Pc3/(c4+tc5)	17,168	0.057	6
4	c0+c1P*c2+c3Pc4+c5Pc6/(c7+c8t*c9)	19,388	0.052	10
5	c0+c1Pc2+c3Pc4+c5softplus(c6t)**c7	18,320	0.040	8

Appendix

Table 10: Top five families of the final buffer, with the whole buffer’s weight summing to

1

. Every family is a sum of powers of

P

and

t

; none contains a saturating term of the kind the ground truth requires.

	in-distribution ( $\mathcal{D}_{\mathrm{test}}$ )	shifted ( $\mathcal{D}_{\mathrm{OOD}}$ )
NMSE	$1.0\times 10^{-8}$	$7.1\times 10^{-4}$
$\mathrm{Acc}_{0.1}$	$0.994$ of points	$1.000$ of points
coverage	$0.886$	$1.000$
coverage gap	$-0.014$	$+0.100$
mean interval width	$1.7\times 10^{-4}$	$3.3\times 10^{-1}$

Appendix

Table 11: Final metrics, nominal level

0.90

. Widths are absolute; the OOD interval is roughly

1{,}900

times wider than the in-distribution one.

Symbol	Meaning
$x_{i}$ , $y_{i}$	input vector and scalar target of data point $i$
$M$ , $M_{k}$	a structure: the shape of an equation with its coefficients left free; $M_{k}$ is candidate $k$
$\theta$ , $c_{0},c_{1},\dots$	the free coefficients of a structure, and the names they carry in proposals
$\hat{\theta}_{k}$	the coefficients of $M_{k}$ fitted on $\mathcal{D}_{\mathrm{fit}}$
$\mathcal{D}_{\mathrm{fit}},\mathcal{D}_{\text{cal-ID}},\mathcal{D}_{\text{cal-EXT}}$	fitting split; labelled calibration split; unlabelled probe region just beyond the data (Table 2 )
$\mathcal{D}_{\mathrm{test}},\mathcal{D}_{\mathrm{OOD}}$	the benchmark’s in-distribution and shifted test splits, used only at the end

Appendix

Table 12: Symbols used in Section 3 , in order of appearance.

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L2G-SR: Learning to Guide LLM-based Symbolic Regression

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Peking University

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Open paper AskPDF arXiv

Symbolic regression searches for an equation that explains observed data. Numerical error is commonly used to guide the search, but it alone can favor an accurate surrogate over a candidate with the right structure. We propose L2G-SR, a framework for LLM-based symbolic regression that guides search with a learned structural verifier. We construct structural supervision signal using tree edit distance (TED), accounting for commutative operators and alternative expression forms. A verifier first aligns numerical observations with the formulas, then learns to score a candidate's structural similarity to the target formula. At search time, the target formula is hidden and the verifier complements numerical error in parent selection. An analyst then uses verifier feedback and diagnostic tools to propose revisions, which a generator converts into refined expressions. On LLM-SRBench, L2G-SR achieves state-of-the-art numerical fit, equation level accuracy and TED across the evaluated LLM backbones and domains. L2G-SR demonstrates the value of learned structural guidance for LLM-based symbolic regression.

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