SURFACEBENCH: A Geometry-Aware Benchmark for Symbolic Surface Discovery
Authors: Sanchit Kabra
Shobhnik Kriplani
Parshin Shojaee
Chandan K. Reddy
Venue: Accepted by TMLR ()

Abstract: Equation discovery from data is a central challenge in machine learning for science, which requires the recovery of concise symbolic expressions that govern complex physical and geometric phenomena. Recent large language model (LLM) approaches have shown promise in symbolic regression, yet existing benchmarks predominantly evaluate low-dimensional scalar functions and rely on string-level or regression-based metrics that fail to capture structural and geometric equivalence. We introduce SURFACEBENCH, the first geometry-aware benchmark for symbolic discovery of three-dimensional surfaces. Unlike scalar curve-fitting tasks, SURFACEBENCH targets surface-level reasoning, where multi-variable coupling, coordinate transformations, and geometric structure must be inferred directly from data. The benchmark comprises 183 analytically constructed, science-inspired surface equations across 15 categories and three representation paradigms: explicit, implicit, and parametric forms. Each task includes variable semantics and synthetically sampled 3D data, and is designed to stress symbolic composition, structural ambiguity, and representational non-uniqueness while mitigating memorization. To evaluate discovery quality, SURFACEBENCH incorporates symbolic equivalence checks with geometric metrics of the object-space (Chamfer and Hausdorff distances) and regression-based error measures, allowing evaluation of functional fidelity beyond algebraic syntax. Empirical evaluation across evolutionary, neural, and LLM-driven frameworks reveals that no current method achieves consistent performance across representation types, with LLM-based approaches exhibiting strong structural priors but limited robustness in parameter calibration and multi-equation reasoning. SURFACEBENCH provides a challenging and diagnostic testbed that bridges symbolic reasoning and geometric reconstruction, enabling principled benchmarking of compositional generalization and structure-aware scientific induction in high-dimensional equation discovery. The code and data are available at this link: https://github.com/deep-symbolic-mathematics/surfacebench.

Published in Transactions on Machine Learning Research (03/2026)
SURFACEBENCH: A Geometry-Aware Benchmark for Symbolic Surface Discovery
Sanchit Kabra Deportment of Computer Science, Virginia Tech
sanchit23@vt.edv
Shobhnik Kriplani Department of Computer Science, Virginia Tech
Parshin Shojace Department of Compuler Science, Virgincia Tech
Chandan K. Reddy Department of Computer Science, Virginia Tech
Reviewed on OpenReview: https: o/perrvveee... net/ ftrum?? id= sHLTzkesSt
Abstract
Equation discovery from data is central challenge in machine learning for science, which requires the recovery of concise symbolic expressions that govern complex physical and geo- metric phenomena. Recent large language model (LLM) approaches have shown promise in symbolic regression, yet existing benchmarks predominantly evaluate -ww-dimeniionl scalar functions and rely on string-level or regression-baet metrics that fail to capt.re structural and geometric equivalence. We introduce SURFACEBENCH, the first geometry-awar bench- mark for symbolic discovery of threo-dimensional surfaces. Unlike scalar curvo-fitting tasks, SUNFACEBENCH targels surface-level reasoning, where multi-variable coupling, coordinate transformations, and geometric structure must be inferred directly from data. The bench- mark comprises 183 analytically constructed, science-inspired surface equations across 15 categories and three representation paradigms: explicit, implicit, and parametric forms. Each task includes variable semantics and synthetically sampled 3D data, and is designed to stress symbolic composition, structural ambiguity, and representational non-umiqueness while mitigating memorization. To evaluate discovery quality, SURFACEBENCH incorpo- rates symbolic equivalence checks with geometric metrics of the object-space (Chamfer and Hausdorff distances) and regression-based error measures, allowing the evaluation of func- tional fidelity beyond algebraic syntax. Empirical evaluation across evolutionary, neural, and L.M--rivee frameworks reveals that no current method achieves consistent. perfor- mance across representation types, with LLM-based approaches exhibiting strong struc- tural priors but limited robustnoss in parameter calibration and multi-equation reason- ing. SURFACEBENCH provides a challenging and diagnostic testbed that bridges sym- bolic reasoning and geometric reconstruction, enabling principled benchmarking of com- positional generalization and structure-aware scientific induction in high-dimensional equa- tion discovery. The code and data are available at this link: github. grtub..co/ddep--ymmblicc /deep-symbolic- mathhenatic///uuacccbenh
1 Introduction
Symbolic regression, or equation discovery, lies al the core of machine learning for scientific discovery (Reddy & Shojaee 2025). Its objective to recover interpretable mathematical expressions that explain observed data and reveal the governing structure of physical, biological, and engineered systems. By transforming raw observations into symbolic form, symbolic regression enables the identification of functional relationships, invariances, and compositional structure that underlie complex phenomena.

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Figure 1: SURFACEBENCH: A benchmark suite for symbolic regression featuring 183 surface equations spanning 15 structurally defined categories. The benchmark representations, illustrating diverse symbolic and geometric challenges.
explicit (red), implicit (blue), and parametric (yellow)
Traditional approaches relied heavily on domain experts to mamually specify relevant variables, transforma- While scientifically grounded, such workflows are
tions, and functional templates (Brunton et al. 2016
labor-intensive and difficult to scale. Automated alternatives based on genetic programming, evolutionary
search, and neural sequence modeling (Cranmer 2023
Biggio et al.j 2021 Shojaee et al., 2023 Petersen
et al. 2019 Landajuela al.) [0022] Kamieeny al.
2022) alleviate manual design but face substantial
combinatorial complexity and often require extensive search to navigate vast hypothesis spaces. More re-
cently, large language models (LLMs) have been introduced structural priors for symbolic discovery. Franeworks such as LLM-SR (Shojaee et al.) 2025a). LaSR (Grayeli el. al. 2024), SGA (Ma el. al., 2024), and OvenEvolve (Sharma 2025) leverage pretrained knowledge to bias search toward plausible functional familics and improve efficiency. However, two fundamental limitations remain. First, LLM-based approaches are susceptible to memorizing canonical expressions rather than reasoning from data. Second, autoregressive generation struggles tightly couple discrete structural exploration with continuous parameter calibration, limiting robustness in complex scientific regimes.
Despite rapid methodological advances, benchmarking symbolic regression remains fundamentally limited. Existing benchmarks are either synthetic (Shojace et al. 2025b) or curated from canonical textbook equations (Udrescu & Tegmark (2020), making them vulnerable to memorization and narrow in structural diversity. Moreover, current benchmarks predominantly evaluate scalar mappings of the form y = f(x). Such tasks fail to reflect. the multivariate coupling, geometric structure, and representational diversily characteristic of real scientific equations. More critically, commonly used evaluation metries, such as string matching and normalized mean squared error (NMSE), are fundamentally inadequate for higher-dimennsonal symbolic objects due to symbolic non- non-uniqqeness. For example, sphere can be written implicitly as x² +²+ R² explicitly
/R² x² y2, or parametrically via trigonometric coordinates; all describe identical geometry yet differ algebraically. Metrics operating solely in symbolic or wointwise regression space therefore fail to capture functional equivalence in geometric settings. As consequence, strong performance on existing benchmarks does not necessarily imply robust symbolic reasoning or structural generalization.
In contrast, surface equations fundamentally alter the nature of equation discovery. Surfaces admit multiple representation paradigms (explicit, implicit, and parametric) each introducing distinct symbolic, geometric,
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and topological challenges. Recovering a surface equation from sampled data (z,g, v. 2) requires reasoning over coupled oulpuls, lalent coordinale systems, invoriances, and structural ambiguilly. Unlike scalar curve fitting, surface-level discovery demands simultaneous reasoning about algebraic structure and spatial consistency, as well as evaluation in object space rather than purely symbolic space.
To address these gaps, we introduce SURFACEBENCH (Figure D. the first systematic benchmark for symbolic surface discovery. SURFACEBENCH comprises 183 surface equations spanning 15 structurally defined cate- gories and three canonical representation forms: explicit, implicit, and parametric. The benchmark reframes equation discovery from scalar regression to geometry-aware multi-output reasoning. Each task provides analytically constructed surfaces inspired by scientific modeling domains, along with variable semantics and syuthetically sampled three-dimensional data. Importantly, SURFACEBENCI incorporates geometry-aware evaluation through object-space metrics such as Chamfer and Hausdorff distances, alongside symbolic equiv- alence checks and regression-bant baressssoo- errors, enabling evaluation of functional fidelity beyond algebraic syntax. The primary contributions of this work are as follows.
We introduce SURPACEBENCH, a large-scale geometry-aware benchmark for symbolic surface discovery comprising 183 surfaces across 15 calegories and three representation paradigms. SURFACEBENCH es- tablishes a new paradigm for equation discovery that moves beyond scalar functions to structured, multi- output, geometry-aware expressions.
We establish a geometry- aware evaluation framework that integrates symbolic equivalence checks with object-space metrics, addressing represcntational non-uniqueness beyond string- comparison. Using this protocol, we evaluated a broad spectrum of classical and LLM-guided symbolic regression methods and find that no approach consistently generalizes across representations, with exact recovery rates of only 4% for LLM-based frameworks and 6% for traditional methods. These results expose substantial headroom in structure discovery, parameter calibration, and multi-output reasoning.
We provide in-depth error taxonomy, decomposing symbolic (structural) versus geometric (shape- level) failure modes. Through largeted ablations, we analyze the impact of representation lype, operalor composition, providing concrete failure diagnostics and generalization breakdowns that offer actionable design insights for future surface reasoning methods.
2 Challenges and Design of SURFACEBENCH
2.1 Benchmark Challenges and Motivation
SURFACEBENCI is symbolic regression benchmark designed to evaluate symbolic reasoning capabilities that are critical for scientific equation discovery. Traditional symbolic regression tasks focus on scalar mappings such as = f(x), where evaluation is limited to one-dimensional dependencies and success is measured by numerical fit. SURFACEBENCH extends this paradigm to governing equations that describe full threo-dimensional surfaces, which encapsulate multi-variable dependencies, invariancos, and geometric constraints. The surfaces themselves are not the end goal, but serve structured substrate through which the benchmark probes symbolic reasoning under coupled geometric and analytic constraints. By represent- ing relationships as explicit, implicit, or parametric surfaces, it tests whether model can recover equations that jointly capture variable interactions and geometric structure. This formulation moves symbolic regres- sion beyond scalar curve fitting toward the more rigorous demands of scientific modeling, where systems are inherently multi-output, coordinate-dependent, and frequently topologically nontrivial. To systemati- cally evaluate these capabilities, SURFACEBENCI is organized around five defining features that collectively characterize the key challenges of symbolic regression:
(1) Multi-output coupling: In conventional benchmarks, outputs are independent, allowing models to fit each dimension separately. SURFACEBENCH introduces multivariate targets that depend on several interact- ing variables for example, 2 = sin(2² 2 + y²), where changes in one variable alter the curvature throughout the surface. This sctup tests whether models can reason jointly over variables that together define a gov- erning law, rather than treating them as separable regressions. This coupling expands the symbolic search space and reflects the interdependencies typical of real-world physical systems (Moyano et al. 2021
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(2) Latent coordinate systems and topology: Many scientific laws admit concise symbolic forms only in transformed coordinate systems such spherical, cylindrical, or other field-aligned representations, while observed data are typically provided in Cartesian space (Champion et al. 2019). SURFACEBENCI examines whether models can infer such latent transformations that reveal underlying invariances. The benchmark also includes surfaces with nontrivial topology, including holes, folds, or disconnected components, that are best represented as implicit level sets f(z, 2) = or as parametric mappings (z(u, v), y(u =u, v)). These formulations challenge symbolic regression pipelines that assume scalar outputs and fixed templates, requiring models to infer both the algebraic law and its structural form.
Multiple algebraically distinct expressions can describe identical be- haviors. A sphere, for example, may appear implicitly as x² +2² + = R² or parametrically as sRs & cos 0, Rsin & sin 8, Rcos o). Trigonometric identities, affine transformations, and reparameteriza- tions multiply the number of equivalent formulations, making string-level comparison unreliable. This repre- sentational non-uniqueness motivates the nced for evaluation criteria that operate in geometric rather than
(3) Symbolic non-uniqueness,
symbolic space (Jiang et al. 2025
(4) Geometry-aware evaluation: To address symbolic non-umiqueness, SURFACEBENCI evaluates pre- dicted and reference equations through their induced geometry. Both are sampled as dense point clouds, aligned under a similarity transform, and scored using the Chamfer Distance (capturing mean geometric fidelity) and Hausdorff Distance (capturing worst-case deviation). This procedure quantifies functional equivalence directly in object space, rewarding models that recover the correct governing law regardless of symbolic form, and explicitly decoupling geometric fidelity from algebraic syntax (Fan et al. 2016).
SURFACEBENCH spans 15 scientifically grounded categorics and 183 vali- dated equations across explicit, implicit, and parametrie formulations. This diversity ensures broad coverage of symbolic operators, compositional patterns, and topological structures, providing a rigorous and repre- sentative testbed for assessing symbolic reasoning across scientific domains.
(5) Diversity and coverage:
2.2 Benchmark Construction Pipeline
Dataset Construction
Seed Selection
Operator Augmentation
Solvability Check
Data Collection
Domain Selection
Operator
Novelty Check
Generation
Distribution
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Figure 2: Dataset curation pipeline for SURFACEBENCH, ensuring a diverse set of seed equations, their transformation to discourage memorization, and rigorous validation through novelty and solvability checks.
The construction of SURFACEBENCH, as visualized in Figure 2 follows a structured multi-phase pipeline de- signed to ensure analytic validity, structural diversity, and robustness against memorization. (1) Domain Selection: We first identify scientifically motivated problem settings (such as optics, fluid dynamics, elec- tromagnetics, materials science, and roboties) that naturally give risc to continuous 3D surfaces expressible through analylic equations, which serve as principled templates for surface construction. (2) Equation Selection: Within each domain, representative problems are selected and their analytic surface equations are obtained in explicit, implicit, or parametric form, providing interpretable analytic scods for transforma- tion. (3) Operator Distribution: The collected corpus is analyzed to estimate the empirical distribution of symbolic operators (e.g., trigonometric, exponential, rational, polynomial), ensuring that later augmenta-
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tions reflect realistic symbolic usage patterns commonly observed in scientific modeling, rather than relying arbitrary statistically ungrounded operator compositions. (4) Compositional Augmentation: To mitigate memorization of canonical forms, we apply controlled symbolic perturbations inspired by LLM- SRBench (Shojace et al. 2025b), including functional nesting (c.g.: sin(x) - sin(2² + y²)), additive and multiplicative term blending, coordinate reparammeterization (affine, polar, or spherical substitutions), and operator substitution (e.g., replacing sin(x) with tanh(x) or (1-e=*)). These augmentations produce non- canonical yet analytically solvable variants that maintain interpretabilitty while forcing models to reason compositionally rather than retrieve memorized templates. (5) Novelty Check: Each generated equation is symbolically simplified and verified for analytic sol solvability, numerical stability, and novelty relative to prior benchmarks (Feynman (Udrescu & Tegmark 2020), SRBench (La Cava et al., 2021)), discarding re- dundant or trivial forms. (6) Stability Check: Candidate surfaces are evaluated under boundary and edge conditions (e.g., F, y=0 or asymptotic limits) to ensure continuity, differentiability; and bounded evaluation domains. (7) Data Generation: For each validated surface, we uniformly sample points across a bounded 3D domain lo produce reference datasets for explicit, implicit, and parametric formulations, using adaptive sampling density in regions of high curvature. (8) Expert Validation: Finally, all surfaces are visually inspected by domain experts to confirm smoothness, completeness, and analytic consistency, removing de- generate or unstable cases. This pipeline yields a benchmark of 183 rigorously validated surfaces with three canonical representations (explicit, implicit, and parametric), spanning 15 structural surface categories.
3
Experimental Setup
3.1 Benchmark Methods
We evaluate SURFACEBENCI (see Figure 3 using representative suite of symbolic regression frameworks that capture both classical and LLM-driven approaches. Together, these baselines span evolutionary search, neural-guided search, reinforcement learning, and LLM-based symbolic hypothesis generation.
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Self Evolving LLMs Workflow
Final Evaluation
Distance and symbolic metrics
Ground Truth Surface
LLM-Predicted Surface
Figure The SURFACEBENCH evaluation pipeline integrates symbolic and geometric metrics to assess equation recovery quality. Given sampled 3D surface data, symbolic regression frameworks, including self-evolving LLM-based methods, generate candidate symbolic expressions. These predictions are compared against the ground truth using three complementary evaluation modes: regression-style errors (NMSE), symbolic accuracy (via equivalence checks), and geometry-aware distance metrics, namely Chamfer and Hausdorff distances.
3.1.1 LLM-Based Equation Discovery Methods
LLM-SR (Shojace et al.; 2025a A program-search framework that expresses candidate equations Python function templates. It fuses the scientific prior knowledge of large language models with multi-island evolutionary search, using data-driven feedback to refine hypotheses.
LaSR (Grayeli et al., 2024) A concopt-learning approach that abstracts high-level semantic descriptions of mathematical relations from previously successful equations. These concepts guide hybrid discovery process that blends T.L--assisted search with evolutionary optimization implemented through PySR
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SGA (Ma et al,, 2024) A bilevel optimization method that alternates between symbolic structures gen- erated by LLM through discrete search and parameter fitting via PyTorch-based numerical optimization, thus coupling symbolic reasoning with continuous parameter refinement.
OvenEvolve (Sharma 2025) An open-source package implementation for the alphaevolve framework (Novikov et al. 2025). is general-purpose LLM- evolutionary framework that uses L.LMs to propose transformation rules, mutation operators, or symbolic search heuristics, while fitness evaluation and selection are performed externally for the given tasks. This design decouples reasoning from optimization, allowing the LLM to serve flexible rule generator that evolves hypotheses efficiently.
3.1.2 Non-LLM-Basee Equation Discovery Methods
TPSR (Shojace et al., 2023) -A Transformer-driven symbolic regression model that integrates Monte Carlo Tree Search (MCTS) a decoding strategy. By incorporating feedback and caching, it accelerates equation generation during inference.
NeSymReS (Biggio et al., 2021) - A neural symbolic regression model pre-trained on large synthetic corpora of equations. maps expressions to latent embeddings and reconstructs symbolic skeletons, followed by gradient-based fitting of numerical constants.
E2E (Kamienny et al., 2022) An end-to-end Transformer trained to output entire equations directly, without intermediate skeletons. Constant values are refined post-prediction using the BFGS optimization algorithm, and scalable inference is achieved through generative sampling.
DSR (Petersen et al., 2019) A reinforcement-leanine baarni symbolic regression framework that di- rectly optimizes expression trees, balancing exploration and parsimony via accuracy-driven reward functions.
uDSR (Landajuela et al., 2022) An extension of DSR that performs variable reduction and adds linear tokens for polynomial construction. It employs large-scale pretraining while remaining rooted in genetic programming principles.
PySR (Cranmer) 2023 A genetic programming engine for symbolic regression that automatically tunes hyperparameters and enforces dimensional consistency by penalizing expressions violating unit constraints.
gplearn (Stephens 2016
scikit-learn compatible genetic programming library using a fit/predict interface, enabling integration with existing ML workflows and hyperparameter tuning pipelines.
3.2 Evaluation Metrics
Evaluating symbolic regression for surface recovery poses unique challenges due to the vast hypothesis space and the existence of multiple algebraically distinct expressions that describe identical geometric manifolds. Prior equation discovery benchmarks typically rely scalar regression metrics (e.g., NMSE) or symbolic string comparisons between candidate and ground-truth formulas. Such approaches are insufficient for surfaces, where functional equivalence is determined by geometric correspondence in object space rather than textual similarity. To address this, SURFACEBENCH introduces a domain-specific evaluation suite that integrates geometry-aware distances, symbolic equivolence checks, and cale-inuariant regression error.
Owing to the nature of surfaces, multiple algebraically distinct expressions can produce the same three- dimensional manifold. These equivalences may arise from representational transformations, reparameteriza- tions, or algebraic rearrangements. Conventional metrics such as normalized MSE or string based similarity would incorrectly penalize such cases. To capture equivalence in functional rather than symbolic space, SURFACEBENCH evaluates candidate and ground-truth surfaces directly in object space. Candidate and reference surfaces are uniformly sampled inlo dense point clouds and aligned under a similarity transform remove translation, rotation, and scale differences. We adopt two standard object-space distances that together capture global and local geometric fidelity:
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Chamfer Distance measures the mean geometric fidelity between two surfaces, emphasizing smooth, global deviations and gradual warping across the shape. It provides stable, differentiable indicator of overall shape fidelity and is robust to minor sampling noise or misalignment. Hausdorff Distance captures the worst-case deviation, making it sensitive to sharp discontinuities, holes, or missing components. is characteristically less smooth and highlights rare but critical geometric failures that the Chamler distance may overlook. Together, they characterize both distributed and localized geometric errors, thus revealing whether discrep- ancies stem from global distortion isolated structural mismatches. Their joint use ensures both global accuracy and local structural integrity in evaluating equation discovery models for surfaces.
Symbolic Accuracy. Following L.M.-RBench (Shojaee et al.] (022bb,, measure Symbolic Accuracy using an LLM-based equivalence check that incorporates algebraic simplifications and parameter rescalings. This provides a principled yet flexible mechanism for assessing symbolic equivalence beyond exact string matching. Additional details and the exact prompt are provided in Appendix [A.I]
Normalized Mean Squared Error (NMSE). To maintain comparability with prior scalar-function benchmarks, include NMSE as regression-style measure of pointwise fit:
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4 Experimental Results
Table 1: Comparison of various symbolic regression methods SurfaceBench. Performance is reported across explicit and implicit forms using Symbolic Accuracy (SA), Normalized Mean Squared Error (NMSE), Chamfer distance, and Hausdorff distance. Higher is better for SA and lower is better for the remaining metrics.
Explicit
Implicit Chamfer J Hausdorff
Base LLM
SA t NMSE
Chamfer & Hausdorff SA
NMSE
SGA
GoTio-mini
0.20
2.86
8,26
16.53
0.06
1.38
2.96
6.72
LJama-3.1 1-8B
0.10
3.73
9.82
18.48
0.05
1.43
3.01
7.81
Qwen3-8B
0.10
4.29
5.19
13.25
0.05
1.57
8.26
LaSR
GPT4o-mini
0.35
2.87
4.30
11.00
0.06
3.48
5.04
10.07
Llama-3.1-88 1-8B
0.30
3.21
14.21
0.10
2.81
9.78
Qwen3-8D
0.30
2.96
4.18
12.84
0.06
3.08
4.92
10.06
LLM-SR
GoT'o-mmni
0.30
2.57
7.08
24.17
0.10
2.54
5.25
Llama-3.1-81
0.20
2.62
7.41
29 .9
0.13
2.71
3.01
9.05
Qwen3-8B
0.25
2.38
6.99
28 83
0.02
2.61
1.51
10.6
OvenEvolve
GPT4o-mini
0.50
0.98
2.69
4.88
0.12
0.71
1.85
4.96
Llama-3.
0.40
0.99
3.17
5.08
0.02
0.99
2.96
5.02
Qwen3-8B
0.40
1.25
3.23
5.82
0.04
0.92
2.35
5.92
Non- LLM Baselines
NeSymReS
0.20
0.59
0.84
1.23
0.05
0.69
2.45
5.95
gplearn
0.15
0.38
0.61
1.09
0.05
0.82
1.83
5.13
F2E
0.30
0.28
0.31
0.88
0.05
0.69
1.83
5.70
DSR
0.15
0.32
0.24
0.87
0.05
0.64
1.74
6.06
uDSR
0.25
0.25
0.30
0.88
0.10
0.61
1.79
5.72
TPSR
0.25
0.21
0.34
0.85
0.05
0.66
2.52
6.88
PySR
0.25
0.18
0.13
0.41
0.05
0.60
1.64
5.53
Table reports results aggregated by representation (explicit and implicit) across four metries: Symbolic Accuracy, NMSE, Chamfer, and Hausdorff. For completeness, we report that exact equation recovery (string- level match up to trivial simplifications) is rare: 4% for LLM-based frameworks and 6% for traditional SR basclines when aggregated across all SurfaceBench tasks. We do not tabulate this Tables 1-2 because the rates are uniformly low across methods and therefore not discriminative. We observe that explicit surfaces yield the highest Symbolic Accuracy, whereas implicit surfaces achieve the lowest geometric distances. Tn

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particular, results explicit surfaces indicate that models often recover the correct structural family, but fail to produce geometrically tight parameterizations. We attribute this to symbolic structures that capture the correct functional family, while the subsequent parameter optimization stage remains under-refined. As a result, Chamfer and Hausdorff distances remain high despite strong Symbolic Accuracy. This exposes a pipeline gap: after structure discovery, a largeled geometric calibration step is needed to refine parameters such as scale and shift, ensuring that structural recovery translates into measurable gains in geometry-based metrics.
In contrast, results for the implicit category show the opposite pattern. Distance-driven search procedures bring the discovered surface equations closer to the ground-truth geomelry even when the recovered algebraic form is not exact, This yields strong Chamfer and Haus- dorff performance despite lower Symbolic Accuracy. Together, these findings high- light the structural tension (and comple- mentarity) between geometric proximity and algebraic fidelity.
Table 2: Evaluation of symbolic regression methods on parametric surface equations.
Model
SA
NMSE
Chamfer
Hausdorff
OpenEvolve
GPT4o-mini
0.07
0.84
1.22
3.98
Llama-3.1-8B
0.02
1.38
2.96
4.87
Qwen3-8B
0.04
1.25
2.35
4.59
Non-LLM Baseline
PySR
0.10
0.61
2.52
5.53
Table reports reports the results for parametric surfaces. Parametric equations remain
the most underexplored representation paradigm in symbolic regression. Among the methods we bench- mark, only OpenEvolve and PySR reliably handle multiple coupled equations, i.e,, multi-output regression within single optimization pipeline. This exposes a major gap in current symbolic discovery methods: few frameworks are explicitly designed jointly learn coupled equation systems, which is essential for modeling parametric surfaces. Finally, results on parametric surfaces show consistently strong performance across all metrics for both non-LLM baselines and the LLM-based OpenEvolve framework.
5 Robustness and Diagnostic Analysis
We conduct a comprehensive set ablations to characterize the robustness and generalization behavior of these methods. As noted previously, the LaSR, SGA, and LLM-SR methods lack algorithmic support for parametric equation discovery; accordingly, we exclude the parametric calegory from these ablation studies.
5.1 Noise sensitivity
To evaluate robustness under realistic data perturbations, we conduct noise sensitivity analysis across two representative L.M backbones: GPT-4o-mini and L.AMA-3.1-8B, We designed an experiment to system- atically investigate how model performance degrades under varying levels of data corruption, simulating real-world scenarios where training data may contain measurement errors, sensor noise, or other sources of uncertainty. We select. 13 representative equations spanning diverse structural families to ensure broad cov- erage of the symbolic hypothesis space. The experimental design employed a factorial approach with three noise levels (1%, 5%, and 10% Gaussian noise). Performance is evaluated using the Chamfer and Hausdorff distances on in-domain test data, providing complementary measures geometric fidelity between predicted and ground truth. Robustness is reflected the degradation slope as Gaussian noise increases from 1% 10%. Traditional non-LLM SR methods exhibit comparatively modest changes in geometric error, suggesting that their performance is primarily constrained by systematic search and fitting limitations rather than input corruption. In contrast, LLM- based methods degrade substantially under increasing noise, indicating higher variance in symbolic hypothesis generation and greater sensitivity to perturbations in the training data. We additionally observe that Hausdorff often increases more sharply than Chamfer, implying that noise primarily manifests itself as localized worst-case defects rather than uniform surface drift. This distinction is critical: two methods may exhibit comparable average alignment (Chamfer) while differing markedly structural brittleness (Hausdorff). Overall, the noise ablation highlights which pipelines maintain geometric fidelity under realistic uncertainty and which fail via. localized breakdowns corruption increases.
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1950
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Hauadorff Distence
Figure 4: Noise sensitivity analysis across Chamfer Distance, Hausdorff Distance, and NMSE. Lower values indicate better performance.
5.2 Out-of- oof-Domain Generalization
We define out-of-domain samples strictly controlled range shift in the inpul domain. If model is fitted imputs sampled from [-5,5] along each axis, OOD tests use the non-overlapping exterior bands [-10, -5 U [5, 10). This isolates extrapolation from interpolation: models must extend learned structure beyond the training support rather than reproduce local trends. Evaluation is conducted directly in object space, using both symbolic and geometrie metrics, ensuring that performance differences reflect genuine extrapolative structure preservation rather than memorized symbolic fragments.
unbe
Dantemia
Chamfer (Explicit)
Hausdorff (Explicit)
Latt
Chamfer (Implicit)
Hausdorff (Implicit)
Figure 5: Chamfer and Hausdorff distance metrics for both in-domain (ID) and out-of-domain (OOD) generalization across explicit and implicit forms.

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As shown in Figure 5] this range-shift protocol induces a consistent generalization gap: methods that per- form strongly in-domain frequently degrade substantially under range extrapolation, indicating that many discovered expressions behave as accurate local interpolants in [-5,5] 5] but fail to preserve correct functional behavior outside the training support. This degradation is particularly pronounced when extrapolation alters the dominant growth behavior of the expression (e.g., growth rates, asymptotes, or oscillatory behavior), where small structural or coefficient errors can rapidly amplify and translate into large geometric deviations.
We also observe that the gap is not uniform between metries. Although Chamfer distances reflect average surface alignment, Hausdorff distances are driven by worst-case discrepancies. Thus, disproportionately larger OOD increases in Hausdorff relative to Chamfer indicate that extrapolation failures often arisc from localized structural breakdowns rather than uniform global drift. Finally, differences between explicit and implicit equations provide an additional diagnostic signal: extrapolation in the explicit forms can be brittle when the recovered functional mapping has unstable out-of-range behavior, whereas implicit forms can sometimes preserve portions of the geometry even when the recovered algebraic form is imperfect.
5.3 Impact of Domain- Prior Prompts
To assess the utility of domain knowledge, we conduct an ablation in which domain priors are injected into prompts as lightweight structural cues describing coordinate systems such as spherical or cylindrical charts, consorvation or symmetry properties, knowledge about the scientific field of the equation, such as optics, electromagnetism, etc. These prompts differ from generic discovery prompts by encoding partial structural knowledge rather than purely data-driven guidance. Although such priors can, in principle, reduce search- space ambiguity, they are not typically available in real-world experimental settings. Moreover, providing incorrect or mismatched priors often leads to severe degradation in reconstruction quality, demonstrating that overly constraining prompts can hinder rather than improve symbolic discovery.
Table 3: Comparison of performance with and without domain priors across explicit and implicit surfaces. Positive A indicates improvement when priors are used.
Explicit
Implicit
Chaaneer w/ priors
Hausdorff w/ prioes
Channer w/ priors
Hausdorff w/ priors
Method
w/o priors
w/o priors
w/o priors
w/o priors
SGA
7.76
6.22
+1.54
16.09
12.22
+3.87
3,01
2.76
+0.25
7.60
6,26
+1.31
LaSR
4.05
4.04
+00.01
12.68
10.04
4.2.4
4.58
4.24
+0.64
9.97
9.24
4-0.73
LLM-SR
7.17
5.77
+1.40
27.43
20.77
+6.66
2.24
2.07
+0.17
8.30
5.07
+3.23
OpenEvolve
2.93
1.16
+1.77
4.98
4.16
+0.82
2.11
1.82
+0.59
4.99
4.82
+0.117
As observed in Table 3 despite being provided with correct priors, LLM-based symbolic regression methods show only marginal improvement over their baseline performance and remain inferior to non-LLM methods. This suggests that, despite narrowing the functional scarch space, current LLM-based approaches do not reliably translate structural cues inlo improved optimization outcomes. These findings molivate a deeper failure analysis, presented in the subsequent section, to identify where such methods falter in integrating structural and geometric cues. The example prompts used for this ablation can be found in Appendix A.5
5.4 Failure Analysis
Figure t summarizes our failure analysis, which investigates why LLM-based symbolic discovery methods underperform relative to traditional approaches. Symbolic regression requires (1) discrete structural search over symbolic expressions and (2) continuous optimization of numerical parameters. These are tightly coupled subproblems that LLM-based models attempt to solve within single generative process, which often makes it difficult to isolate specific failure modes. To disentangle these effects, we classify errors search failures and equation fitting failures. (i) A search failure occurs when the discovered equation includes terms from the incorrect functional family (for example, polynomials instead of trigonometric terms), indicating a breakdown in symbolic term retrieval. (ii) An equation-fite fitting failure, in contrast, arises when the model correctly identifies the relevant symbolic familics but fails to assemble them in the correct structural order or infer accurate constants, indicating a lack of optimization quality. To ground the analysis, we evaluate LLM-SR and OpenEvolve on ten representative target equations containing trigonometric or exponential
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terms; accordingly, legend entries such Trigonometric and Exponential indicate the target family of cach example rather than distinct error types. In Figure G the yellow box corresponds to search failures implying incorrect family retrieval, whereas the green box corresponds to equation-fitiing failures, corresponding to correct family retricval but poor optimization.
Non-I.I.M based methods perform thousands of evaluation and mutation cycles within min- utes, guided by explicit loss-driven feedback. We attribute this limitation to the autoregres- sive generation paradigm of LLM-based ap- proaches, which introduces computational la- tency and lacks explicit iterative optimization mechanisms once a candidate structure is pro- posod. We also observe that LLM- basod meth- ods succeed primarily when they identify the correct functional categories early in the search process. After approximately 200 iterations, it is rare for these models to meaningfully recover toward the correct structural family, suggesting limited corrective or self-refimemen capability once an initial hypothesis diverges from the target function. In complete failure cases, the issue originates carlier, where the models fail to identify correct functional prim- itives at all, leading to collapse at the sym- bolic search stage rather than during optimiza- tion. While LLMs offer early-stage efficiency through linguistic priors, the absence of tightly coupled iterative optimization and feedback mechanisms limits robust convergence, espe- cially under multi-oulput constraints or com-
Score drop = equation fitting errors
Figure Failure modes of LLM-based symbolic regres- sion (LLM-SR, OpenEvolve).WK analyze two error modes: (i) search space errors (the generated expression uses prim- itives from incorrect functional family) and (ii) equation- fitting errors (the correct families are retrieved but constants and/or composition are not oplimized). Legend entries such as Trigonometric and Exponential denote the ezample equation families used in this analysis (i.e., representative targets contain- ing trigonometric or exponential terms only). The green boxed region highlights equation-fttingg failures while the yellow boxed region highlights search failures.
plex compositional structures. These observations suggest that future LLM- based symbolic regression frame- works should incorporate stronger post-structure calibration and iterative optimization mechanisms to mit- igate the dominant equation-fite -iittnnn--iiigg failures exposed by SURFACEBENCH.
6 Related Work
6.1 Symbolic Regression
Farly symbolic regression (SR) research was primarily centered on genetic programming (GP), which evolves tree-structured equations using crossover and mutation, allowing discovery of interpretable analytic forms
directly from data. (ozza 1992). Later variants of GP improved efficiency through regularization strategics
and lexicographic tournament selection (Schmidt & Lipson 2009 Vladislavleva et al.] 2009 Agapitos et al.
2012). while others incorporated cannensional-conisttenn success on low-dimensional problems, classical GP methods suffer from exponential search complexity and
constraints (Udrescu & Tegmark
2020
Despite
limited scalability under noisy observations or higher-order nonlinear dynamics (McConaghy 201
La Cava
et al. 2021 ). In physics-guided SR, partial differential and variational approaches discover governing equa- tions by enforcing physical constraints such as smoothness or conservation; examples include SINDy (Bunton et al. 2016) and PDE-Net (Long et al. 2018), which identify sparse terms from predefined operator libraries. Differentiable and neural SR. frameworks such as DSR (Petersen et al. 2019 and NeSymReS (Biggio et al. 2021) combine neural function approximators with symbolic optimization, blending gradient-based fitting with discrete equation search to improve sample efficiency and search stability. Recently, large language models (LLMs) have been incorporated into SR pipelines, with approaches such as LLM-SR (Shojaee et al., 2025a), LaSR (Grayeli et al. 22024), and OpenEvolve (Sharma) (2025). These methods use L.LMs to gener- ate symbolic hypotheses, refine them using tool-assisted feedback, and apply self-reflective reasoning loops. While they exploit pretrained scientific priors to propose meaningful candidate expressions, these meth-
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ods contimue to struggle with coupled multi-equation systems, implicit and parametric representations, and topology-aware surface reconstruction.
6.2
Benchmarking Symbolic Regression
Farly benchmarks such as AT Feymman (Udrescu & Tegmark] 2020). SRSD (Matsubara et al. 2024). and SR- Bench (La Cava et al. 2021 have been widely used to evaluate symbolic regression methods. However, these benchmarks are vulnerable to memorization, where LLMs often reproduce canonical formulas directly rather than reasoning from data (Shojace et. al., 2025b). To address this, L.M.SRBench (Shojaee et al. 2025b) introduced large-scale set of synthetic equations specifically designed to mitigate memorization. However, LLM-SRBench remains restricted to scalar mappings and evaluates discovery quality through algebraic fit alone, without accounting for geometric equivalence or representational non-uniqueness. Additionally, its tasks are structural variations of a limited set of formula families, offering narrow coverage of the compo- sitional and multi-output challenges encountered in scientific modeling. SURFACEBENCII extends beyond this paradigm by targeting multi-output surface equations across three representation types and introducing geometry-anee evaluation that captures functional fidelity in object space.
6.3 Surface Equation Discovery
The task of recovering closed-form equations that define 3D surfaces has long history across computer graphics, geometry processing, and tearningbassed methods. Early work focused on fitting smooth implicit
functions to point clouds using radial basis functions (Carr et al., 2001] or solving Poisson-type PDEs to
generate indicator fields aligned with imput normals (Kazhdan et al. 2006 Kazhdan & Hoppe 2013 ) These
methods yield robust roconstructions, but the resulting oquations are either constraincd to a prodefined
basis or lack closed-form bnterpretability. Neural implicit models such as DeepSDF (Park et al. 2019 and
Occupancy Networks (Mescheder et al. 2019) generalize this idea by learning continuous fields whose level sets define geometry, achieving high fidelity but producing non-symbolie, black-box representations. Hybrid methods such as AtlasNet (Groucix et al.] (018)) learn parametric surface patches, offering some structural insight but stiil lacking explicit symbolic representations amenable to equation-level reasoning. Collectively, these three lines of work reveal a gap at their intersection that SURFACE.BENH addresses directly: symbolic regression methods have not been evaluated on surface-level discovery tasks, existing SR benchmarks remain restricted to scalar mappings, and surface reconstruction methods have not pursued symbolic representations.
7 Conclusion
We introduce SURFACEBENCH, the first geometry-aware benchmark for LLM-driven symbolic discovery of 3D surfaces, comprising 183 tasks across 15 scientifically grounded categories and three representation paradigms:
explicit, implicit, and parametric. Beyond scale, SURFACEBENCH redefines evaluation for equa- tion discovery by integrating symbolic equivalence checks with object-space geometric metrics (Chamfer and Hausdorff distances) and regression-basse error measures into a unified protocol that accounts for represen- tational non-uniqueness and structural ambiguity. Unlike prior symbolic regression datasets that emphasize scalar or low-dimensional mappings, SURFACEBENCH targets surface-level discovery, where variablle coupling, coordinate transformations, and geometric structure must be inferred directly from data. The benchmark consists of analytically constructed, science-inspired surfaces designed to stress symbolic compo- sition, multi-output coupling, topology, and goometry-aware reasoning under structural complexity.
Our empirical study across evolutionary, neural, and LLM-driven discovery frameworks reveals a consistent pattern: although modern methods can occasionally recover correct functional families, none demonstrate robust performance across representation types or evaluation regimes. In particular, LLM-based approaches exhibit strong early structural priors but struggle with iterative refinement and parameter calibration, high- lighting fundamental tension between autoregressive symbolic generation and continuous optimization. Performance remains far from saturation, underscoring the need for tighter integration between discrete structure scarch, geometric alignment, multi-cquation coupling, and differentiable parameter estimation. By releasing both the curated dataset and the accompanying geomel tablish SURFACEBENCH as a community benchmark for symbolic surface discovery. We anticipate that it will catalyze advances at the intersoction of symbolic regression, geometric learning, and scientific induction, and serve as a standardized platform for assessing structure-aware compositional reasoning in dimeesiim equation discovery.
evaluation pipeline, we aim to es-
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8 Acknowledgments
This research was partially supported by the U.S. National Science Foundation (NSF) under Grant No. 2416728 and Autodesk Research.
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A Appendix
A.1 Symbolic Accuracy
In addition geometry-based metrics, we also report Symbolic Accuracy to provide complementary view of equation discovery performance. We adopt the evaluation methodology introduced in (Shojace al. 2025b), which leverages GPT-4o automated evaluator to assess the mathematical equivalence between the predicted and ground-truth hypotheses. Traditional exact-match metrics (c.g.. recovery rate, tree edit distance) are insufficient in our selting, as many surface equations admit multiple algebraically equivalent representations. The LLM-based evaluator provides a more flexible and semantically meaningful assessment of symbolic equivalence, operating in diverse formats (strings, trees, and executable forms).
We follow the same preprocessing pipeline as (Shojace et al. 2025b), including normalization of constants and removal of auxiliary information, and rely GPT-4o's j judgement of equivalence. This ensures comparability primary focus on geometric fidelity,
with prior symbolic regression benchmarks, while complementing As shown in Figure 7 the symbolic evaluation provides a complementary view of the performance of the discovery of equations.
A.2 Memorization Resistance
To quantify the novelty of SurfaceBench relative to existing symbolic regression benchmarks, we compare each SurfaceBench equation against all equations in Al-Feynman, SRBench, and LLM-SRBench using two complementary similarity measures.
Operator-sct similarity measures overlap between the sets of symbolic operators appearing in two ex- pressions (e.g., (+, x, sin, exp, pow)), capturing commonality in operator usage while ignoring compositional structure. AST-signature similarity is computed by extracting normalized abstract syntax tree (AST) signature that summarizes structural properties of an expression, including operator composition patterns, function types, tree depth, and node counts. Similarity between signatures is then computed using a nor- malized similarity seore.
For each SurfaceBench equation, we compute both metrics against all equations in each prior benchmark and report the mazimum similarity observed under each metric. Using this protocol, we observe consistently low overlap with existing benchmarks: operator-set similarity has mean of 0.284 across SurfaceBench, with
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Prompt for Symbolic Accuracy Evaluation
Question: Given the ground truth mathematical expression A and the hypothesis B, determine if there exist any constant paramcter values that would make the hypothesis equivalent the given ground truth expression.
Let's think step by step. Explain your reasoning and then provide the final answer
(A): sqrt
(B): Hypothesis as Program
LLM (GPT-40) Judgement:
Reasoning: "The expressions can match if params[0] * params[[] = 1 and params|2] = 1, this aligns both the scalar and constant factors appropriately."
Answer: Yes
Figure 7: Prompt used for symbolic accuracy evaluation.
equations exceeding 0.7, and signsigntture similarity has a mean of 0.407, with no equations exceeding 0.65. No exact near-duplicate equations are observed.
Table summarizes the average similarities by benchmark. These results indicate that SurfaceBench equa- tions are structurally distinct from those of prior symbolic regression benchmarks, supporting that the benchmark resists direct memorization and near-match retrieval by LLM-based mothods.
Table Average operator-set and AST-signature similarity between SurfaceBench equations and prior symbolic regression benchmarks.
Benchmark
Operator Similarity
AST Similarity
AT-Feynman
0.118
0.191
SRBench
0.138
0.272
L.LM-RRBench
0.284
0.407
A.3
Implementation Details
For a comprehensive evaluation, we implement three state-of-the-art LLM-guided scientific equation discovery baselines, each tested on the SURFACEBENCH datasets using three different LLM backbones: open-source models (Llama-3.1 1i-ma--.--B--nnstruche (Qwen3-8B), and closod-source model (GPT-40-mini),
A.3.1 Parameters
Table presents the key implementation details for each discovery agentic method. We adopt most of the hyperparameters from the original implementation for these methods. We have only changed some hyperparameters in different baselines that affect the number of LLM calls in the search framework. This is make sure we have fair comparison across baseline discovery frameworks with the same access budget to LLMs. In our experiments, all baseline frameworks have 1k calls LLMs (per problem) through the discovery process and an equivalent number of calls to the Non-LLM method.
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Table Implementation details of LLM- based scientific equation discovery methods.
Method
Parameters
OpenEvolve
Temperature 0.8
equation program hypothcses sampled from LLM for initial prompt No access data for data-driven refinement
Time limit T per program hypothesis execution,
BFGS optimizer from Scipy for parameter optimization oquation skeletons
SGA
Py'oochhbbsee implementation of model and torch na Module class
Mean square error loss for data-driven feedback in agentic search
Adam optimizer in PyTorch for differential parameter optimization of equation skeletons
LaSR
Iterations 25
Cyclos per iteration 550
Populations 10
Population size
Maximum
Operators: 1, +, / A, exp, log, sqrt, sim, cos, tan, cosh
LLM weights: Ilm mutate 0.005, Ihm crossovr =0.005, Ilm gen random 0.005 Top-K 20 coucepts from library
Default configuration of PySR for parameter optimization
LLM-SR
Temperature 0.8
Batch size 4 equation programs per prompt
4 parallel evalustors
Time limit T = 30s per program hypothesis,
Memory limit M = 2GB
10 islands for population diversity through search k= in-context examples per prompt
Maximum 10 parameters per equation skeleton
BFGS optimizer from Scipy for parameter optimization equation skeletons
A.4 Task Pipelines
Implicit surfaces (f(x,v,2) = 0). We cast implicit surface discovery scalar regression. We sample 3D points v.) from fixed benchmark domain and use ground-truth values f(x, as regression targets. Each symbolic regression baseline produces scalar expression f(z, y, by minimizing MSE on these samples. We then extract the predicted surface as the zero-level set f(x, U, 2) via dense sampling (c.g., grid-based extraction) and evaluate geometric fidelity against the ground-truth surface using Chamfer and Hausdorff distances.
Parametric surfaces ((r(u, v), y(u 2(u, 2))). Parametric surfaces require discovering multiple coupled coordinate functions, which most traditional SR baselines do not reliably support. We therefore evaluate only PySR and Ovenevolve on parametric tasks. OpenEvolve natively supports multi-equaaiinn programs and jointly optimizes (i(u, v). ((u, u), (u,, v)). PySR is applied by fitting ă(u, v), (u, v), and B(u, v) indo- pendently from identical (u, v) samples and operator sets. Other SR baselines are excluded due the lack of stable multi-output support.
A.5 Prompt templates for domain informed ablation
Section 5.3 describes explicit and implicit domain priors qualitatively. For reprodueibility, we provide (i) the exact prompt text used for domain priors, (ii) the exact equation (reported outside the prompt), and (iii) the structured inputs provided alongside the prompt: coordinate hints, domain bounds, domain label, and symmetry hints.
Inputs provided to the LLM (in addition to the prompt text). For each instance, we provide: [DOMAIN [DOMAIN_LABEL], [DOMAIN BOUNDS], faces, [SAMPLES] JSON array of (a,, (r.8.2) tuples. For implicit surfaces, [SAMPLES] provided as [SAMPLES_ON] (on-surface (x, y. 2) points) and optionally [SAMPLES_NEAL] (near-surface (z, y. 2, =1) tuples). The prompt text below is unchanged.
SEOMMTRYYHINTTSS and [SAMPLES]. For explicit sur-
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Example E1: Parabola Sine Pieceeiie:
x2
x < 0,
sin(y),
x 2
Structured inputs (provided alongside the prompt text)
JOOMAIK_LAEEL Parabola, Piecevise
[DGNAIK_BOUG BOUNDS]
(-55(,
[=";
}
[CDORD_ HINTS] "cartesian",
"piecevise split x",
"allow trig in
[SYMMETRY HINIS]
"y-axia symmatzy"
[SAMPLES] JSON array of (2,y,2) tuples
Domain-prior prompt text
The surface exhibits distinct transition behavior based the x-axis, with parabolic profile for negative x-values and sinusoidal form for non-negative x-values. Tt closed and lacks voids nels, maintaining continuous structure without self-inte seff-intersections. The surface demonstrates sym- melry about the y-axis, with periodic features along the y-axis in the sinusoidal region. Anisotropic characteristics are evident, with curvature concentration varying significantly between the two defined regions.
Figure Prompt E1 for the hybrid piecewise surface Parabola Sine Piecewise (= = x² for x < 0, & = sin(y) for Lttp__SiiieenniiieeeeeeeeeeSSS 0). The figure presents the full prompt construction for hybrid multi-modal symbolic surface whose governing expression changes across a piecewise boundary at = 0. As in our prompt design, the specification combines (i) a structured input block and (ii) domoin-prior natural-language description.
Example I1 (implicit): Quartic_Sphere,
2 1.
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Published in Transactions on Machine Learning Research (03/2026)
Structured inputs (provided alongside the prompt text)
[DONAIN,
[DONAIK_LABEL] LABEL]
Spperic_Sphera
[DONAIK_BOUNDS)
"x": [-),6],
[=,,6,
[COORD_HINTS] =
"ixplicit: 1(z,,,2)=00,
"polynonial basis: x",
[SYNMETRY SINTS] "sign-flip invariance", "pernutation invariance"
[SANPLES] JSDN array of (x,y22) on-aurface points
Domain-prior prompt text
This surface closed and bounded, exhibiting a smooth topology without voids tunnels. It pos- sesses symmetrical propertics about all principal axes, maintaining invariance under sign-flip and per- mutation. The surface features lobes and pinches, with curvature concentration evident in certain re- gions. Anisotropic scaling trends are observed as one moves away from the origin, affecting the sur- face's profile along the axes.
Figure 9: Prompt 11 for the implicit surface Qhartic_Sphers (x* + y² + xa 1). The figure shows the complete prompt specification used for this example, combining (i) structured input block and (ii) domain-prior natural- language description.
A.6 Dataset Statistics
The Table D
details the split of each domain by category.
Table 6: SurfaceBench Dataset Statistics. The benchmark spans 183 analytically valid, science-inspired symbolic surface equations across 15 structural categories and 3 representation types (explicit, implicit, and parametric). Each surface includes 5k training, 500 test, and 500 out-of-dommin (OOD) samples, enabling evaluation across symbolic, geometric, and regression modalities. While these surface equations are inspired by forms commonly encountered in scientific modeling, SurfaceBeuch focuses on analytically constructed surfaces designed probe symbolic compusition, multi-output coupling, and goomelry-aware reasoning rather than reproducing full governing physical laws.
Category
Count
Highlights
Nonlinear Analy Lic Composition Surfaces
Nested analy expressious with trigomometcic, hyperbolic, exponenlial operators Couditioaal logic, brauching belavior, discontimutiies
Piesewise Surfaces
10
Mixed Transcendsntal Analytie Sur Surfires
Combinations trigononetriie, expomential, and loguetthmic operators with symbolic constants
Contitional Multi-Regime Surfaces
Piecewise definitions with distinct operator familes across regimes
Oscillatory Composite Surfaces
11
Nested and summed trigonometric compositions with vurying froquencies Smooth bounded surfacee combining trigonometric and exponentio inl operators
Trigomometric Exponeutial Compotition Surfxces
10
Multi- Operator Composite Surfaces
10
Mixed polynomiial, trigonometric, hyperbolic, and exponemcial compositions Staodard two- woo-varialle analytic functions with simple operator structure
Elementary Bivariate Surfaces
10
Discrete Inleger-Grid Surfacre
10
Surfaces define on integer indicss using modulo, sign, and continuou operators
Nonlinear Coupled Surfaces
10
Variable coupling through products, nested functions, and nonlincur mixing
Sxpoaertiall--Moouiseee Modulated Surfaces
10
Trgooometrie functions moculated by exponential or Gaussian envelopes
Hadially Decaying Surfaces
10
Spatially loualized functions with Gaassion sigmoidal, or bounded decay profiles Polynomial terms combiued with transcendenaal functions
Polynomisl Transendenial Mixtares
Implicit Suffaces
24
Higb-order polyacuial and transcndeutal implici equations f(r, Multi-ontpar equations latent coordinates (ar(u, v), pfa, r),
x)
Paramettric Multi-Outpat Surfaces
30
A.7 SurfaceBench equations for each scientific categories
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Published in Transactions on Machine Learning Research (03/2026)
Table 7: Exact equations used in SURFACEBENCH after applying transformations. by categories of scientific domains.
This table provides the equations
Category
ID
Equations
Nonlinear Analytic
NACS1
sin(x² + + + x2² + y²
Composition Surfaces
NACS2
/-1 x²2 +²² y²
NACS3
atan2 (Ir, v) exp( - (xx² + t2))
NACS4
(anh(sin(ar))
NACS5
log(1 + x.2 + x²) sin(w y)
NACS6
exp( app(sina² + 1²))
NACS7
cos(2² + u)/1 + |zyl
NACS8
sinh(zy) exp(-x²)
NACS9
sin (i(2²2 x² + +322/1og(( + x²)
NACS10
x exp( x² - y²² cos
NACS11
sin(ry)/1 x= y²
Piccewise Surfaces
PDS1
x² if x < elsc y²
PDS2
sin(x) if else exp(y)
PDS3
xy if xy > else zy
PDS4
x² +²+ y² if y else x² y2
PDS5
cos(x) if [ < else exp(-1²²
PDS6
13 if y else - y²
PDS7
xz - yl + sin(x)
PDS8
sin(x 3) if x2 y2² clse
PDS9
tanh(x) if x > y else cos(y)
PDS10
xy if [r ul < 0.5 else sin(xr - y)
Mixed Transcendental Analytic Surfaces
MTAS1
sin(x) exp(-1²)
MTAS2
tanh(ry) + x²
MTAS3
exp(-x² - 1²²) cos(3ar)
MTAS4
asin(Bx) + log(1 + y²)
MTAS5
sinh(x) tanh(y)
MTAS6
sin(x² + +y²) exp(- 1ep(-v/a x2 + y2)
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Table 7 continued from previous page
Category
ID
Equations
MTAS7
tanh(r) log(1 + y²)
MTAS8
cos(zy) exp(-a² y)
MTAS9
3cos(2) + +in((y
Conditional
CMRS1
x² if x else sin(y)
Multi- Mallii Surfaces
CMRS2
log(1 + [x|) if < 0 clsc cxp(-1²²)
CMRS3
x² sin if ay > clse (- x²2
cos y)
CMRS4
tanh(r - y) if x > else 0
CMRS5
[zyl + +in((r - y)
CMRS6
x² if else cos(y²)
CMRS7
sin(xy) xf + 3²² < else log(1 + x²)
CMRS8
tanh(r + y) if xy 0 else sin(æ y)
CMRS9
x if x > y else y² + sin(x) sin(5z) cos(5y)
Oscillatory Composite
OCS1
Surfaces
OCS2
cos(2² cos(z2y²) + 0.2 sin (5 (5 |>| + lvl)
OCS3 sin(xy) + 0.5 sin + 5y)
OCS4
-0.11(0+ sin(
OCS5
sin(x3 + +y²
OCS6
ty cos Vx² + ys²)
OCS7
sin(2" cos (2 y)
OCS8
e -1=-v sin (3(x + 8)
sin(2'x)
OCS9
OCS10
tanh(ry) cos (V x22 + y22
OCS11
sin(2x) + a exeeeppp19 (-y²) sin (222 1/22+0 + y2)
Trigonometric- Exponential Composition Surfaces
TECS1
TECS2
exp(- 2x² - yy2) cos(3x)
TECS3
tanh(x + 2) sin(ry)
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Table 7 continued from previous page
Category
ID
Equations
TECS4
log(1 - x² + (²) cos(ay)
TECS5
x² y2 -sin(22 + 2y)
TECS6
c0s(2.r) cos(2y)
TECS7
sin(x² + /²²)/ / + x²² +
TECS8
tanh(1² - )²)
TECS9
exp( - (ryl) sin(x + y) cos(zy) 0.1 (x² + x²)
TECS10
Aulti-Operator
MOCS1
log (1 + x² + y²) cos(æ - y)
Composite Surfaces
MOCS2
sin(x) cos(y) / +x² + y²
MOCS3
e -.11/y tanh + y)
MOCS4
a²y y² /1 + x²
MOCS5
v1 + x² + sin(xy)
MOCS6
e" + 1/1 |x yl
x² +y² if + < 0,
MOCS7
sin(x + +J),
if x + +2
MOCS8
cos( xx2 y22)
zy
MOCS9
sinh(x² - -2²) - 01)
arctan(sy) + 0.2 22
MOCS10
Elementary Bivariate
EBS1
x² +y²
Surfaces
EBS2
sin(x) cos(y)
EBS3
exp( cxp(-x2 - y²)
EBS4
xy
EBS5
tanh(r: + y)
EBS6
cos(x² +2²2)
FBS7
log(1 + +x² + y²)
EBS8
x² y2
EBS9
sin(x y)
EBS10
exp( - |x - 21)
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Table 7- continued from previous page
Category
ID
Equations
Discrete Integer-Grid
DIGS1
sin(t) + cos(j)
Surfaces
DIGS2
(-1)
DIGS3
mo, + mod(j.
DIGS4
Vi2 +
DIGS5
sin(ij) + j
DIGS6
cos(i +
DIGS7
mod(i² + 5)
DIGS8
tanh(i- -j)
DIGS9
[sin(12 +
DIGS10
mod(ij, 4)
Nonlinear Coupled
NCS1
cosh 0osh(001(r - 4)) c0.300 + +))
Surfaces
c-0.05(u²+y3) (x² y) cos(y)
NCS2
NCS3
log(1 + 2²) sin(y) log(1 + y²) cos(x)
NCS4
V1+0.1 + x2) sin (0.5(x v)) tanh(0.2(22= - -2)))
NCS5
0.3(zy) 0.2 sin + 0.05(22+0))
NCS6
NCS7
x² siin(y)/1++ 0.2y²
NCS8
sinh(0.22z) -01y -0.1yy
NCS9
arctan(ry) 0.3 sin(2 - y)
NCS10
arctan(ry) + sin(x +
Exponentially
EMTS1
3e 0.05(22++0) cos(0.2ry) + 0.1x
Modulated Surfaces
EMTS2
2.2 (0.33r + 0.2y) (1 e 0..1.
EMTS3
1.8 L.8cos(O..5) (0.40)))))) e 0.1. + 0.3y"
EMTS4
20in(0.7.)) c eu.0sy" 0.5zy
EMTS5
3 - 0.11 -0.15- cos(0.3g) + 0.2x
EMTS6
2.5 tanli 0.4 0.4 s0. y)
1.5e (00-0001==++))) sine=*+""ssinn.... sin(0.62r) + 0.3y
EMTS7
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Table 7 continued from previous page
Category
ID
Equations
EMTS8
cc.s(0.42) (1 e -.055)) + 0.1x²
EMTS9
2x + 1.5 (0..3r))))
EMTS10
(0in(0.5 0.1y" + 0.2xy cos
e --..((=*+)))
Sediaall-Decaying
LRDS1
Surfaces
x² -2cc-(7to7))
LRDS2
LRDS3
LRDS4
--.,(6"+00) (1.1 cos(5x))
(L.5( cos (1,5z))² 2
LRDS5
sin (3 aarctann +)) E- -222+y²
LRDS6
LRDS7
1 tanh(x² 2 + + - 4)
(x² -- e c0.7(=++
LRDS8
LRDS9
sin (xr + y) cos (r y)
LRDS10
1 x² /1 + (x² y²)² x² + n² - -3ay + sin(x)
Polynomial
PTM1
Transcendental Mixtures
PTM2
log (1 x² + +²) tanh(x y)
e -- sin (2.r + y)
РТM3
PTM4
arctan(r) + arctan(y)
PTM5
sin(x) cos()) + 0.1 ay
PTM6
3 sin(0.4x) 0.05y² + 0.2xy
PТM7
2 sin(x + + 0.5z
+ n²
PTM8
tanh(ry) + 0.5 sin (0.5xx) y
РТм9
1.5x² cos(0.2y) + 0.3 0.12
High-Degree Implicit
HDIS1
13 + 3 + 2 3ryz
Surfaces
HDIS2
y + y² z + 2²xx = 0
HDIS3
15 + +0 +25 25
+
xyz= 0
HDIS4
I'y - 26 + sin(xz) = 1
HDIS5
25 23y1 e" 0
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Table 7 continued from previous page
Category
ID
Equations
HDIS6
x y²²² + tan(2) 2
HDIS7
2d x²y²² cos(æ)
HDIS8
x²²² - 2² sin(()) 0
HDIS9
x²1²²2 x² sin(x)
HDIS10
2 + x²y e² + zy² 0
HDIS11
x² 2-222² 25 + tan(2) =
HDIS12
2 + x²y 3/ -cos(y)
HDIS13
x x0 23 + te"
HDIS14
x² x²y sin(xz) -2
HDIS15
x² + x²²² - e² + cos(xr) = JOy 2 sin(yz)
HDIS16
HDIS17
x² - x²y + e² + 2ay x² +-y² cos(xz)
HDIS18
HDIS19
x² - -=²²²² ? + tan(z) = 2²1²² 20 sin(x2)
HDIS20
HDIS21
23 + x'y 2e + -xys = 0
HDIS22
u + cos(ary =
HDIS23
x3 + 1f= (zn + 0
HDIS24
25 + 2² 22 + sin(y)
Parametric Multi-Output Surfaces
PMOS1
(sinh(u//5,, cosh(ur/10), sin(u + v) log(1 + 0²))
PMOS2
cos v, o² sin u, tanh(ur))
PMOS3
(e(1"-0)/10 /inp:////iii.. sinw, cos(ur), u² + v²)
PMOS4
(lanh(w + v²), sin(u?u), cos(u - u²) log(1 (1 + x²))
PMOS5
(urossv)), usin sin(u² + v))
PMOS6
(sinh(uv/5) cos(u - x²), e 12/10 tanh
PMOS7
(usin e²), v u, log(1 + u² + u²) sin u) (tanh + v) cos u, sin(uv²), u²e v//3)
PMOS8
PMOS9
(c(u-0)/5 sin(u²), cos(v² - u), utanh(?²))
Continued on next page
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Table 7 continued from previous page
Category
ID
Equations
PMOS10
(sin( "²v), v log(1 + u²) sinh(u/5))
10
PMOS11
(log(1 u²) cos(v²), sin(u + u), /200
PMOS12
('aenn, cos(ur²), tanh(u x²))
PMOS13
(sin( ( cos u, sinh(1²))
PMOS14
((5 + cos( u / 2)) sin u, (5+
v (cos(u/2) cos usin(u/2))
PMOS15
((5 + sin(ue)) cos u sin (5 sin(uv)) sin sin v, (5 + (in((u)))))))))))))))))()))) cos
PMOS16
(cos usn sin a sin v, cos v + 2)
PMOS17
(u, 0, sin V u² + x² cos cosusing
PMOS18
(cos (5 + sin 3r), sin u (5 + sin 3e), cos(3v) + u/2)
PMOS19
(sin(2u) cos? u, cos(2u) sin v, sin cos
PMOS20
(sinh(u/5) cos cosh (u/5) sin v, tanh(v) cos u)
PMOS21
(sin(u² + v) e",, cos(uv) log(1 + (u]), sin(ue") Itu"
PMOS22
(cosh(u + o²), sinh(wv), tanh(u² v) cos
PMOS23
(e" sinu, u² cos log(1 + x² + +2)) (sin(u") v, cos(v²) u, ue=²)
PMOS24
PMOS25
(u - v², cos(ur²), tanh(u - log(1 + +²))
PMOS26
((u² + u²) sin u, (u 1²) cos v, Vua va cos / uu + v²)
PMOS27
(log(1 u²) cos v sin(u v²), u² tanh v)
(tanh(u²) sin u, ue -v² cos(uv²) lta²
PMOS28
PMOS29
(cos(u² u) w/5 sin(v² u), log(1 v²) tanh u)
(u cos(e²), usin u, elu-
PMOS30
26


Reviews:
[{"review_id": "9voQL21L3D", "reviewer": "Reviewer_vp37", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper claims that existing benchmarks for LLM-based symbolic regression (SR) methods often focus on scalar algebraic equations while not capturing the multivariate and geometrically structured nature of equations common in practical scientific tasks. Therefore, it introduces a comprehensive benchmark that includes surface equations from scientific modeling domains. Existing LLM-based methods are evaluated on this new benchmark and are shown to suffer from unsatisfactory overall performance. As a result, the proposed benchmark is a meaningful contribution, highlighting the many unaddressed challenges in LLM-based symbolic regression.", "claims_and_evidence_yn": "Yes", "claims_explanation": "The experimental findings are described in detail. The proposed datasets and the evaluated methods are clearly documented. Overall, the results demonstrate the main argument that existing LLM-based SR frameworks generally cannot perform well across multiple types of equation representations and evaluation metrics.", "audience_yn": "Yes", "audience_explanation": "The paper raises concerns in the field of LLM-based symbolic regression, an increasingly popular direction in the past one to two years. The proposed benchmark provides a meaningful challenge for future work in this domain to target.", "requested_changes": "**The symbolic accuracy metric.** The paper now uses an LLM-based equivalence check following LLM-SR, which is fine. However, from my own experience, the LLM evaluator is prone to errors, e.g., when the equation contains long textual parts that are not mathematically significant (1e-8 * (a-really-complicated-function), for example). I request that the authors look into some evaluation results and comment on the overall quality of the LLM symbolic evaluator, and also possibly discuss the potential alternatives for this metric, e.g., by purely symbolic calculations.", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "6FTULhn5wf", "reviewer": "Reviewer_PWKE", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper introduces SurfaceBench, a benchmark for symbolic regression / equation discovery where the target is a 3D surface rather than a scalar-valued function. Tasks span explicit, implicit, and parametric surface representations and multiple scientific categories. The authors emphasize that surface discovery introduces stronger challenges than typical scalar SR, including multi-output coupling, latent coordinate systems/topology, and symbolic non-uniqueness. SurfaceBench evaluates candidates in object/geometric space by sampling predicted and ground-truth surfaces into point clouds and using Chamfer/Hausdorff distances as metrics.", "claims_and_evidence_yn": "Yes", "claims_explanation": "* In page 3, point 2 \"with equation recovery rates of only 4% for LLM-based frameworks and 6% for traditional symbolic regression methods.\", I did not find these numbers in the result tables. Are these numbers supported by experiments or did I miss something?\n* The authors claim that \"our tasks reflect surface-level structure, resist LLM memorization through novel symbolic compositions\", however, as I can see from the appendix, the equations in the benchmark are **quite common**, there is no concrete evidence shown like: overlap statistics (how many equations match/near-match AI Feynman / SRBench), similarity analyses, leakage tests. These could be difficult, but without these, “resists memorization” remains plausible but not convincingly evidenced.", "audience_yn": "Yes", "audience_explanation": "The move to 3D surfaces and geometry-aware evaluation seems compelling and likely to be useful for the symbolic regression community.", "requested_changes": "* Page 4 Line 4: For the statement “evaluation criteria that operate in geometric rather than symbolic space”. There are two non-uniqueness here, I suppose, (1) different ways of parameterising the same function (2) different functions but same or very close in the evaluation domain, which relies on the domain for the samples. I guess as a benchmark, the domain (e.g., for train/test/ood) should also be defined for the user, otherwise, people might use different settings to benchmark their models and results in unfair comparison.\n* For Section 5.1 and Section 5.2, there is only description for the resulting data, it will be useful to add your interpretations after the result descriptions, i.e., how do you interpret the results? \n* Section 5.2, what do you mean by “If a model is trained on inputs sampled from [-5,5] along each axis” which results are from fine-tuned models, which are from direct query?", "broader_impact_concerns": "N/A", "additional_comments": "----------- After Rebuttal ---------------\n\nI would like to thank the authors for addressing my questions and updating the paper for clarification. I have no further questions", "extra_scores": {}, "extra_text": {}}, {"review_id": "XGvSQ2bpRs", "reviewer": "Reviewer_w8TU", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The authors argue that LLM-based approaches for Symbolic Regression (SR) significantly underperform compared to traditional techniques. To test this hypothesis, the paper introduces SURFACEBENCH, a large-scale benchmark designed to evaluate equation discovery for 3D surfaces. The study assesses various state-of-the-art frameworks and indicates that LLM-based methods struggle with geometric reconstruction and generalization across diverse surface representations.\n\n### Strengths:\n\n- The paper contains a wider variety of equations considered compared to previous Symbolic regression benchmarks. It also incorporates explicit, implicit, and parametric representations, which is a significant expansion over previous work.\n- Includes various ablation studies (such as the effect of noise, OOD generalization, Priors as prompts to LLM-methods).\n\n### Weaknesses:\n\n- Some metric descriptions are easy to miss. Symbolic accuracy is first defined briefly in Section 3.2 as using an LLM based equivalence check that follows LLM SRBench, and more details appear later in Appendix A.1 with an example prompt. A link to Appendix A.1 in the main text would help.\n- In Tables 1 and 2, it is hard to visually scan which methods perform best among LLM based methods versus non LLM baselines for each metric. Some simple visual cues, such as bolding the best method within each block, would make the tables easier to parse (similar to Table 1 in LLM-SRBench).\n- The legend of Fig. 6 is poorly explained. A short explanation in the caption that maps colors or line styles to specific error categories or example equations would improve readability.", "claims_and_evidence_yn": "Yes", "claims_explanation": "- The experiments on SURFACEBENCH effectively highlight the current limitations of LLM-based symbolic regression. The data shows that these methods achieve low recovery rates compared to non-LLM baselines even when provided with domain priors.\n- For LLM-based methods Chamfer and Hausdorff distances remain relatively large in many settings, which confirms that even when the functional family is roughly correct, the fitted geometry can be quite far from the ground truth.\n\nThese results support the claim that the benchmark is challenging and that current methods are far from solving it.", "audience_yn": "Yes", "audience_explanation": "SurfaceBench sits at a very active intersection of topics that are central to TMLR. It combines symbolic regression, scientific machine learning, and agentic or self evolving LLMs. The benchmark is carefully documented, and the authors plan to release the dataset and evaluation pipeline, which can become a shared testbed for many future methods.", "requested_changes": "1. (Critical) Presentation: Figure 1 currently mixes different font types/embedded text between the plots and the labels. Please standardize the fonts to ensure a consistent and professional look.\n2. (Critical) Presentation: Please revise the legend or the supporting text for Figure 6. The legend currently lists specific terms like \"Trignometric\" and \"Search Exponential\" and uses green and yellow rectangular regions that are not defined in the main text. \n3. (Critical) Section 5.3 describes explicit and implicit domain priors in qualitative terms. For reproducibility, please spell out the exact prompt templates and the inputs given to each LLM-based method, at least for a few representative explicit and implicit surfaces. This should include how coordinate system hints, domain labels, and symmetry hints are encoded.", "broader_impact_concerns": "I do not see major negative societal impacts specific to this work. The benchmark and code will likely encourage more rigorous evaluation and improve the reliability of symbolic regression and equation discovery methods in scientific domains.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "0SKRDgkUnV", "reviewer": "Reviewer_xQZ2", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "Summary of Contributions:\n\n- New benchmark SurfaceBench for symbolic surface discovery with 183 tasks across 15 categories. Tasks include explicit, implicit, and parametric equations.\n- The authors consider Chamfer and Hausdorff distances to evaluate the quality of discovered equations.\n- Comprehensive comparison of LLM-based vs traditional symbolic regression methods.\n\nStrengths:\n\n- The benchmark is well-designed to test the capabilities of symbolic regression methods in discovering equations for 3D surfaces.\n- Interesting use of geometry-aware metrics for evaluation.\n- Interesting results showing the limitations of current LLM-based methods.\n\nWeaknesses:\n\n- Limited novelty: In essence, a new table of symbolic equations (Table 6) are proposed, and the rest of the paper is an experimental comparison of existing methods on this benchmark.", "claims_and_evidence_yn": "Yes", "claims_explanation": "The authors provide a comprehensive set of experiments comparing various LLM-based and traditional symbolic regression methods on their proposed benchmark. The results are presented clearly, with appropriate metrics (Chamfer and Hausdorff distances) to evaluate the quality of the discovered equations. These comparisons are very relevant to the symbolic regression community and provide convincing evidence for the claims made in the paper.", "audience_yn": "Yes", "audience_explanation": "The community interested in symbolic regression and equation discovery would find this paper relevant. The proposed benchmark, SurfaceBench, addresses a gap in existing benchmarks by focusing on 3D surfaces and incorporating geometry-aware metrics. The comprehensive comparison of LLM-based and traditional methods provides valuable insights into the current state of the field and highlights areas for future research.", "requested_changes": "- Please provide more details on how the traditional symbolic regression methods are run for the implicit and parametric equations. This would help readers better understand the experimental setup and results.", "broader_impact_concerns": "No concerns.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)