SurfaceBench: Can Self-Evolving LLMs Find the Equations of 3D Scientific Surfaces?

Sanchit Kabra, Shobhnik Kriplani, Parshin Shojaee, Chandan K. Reddy

Keywords: Symbolic Regression, Equation Discovery, Large Language Models, Evolutionary Search

TL;DR: We introduce SurfaceBench, a comprehensive benchmark for evaluating LLM-based self-evolving frameworks beyond data contamination via the symbolic discovery of 3D scientific surfaces .

Abstract: Equation discovery from data is a core challenge in machine learning for science, requiring recovery of concise symbolic expressions that govern complex physical and geometric phenomena. Recent approaches with large language models (LLMs) show promise in symbolic regression, but their success often hinges on memorized formulas or simplified functional forms. Existing benchmarks exacerbate this limitation: they focus on scalar functions, ignore domain grounding, and rely on brittle string-matching metrics that fail to capture scientific equivalence. We introduce SurfaceBench, the first comprehensive benchmark for symbolic surface discovery. SurfaceBench comprises 199 surfaces across 18 categories of symbolic complexity, spanning explicit, implicit, and parametric forms. Each task includes ground-truth equations, variable semantics, and synthetically sampled 3D data. Many surfaces are novel or synthetically constructed to resist memorization, yet remain grounded in scientific domains such as fluid dynamics, robotics, electromagnetics, and geometry. To evaluate discovery quality, we pair symbolic checks with geometry-aware metrics (Chamfer, Hausdorff), ensuring models are judged by the structures they reproduce rather than their algebraic syntax. Our experiments show that state-of-the-art frameworks, while occasionally successful on specific families, fail to generalize consistently across representations. SurfaceBench thus establishes a challenging, diagnostic testbed for equation discovery, enabling principled progress in symbolic generalization, data-driven induction, and geometry-aware reasoning with LLMs.

Primary Area: datasets and benchmarks

Code Of Ethics: I acknowledge that I and all co-authors of this work have read and commit to adhering to the ICLR Code of Ethics.

Submission Guidelines: I certify that this submission complies with the submission instructions as described on https://iclr.cc/Conferences/2026/AuthorGuide.

Anonymous Url: I certify that there is no URL (e.g., github page) that could be used to find authors’ identity.

No Acknowledgement Section: I certify that there is no acknowledgement section in this submission for double blind review.

Submission Number: 22609