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

2025-11-13

AI Summary

Symbolic regression methods for discovering 3D scientific surface equations suffer from weak generalization, heavy reliance on predefined functional forms, and memorization bias. Method: We introduce SurfaceBench—the first comprehensive, surface-oriented benchmark—comprising 15 levels of symbolic complexity and 183 tasks, supporting explicit, implicit, and parametric equation representations while integrating physical semantics and geometric structure. It features three innovations: (i) scientific grounding-based construction, (ii) memorization-resistant symbolic generation, and (iii) geometry-aware evaluation via Chamfer/Hausdorff distances coupled with symbolic verification—moving beyond scalar regression. Contribution/Results: Experiments reveal significant performance degradation of current SOTA methods under cross-representation transfer and high-complexity regimes. SurfaceBench establishes a diagnostic, challenging evaluation platform for large language models in geometry-aware reasoning, compositional generalization, and data-driven scientific discovery.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Cognitive Modeling & Cognitive Systems: Symbolic Representations
• Computer Vision: Visual Reasoning & Symbolic Representations

Application Category

• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs

Abstract

Equation discovery from data is a core challenge in machine learning for science, requiring the 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 overly simplified functional forms. Existing benchmarks exacerbate this limitation: they focus on scalar functions, ignore domain grounding, and rely on brittle string-matching based metrics that fail to capture scientific equivalence. We introduce SurfaceBench, first comprehensive benchmark for symbolic surface discovery. SurfaceBench comprises 183 tasks across 15 categories of symbolic complexity, spanning explicit, implicit, and parametric equation representation forms. Each task includes ground-truth equations, variable semantics, and synthetically sampled three dimensional data. Unlike prior SR datasets, our tasks reflect surface-level structure, resist LLM memorization through novel symbolic compositions, and are grounded in scientific domains such as fluid dynamics, robotics, electromagnetics, and geometry. To evaluate equation discovery quality, we pair symbolic checks with geometry-aware metrics such as Chamfer and Hausdorff distances, capturing both algebraic fidelity and spatial reconstruction accuracy. Our experiments reveal that state-of-the-art frameworks, while occasionally successful on specific families, struggle to generalize across representation types and surface complexities. SurfaceBench thus establishes a challenging and diagnostic testbed that bridges symbolic reasoning with geometric reconstruction, enabling principled benchmarking of progress in compositional generalization, data-driven scientific induction, and geometry-aware reasoning with LLMs. We release the code here: https://github.com/Sanchit-404/surfacebench

Problem

• Discovering symbolic equations governing 3D scientific surfaces from data
• Addressing limitations of existing benchmarks that focus on scalar functions
• Evaluating LLMs' ability to generalize across surface representation types

Innovation

• SurfaceBench benchmark for symbolic surface discovery
• Geometry-aware metrics replacing string-matching evaluation
• Novel symbolic compositions resisting LLM memorization