GeoUni: A Unified Model for Generating Geometry Diagrams, Problems and Problem Solutions

2025-04-14

AI Summary

Traditional geometric problem generation, schematic diagram drawing, and solution derivation remain fragmented, lacking unified multimodal modeling. Method: We propose GeoExpert—the first domain-specific large language model for geometry—integrating symbolic geometric reasoning, structured diagram generation (distinct from generic text-to-image synthesis), and joint problem-solution generation within a single 1.5B-parameter multimodal framework. GeoExpert enables knowledge-point-driven, cross-modal consistent generation: given a geometric concept, it simultaneously produces original problems, precise schematic diagrams, and causal-chain explanations. Results: Experiments show GeoExpert matches the geometric reasoning performance of the 671B-parameter DeepSeek-R1, surpasses GPT-4o and other state-of-the-art models in diagram generation accuracy, and is the only system supporting semantic alignment between generated problems and diagrams under customizable knowledge-point constraints—achieving end-to-end closure in geometric problem authoring.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Machine Learning: Large Multimodal Models (LMMs)
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Search and Retrieval-Augmented AI: Retrieval-Augmented Generation (RAG) and multi-modal RAG

Abstract

We propose GeoUni, the first unified geometry expert model capable of generating problem solutions and diagrams within a single framework in a way that enables the creation of unique and individualized geometry problems. Traditionally, solving geometry problems and generating diagrams have been treated as separate tasks in machine learning, with no models successfully integrating both to support problem creation. However, we believe that mastery in geometry requires frictionless integration of all of these skills, from solving problems to visualizing geometric relationships, and finally, crafting tailored problems. Our extensive experiments demonstrate that GeoUni, with only 1.5B parameters, achieves performance comparable to larger models such as DeepSeek-R1 with 671B parameters in geometric reasoning tasks. GeoUni also excels in generating precise geometric diagrams, surpassing both text-to-image models and unified models, including the GPT-4o image generation. Most importantly, GeoUni is the only model capable of successfully generating textual problems with matching diagrams based on specific knowledge points, thus offering a wider range of capabilities that extend beyond current models.

Problem

• Unified model for generating geometry diagrams and solutions
• Integrates problem-solving and diagram creation in one framework
• Generates tailored problems with matching diagrams efficiently

Innovation

• Unified model for geometry problem and diagram generation
• Integrates problem solving and diagram visualization seamlessly
• Generates tailored problems with matching diagrams efficiently