CogFlow: Bridging Perception and Reasoning through Knowledge Internalization for Visual Mathematical Problem Solving
Shuhang Chen, Yunqiu Xu, Junjie Xie, Aojun Lu, Tao Feng, Zeying Huang, Ning Zhang, Yi Sun, Yi Yang, Hangjie Yuan
2026-01-07
Summary
This paper focuses on improving how well AI models can solve math problems presented with images, like those found in textbooks. Current models struggle with this, even though they've gotten good at other tasks involving both text and images.
What's the problem?
While recent attempts have tried to help AI better 'see' the math problems in images, they haven't addressed a crucial issue: even if the AI correctly identifies the visual elements, it might not actually *use* that information correctly when solving the problem. The AI could find patterns that seem right but aren't actually based on the image, essentially taking shortcuts and ignoring the visual information.
What's the solution?
The researchers created a new system called CogFlow, which mimics how humans think when solving visual math problems. It works in three steps: first, it carefully 'perceives' the image, improving how it recognizes both symbols and diagrams using a new reward system. Second, it 'internalizes' the information, making sure the AI understands the connection between what it sees and the math it needs to do, again using a special reward. Finally, it 'reasons' through the problem, and a new technique called Visual-Gated Policy Optimization ensures the AI’s steps are actually based on the image and not just random guesses. They also created a large new dataset of math problems with detailed explanations to help train the AI.
Why it matters?
This work is important because it tackles a fundamental flaw in current AI models for visual math reasoning. By forcing the AI to truly integrate visual information into its problem-solving process, it can lead to more accurate and reliable solutions, bringing us closer to AI that can genuinely understand and solve complex math problems presented in a visual format.
Abstract
Despite significant progress, multimodal large language models continue to struggle with visual mathematical problem solving. Some recent works recognize that visual perception is a bottleneck in visual mathematical reasoning, but their solutions are limited to improving the extraction and interpretation of visual inputs. Notably, they all ignore the key issue of whether the extracted visual cues are faithfully integrated and properly utilized in subsequent reasoning. Motivated by this, we present CogFlow, a novel cognitive-inspired three-stage framework that incorporates a knowledge internalization stage, explicitly simulating the hierarchical flow of human reasoning: perceptionRightarrowinternalizationRightarrowreasoning. Inline with this hierarchical flow, we holistically enhance all its stages. We devise Synergistic Visual Rewards to boost perception capabilities in parametric and semantic spaces, jointly improving visual information extraction from symbols and diagrams. To guarantee faithful integration of extracted visual cues into subsequent reasoning, we introduce a Knowledge Internalization Reward model in the internalization stage, bridging perception and reasoning. Moreover, we design a Visual-Gated Policy Optimization algorithm to further enforce the reasoning is grounded with the visual knowledge, preventing models seeking shortcuts that appear coherent but are visually ungrounded reasoning chains. Moreover, we contribute a new dataset MathCog for model training, which contains samples with over 120K high-quality perception-reasoning aligned annotations. Comprehensive experiments and analysis on commonly used visual mathematical reasoning benchmarks validate the superiority of the proposed CogFlow.