# CogFlow: Bridging Perception and Reasoning through Knowledge Internalization for Visual Mathematical Problem Solving

arXiv: https://arxiv.org/abs/2601.01874 · HTML: https://mlsanity.com/paper/2601.01874/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Placed in Multimodal reasoning, one of 17 directions in LLM reasoning.
- Builds on 27 of its 80 references that are in the map.
- First posted on arXiv 5 Jan 2026.

## Placed in

- LLM reasoning / [Multimodal reasoning](https://mlsanity.com/lr/multimodal-reasoning/)
- Multimodal LLMs / vision-language models / [VLM benchmarks and visual reasoning](https://mlsanity.com/mllm/vlm-benchmarks-and-visual-reasoning/)

## Builds on (most cited first, 27 of 27)

- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [Qwen2.5-VL Technical Report](https://mlsanity.com/paper/2502.13923/) (2025)
- [Qwen-VL: A Versatile Vision-Language Model for Understanding, Localization, Text Reading, and Beyond](https://mlsanity.com/paper/2308.12966/) (2023)
- [Expanding Performance Boundaries of Open-Source Multimodal Models with Model, Data, and Test-Time Scaling](https://mlsanity.com/paper/2412.05271/) (2024)
- [Gemini: A Family of Highly Capable Multimodal Models](https://mlsanity.com/paper/2312.11805/) (2023)
- [Group Sequence Policy Optimization](https://mlsanity.com/paper/2507.18071/) (2025)
- [MiniGPT-v2: large language model as a unified interface for vision-language multi-task learning](https://mlsanity.com/paper/2310.09478/) (2023)
- [LLaMA-Adapter V2: Parameter-Efficient Visual Instruction Model](https://mlsanity.com/paper/2304.15010/) (2023)
- [VLM-R1: A Stable and Generalizable R1-style Large Vision-Language Model](https://mlsanity.com/paper/2504.07615/) (2025)
- [GLM-4.5V and GLM-4.1V-Thinking: Towards Versatile Multimodal Reasoning with Scalable Reinforcement Learning](https://mlsanity.com/paper/2507.01006/) (2025)
- [Seed1.5-VL Technical Report](https://mlsanity.com/paper/2505.07062/) (2025)
- [InternLM-XComposer2: Mastering Free-form Text-Image Composition and Comprehension in Vision-Language Large Model](https://mlsanity.com/paper/2401.16420/) (2024)
- [LogicVista: Multimodal LLM Logical Reasoning Benchmark in Visual Contexts](https://mlsanity.com/paper/2407.04973/) (2024)
- [TinyLLaVA: A Framework of Small-scale Large Multimodal Models](https://mlsanity.com/paper/2402.14289/) (2024)
- [Seed1.5-Thinking: Advancing Superb Reasoning Models with Reinforcement Learning](https://mlsanity.com/paper/2504.13914/) (2025)
- [MultiMath: Bridging Visual and Mathematical Reasoning for Large Language Models](https://mlsanity.com/paper/2409.00147/) (2024)
- [Advancing Multimodal Reasoning: From Optimized Cold Start to Staged Reinforcement Learning](https://mlsanity.com/paper/2506.04207/) (2025)
- [Skywork R1V2: Multimodal Hybrid Reinforcement Learning for Reasoning](https://mlsanity.com/paper/2504.16656/) (2025)
- [Theorem-Validated Reverse Chain-of-Thought Problem Generation for Geometric Reasoning](https://mlsanity.com/paper/2410.17885/) (2024)
- [We-Math 2.0: A Versatile MathBook System for Incentivizing Visual Mathematical Reasoning](https://mlsanity.com/paper/2508.10433/) (2025)
- [X-Reasoner: Towards Generalizable Reasoning Across Modalities and Domains](https://mlsanity.com/paper/2505.03981/) (2025)
- [Slow Perception: Let's Perceive Geometric Figures Step-by-step](https://mlsanity.com/paper/2412.20631/) (2024)
- [WeThink: Toward General-purpose Vision-Language Reasoning via Reinforcement Learning](https://mlsanity.com/paper/2506.07905/) (2025)
- [Describe-then-Reason: Improving Multimodal Mathematical Reasoning through Visual Comprehension Training](https://mlsanity.com/paper/2404.14604/) (2024)
- [Open Eyes, Then Reason: Fine-grained Visual Mathematical Understanding in MLLMs](https://mlsanity.com/paper/2501.06430/) (2025)
- [MathFlow: Enhancing the Perceptual Flow of MLLMs for Visual Mathematical Problems](https://mlsanity.com/paper/2503.16549/) (2025)

## Built on by (most cited first, 5 of 5)

- [Zooming without Zooming: Region-to-Image Distillation for Fine-Grained Multimodal Perception](https://mlsanity.com/paper/2602.11858/) (2026)
- [CoLVR: Enhancing Exploratory Latent Visual Reasoning via Contrastive Optimization](https://mlsanity.com/paper/2605.08802/) (2026)
- [StemBind: When MLLMs Get Lost Between Rules and Instances in Abstract Visual Reasoning](https://mlsanity.com/paper/2606.00148/) (2026)
- [ROSE: Benchmarking the Perception-to-Action Gap in Multimodal Models](https://mlsanity.com/paper/2606.19965/) (2026)
- [Visual Evidence Under Cross-Examination: Evaluating and Controlling Decision-Level Evidence Use in Vision-Language Models](https://mlsanity.com/paper/2610.09550/) (2026)

## Abstract (arXiv)

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. In line 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. Project page: https://shchen233.github.io/cogflow.